# Gröbner–Shirshov bases and their calculationAcademic research paper on "Mathematics"

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## Academic research paper on topic "Gröbner–Shirshov bases and their calculation"

﻿Bull. Math. Sci. (2014) 4:325-395 DOI 10.1007/s13373-014-0054-6

Grobner-Shirshov bases and their calculation

L. A. Bokut • Yuqun Chen

Received: 3 December 2013 / Revised: 3 July 2014 / Accepted: 13 August 2014 / Published online: 9 September 2014

Abstract In this survey we give an exposition of the theory of Grobner-Shirshov bases for associative algebras, Lie algebras, groups, semigroups, ^-algebras, operads, etc. We mention some new Composition-Diamond lemmas and applications.

Keywords Grobner basis ■ Grobner-Shirshov basis ■ Composition-Diamond lemma ■ Congruence ■ Normal form ■ Braid group ■ Free semigroup ■ Chinese monoid ■ Plactic monoid ■ Associative algebra ■ Lie algebra ■ Lyndon-Shirshov basis ■ Lyndon-Shirshov word ■ PBW theorem ■ ^-algebra ■ Dialgebra ■ Semiring ■ Pre-Lie algebra ■ Rota-Baxter algebra ■ Category ■ Module

Mathematics Subject Classification 13P10 ■ 16-xx ■ 16S15 ■ 16S35 ■ 16W99 ■ 16Y60 ■ 17-xx ■ 17B01 ■ 17B37 ■ 17B66 ■ 17D99 ■ 18Axx ■ 18D50 ■ 20F05 ■ 20F36 ■ 20Mxx ■ 20M18

Communicated by Efim Zelmanov.

Supported by the NNSF of China (11171118), the Research Fund for the Doctoral Program of Higher Education of China (20114407110007), the NSF of Guangdong Province (S2011010003374) and the Program on International Cooperation and Innovation, Department of Education, Guangdong Province (2012gjhz0007). Supported by RFBR 12-01-00329, LSS-3669.2010.1, SB RAS Integration Grant No. 2009.97 (Russia) and Federal Target Grant "Scientific and educational personnel of innovation Russia" for 2009-2013 (government contract No.02.740.11.5191).

L. A. Bokut (B)

Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk 630090, Russia e-mail: bokut@math.nsc.ru

L. A. Bokut • Y. Chen

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, People's Republic of China e-mail: yqchen@scnu.edu.cn

Abbreviations

CD-lemma Composition-Diamond lemma

GS basis Grobner-Shirshov basis

LS word (basis) Lyndon-Shirshov word (basis)

ALSW(X) The set of all associative Lyndon-Shirshov words in X

NLSW(X) The set of all non-associative Lyndon-Shirshov words in X

PBW theorem The Poincare-Birkhoff-Witt theorem

X* The free monoid generated by X

[X] The free commutative monoid generated by X

X** The set of all non-associative words (u) in X

gp(X | S} The group generated by X with defining relations S

sgp(X |S} The semigroup generated by X with defining relations S

k A field

K A commutative algebra over k with unity

k( X} The free associative algebra over k generated by X

k( X| S} The associative algebra over k with generators X and defining relations S

Sc A Grobner-Shirshov completion of S

Id(S) The ideal generated by a set S

s The maximal word of a polynomial s with respect to some ordering <

Irr(S) The set of all monomials avoiding the subword s for all s e S

k[X] The polynomial algebra over k generated by X

Lie(X) The free Lie algebra over k generated by X

LieK(X) The free Lie algebra generated by X over a commutative algebra K

1 Introduction

In this survey we review the method of Grobner-Shirshov1 (GS for short) bases for different classes of linear universal algebras, together with an overview of calculation of these bases in a variety of specific cases.

Shirshov (also spelled Sirsov) in his pioneering work ([207], 1962) posed the following fundamental question:

How to find a linear basis of a Lie algebra defined by generators and relations? He gave an infinite algorithm to solve this problem using a new notion of the composition (later the's-polynomial' in Buchberger's terminology [65,66]) of two Lie

1 Though Shirshov [207] 1962 was the first to come up with the idea of a 'Grobner-Shirshov basis' for Lie and non-commutative polynomial algebras, his paper became practically unknown outside Russia. In the meantime, Buchberger's 'Grobner basis' (Thesis 1965 [65], paper 1970 [66]) for (commutative) polynomials became very popular in science. As a result, the first author suggested the name 'Grobner-Shirshov basis' for non-commutative and non-associative polynomials. For (commutative) differential polynomials an analogous, or better to say, closely related 'basis' is called a Ritt-Kolchin characteristic set, due to Ritt [193] 1950 and Kolchin [140] 1973, and rediscovered by Wu [219] 1978.

polynomials and a new notion of completion of a set of Lie polynomials (adding nontrivial compositions; the critical pair/completion (cpc-) algorithm in the later terminology of Knuth and Bendix [138] and Buchberger [67,68]).

Shirshov's algorithm goes as follows. Consider a set S c Lie(X) of Lie polynomials in the free algebra k(X} on X over a field k (the algebra of non-commutative polynomials on X over k). Denote by S' the superset of S obtained by adding all non-trivial Lie compositions ('Lie s-polynomials') of the elements of S. The problem of triviality of a Lie polynomial modulo a finite (or recursive) set S can be solved algo-rithmically using Shirshov's Lie reduction algorithm from his previous paper [203], 1958. In general, an infinite sequence

S c S' c s" c ••• c S(n) c ...

of Lie multi-compositions arises. The union Sc of this sequence has the property that every Lie composition of elements of Sc is trivial modulo Sc. This is what is now called a Lie GS basis.

Then a new 'Composition-Diamond lemma2 for Lie algebras' (Lemma 3 in [207]) implies that the set Irr(Sc) of all Sc-irreducible (or Sc-reduced) basic Lie monomials [u] in X is a linear basis of the Lie algebra Lie(X|S) generated by X with defining relations S. Here a basic Lie monomial means a Lie monomial in a special linear basis of the free Lie algebra Lie(X) c k (X}, known as the Lyndon-Shirshov (LS for short) basis (Shirshov [207] and Chen-Fox-Lyndon [72], see below). An LS monomial [u] is called Sc-irreducible (or Sc-reduced) whenever u, the associative support of [u], avoids the word s for all s e S, where s is the maximal word of s as an associative polynomial (in the deg-lex ordering). To be more precise, Shirshov used his reduction algorithm

at each step S, S', S",____Then we have a direct system S ^ S' ^ S" ^ ... and

Sc = limS(n) is what is now called a minimal GS basis (a minimal GS basis is not unique, but a reduced GS basis is, see below). As a result, Shirshov's algorithm gives a solution to the above problem for Lie algebras.

Shirshov's algorithm, dealing with the word problem, is an infinite algorithm like the Knuth-Bendix algorithm [138], 1970 dealing with the identity problem for every variety of universal algebras.3 The initial data for the Knuth-Bendix algorithm is the defining identities of a variety. The output of the algorithm, if any, is a 'Knuth-Bendix basis' of identities of the variety in the class of all universal algebras of a given signature (not a GS basis of defining relations, say, of a Lie algebra).

Shirshov's algorithm gives linear bases and algorithmic decidability of the word problem for one-relation Lie algebras [207], (recursive) linear bases for Lie algebras with (finite) homogeneous defining relations [207], and linear bases for free products of Lie algebras with known linear bases [208]. He also proved the Freiheitssatz (free-ness theorem) for Lie algebras [207] (for every one-relation Lie algebra Lie(X| f),

2 The name 'Composition-Diamond lemma' combines the Neuman Diamond Lemma [172], the Shirshov Composition Lemma [207] and the Bergman Diamond Lemma [11].

3 We use the standard algebraic terminology 'the word problem', 'the identity problem', see Kharlampovich,

Sapir [136] for instance.

the subalgebra (X\{xi0}}, where xi0 appears in f, is a free Lie algebra). The Shir-shov problem [207] of the decidability of the word problem for Lie algebras was solved negatively in [21]. More generally, it was proved [21] that some recursively presented Lie algebras with undecidable word problem can be embedded into finitely presented Lie algebras (with undecidable word problem). It is a weak analogue of the Higman embedding theorem for groups [115]. The problem [21] whether an analogue of the Higman embedding theorem is valid for Lie algebras is still open. For associative algebras a similar problem [21] was solved positively by Belyaev [10]. A simple example of a Lie algebra with undecidable word problem was given by Kukin [142].

Actually, a similar algorithm for associative algebras is implicit in Shirshov's paper [207]. The reason is that he treats Lie(X) as the subspace of Lie polynomials in the free associative algebra k(X}. Then to define a Lie composition ( f, g}w of two Lie polynomials relative to an associative word w = lcm(f, g), he defines firstly the associative composition (non-commutative 's-polynomial') ( f, g)w = fb — ag, with a, b e X *. Then he inserts some brackets ( f, g}w = [fb] f — [ag]g by applying his special bracketing lemma of [203]. We can obtain Sc for every S c k (X} in the same way as for Lie polynomials and in the same way as for Lie algebras ('CD-lemma for associative algebras') to infer that Irr(Sc) is a linear basis of the associative algebra k( X| S} generated by X with defining relations S. All proofs are similar to those in [207] but much easier.

Moreover, the cases of semigroups and groups presented by generators and defining relations are just special cases of associative algebras via semigroup and group algebras. To summarize, Shirshov's algorithm gives linear bases and normal forms of elements of every Lie algebra, associative algebra, semigroup or group presented by generators and defining relations! The algorithm works in many cases (see below).

The theory of Grobner bases and Buchberger's algorithm were initiated by Buch-berger (Thesis [65] 1965, paper [66] 1970) for commutative associative algebras. Buchberger's algorithm is a finite algorithm for finitely generated commutative algebras. It is one of the most useful and famous algorithms in modern computer science.

Shirshov's paper [207] was in the spirit of the program of Kurosh (1908-1972) to study non-associative (relatively) free algebras and free products of algebras, initiated in Kurosh's paper [143], 1947. In that paper he proved non-associative analogs of the Nielsen-Schreier and Kurosh theorems for groups. It took quite a few years to clarify the situation for Lie algebras in Shirshov's papers [200], 1953 and [207], 1962 closely related to his theory of GS bases. It is important to note that Kurosh's program quite unexpectedly led to Shirshov's theory of GS bases for Lie and associative algebras [207].

A step in Kurosh's program was made by his student Zhukov in his Ph.D. Thesis [226], 1950. He algorithmically solved the word problem for non-associative algebras. In a sense, it was the beginning of the theory of GS bases for non-associative algebras. The main difference with the future approach of Shirshov is that Zhukov did not use a linear ordering of non-associative monomials. Instead he chose an arbitrary monomial of maximal degree as the 'leading' monomial of a polynomial. Also, for non-associative algebras there is no 'composition of inter-

section' ('s-polynomial'). In this sense it cannot be a model for Lie and associative algebras.4

Shirshov, also a student of Kurosh's, defended his Candidate of Sciences Thesis [199] at Moscow State University in 1953. His thesis together with the paper that followed [203], 1958 may be viewed as a background for his later method ofGS bases. In the thesis, he proved the free subalgebra theorem for free Lie algebras (now known as Shirshov-Witt theorem, see also Witt [218], 1956) using the elimination process rediscovered by Lazard [149], 1960. He used the elimination process later [203], 1958 as a general method to prove the properties of regular (LS) words, including an algorithm of (special) bracketing of an LS word (with a fixed LS subword). The latter algorithm is of some importance in his theory of GS bases for Lie algebras (particularly in the definition of the composition of two Lie polynomials). Shirshov also proved the free subalgebra theorem for (anti-) commutative non-associative algebras [202], 1954. He used that later in [206], 1962 for the theory of GS bases of (commutative, anti-commutative) non-associative algebras. Shirshov (Thesis [199], 1953) found the ('Hall-Shirshov') series of bases of a free Lie algebra (see also [205] 1962, the first issue of Malcev's Algebra and Logic).5

TheLS basis is a particular case of the Shirshov or Hall-Shirshov series of bases (cf. Reutenauer [190], where this series is called the 'Hall series'). In the definition of his series, Shirshov used Hall's inductive procedure (see Ph. Hall [114], 1933, Hall [113], 1950): a non-associative monomial w = ((u)(v)) is a basic monomial whenever

(1) (u), (v) are basic;

(2) (u) > (v);

(3) if (u) = ((ux)(u2)) then (u2) < (v).

However, instead of ordering by the degree function (Hall words), he used an arbitrary linear ordering of non-associative monomials satisfying

((u)(v)) > (v).

4 After his Ph.D. Thesis of 1950, Zhukov moved to the present Keldysh Institute of Applied Mathematics (Moscow) to do computational mathematics. Godunov in 'Reminiscence about numerical schemes', arxiv.org/pdf/0810.0649, 2008, mentioned his name in relation to the creation of the famous Godunov numerical method. So, Zhukov was a forerunner of two important computational methods!

5 It must be pointed out that Malcev (1909-1967) inspired Shirshov's works very much. Malcev was an official opponent (referee) of his (second) Doctor of Sciences Dissertation at MSU in 1958. The first author, Bokut, remembers this event at the Science Council Meeting, chaired by Kolmogorov, and Malcev's words "Shirshov's dissertation is a brilliant one!". Malcev and Shirshov worked together at the present Sobolev Institute of Mathematics in Novosibirsk since 1959 until Malcev's sudden death at 1967, and have been friends despite the age difference. Malcev headed the Algebra and Logic Department (by the way, the first author is a member of the department since 1960) and Shirshov was the first deputy director of the institute (whose director was Sobolev). In those years, Malcev was interested in the theory of algorithms of mathematical logic and algorithmic problems of model theory. Thus, Shirshov had an additional motivation to work on algorithmic problems for Lie algebras. Both Maltsev and Kurosh were delighted with Shirshov's results of [207]. Malcev successfully nominated the paper for an award of the Presidium of the Siberian Branch of the Academy of Sciences (Sobolev and Malcev were the only Presidium members from the Institute of Mathematics at the time).

For example, in his Thesis [199], 1953 he used the ordering by the content of monomials (the content of, say, the monomial (u) = ((x2x1)((x2x1)x1)) is the vector (x2, x2, xi, xi, xi)). Actually, the content t of (u) may be viewed as a commutative associative word that equals u in the free commutative semigroup. Two contents are compared lexicographically (a proper prefix of a content is greater than the content).

If we use the lexicographic ordering, (u) >- (v) if u >- v lexicographically (with the condition u >- uv, v = 1), then we obtain the LS basis.6 For example, for the alphabet x1, x2 with x2 >- x1 we obtain basic Lyndon-Shirshov monomials by induction:

x2, x1, [x2x1], [x2[x2x1 ]] = [x2x2x1], [[x2x1]x1] = [x2x1 x1 ], [x2[x2x2x1 ]] = [x2x2x2x1], [x2[x2x1 x1]] = fex2x1 x1], [[x2x1 x1]x1] = [x2x1 x1 x1], [[x2x1][x2x1 x1]] = [x2x1 x2x1 x1],

and so on. They are exactly all Shirshov regular (LS) Lie monomials and their associative supports are exactly all Shirshov regular words with a one-to-one correspondence between two sets given by the Shirshov elimination (bracketing) algorithm for (associative) words.

Let us recall that an elementary step of Shirshov's elimination algorithm is to join the minimal letter of a word to previous ones by bracketing and to continue this process with the lexicographic ordering of the new alphabet. For example, suppose that x2 >- x1. Then we have the succession of bracketings

x2 x1 x2 x1 x1 x2 x1 x1 x1 x1 x2 x1 x1, [x2 x1][x2 x1 x1][x2 x1 x1 x1][x2 x1 x1], [x2 x1][[x2 x1 x1][x2 x1 x1 x1]][x2 x1 x1 ], [[x2 x1][[x2 x1 x1][x2 x1 x1 x1]]][x2 x1 x1], [[[x2 x1][[x2 x1 x1][x2 x1 x1 x1]]][x2 x1 x1]]; x2 x1 x1 x1 x2 x1 x1 x2 x1 x2 x2 x1 , [x2 x1 x1 x1][x2 x1 x1][x2 x1]x2[x2 x1], [x2 x1 x1 x1][x2 x1 x1][x2 x1][x2 [x2 x1]]; x2 x1 x1 x1 -< x2 x1 x1 -< x2 x1 -< x2 x2 x1.

By the way, the second series of partial bracketings illustrates Shirshov's factorization theorem [203] of 1958 that every word is a non-decreasing product of LS words (it is often mistakenly called Lyndon's theorem, see [12]).

The Shirshov special bracketing [203] goes as follows. Let us give as an example the special bracketing of the LS word w = x2x2x1 x1 x2x1 x1 x1 with the LS subword u = x2x2x1. The Shirshov standard bracketing is

[w] = [x2[[[x2x1]x1 ][x2x1 x1 x1]]].

6 The Lyndon-Shirshov basis for the alphabet x1 , x2 is different from the above Shirshov content basis starting with monomials of degree 7.

The Shirshov special bracketing is

[w]u = [[[u]Xi][X2Xi Xi Xi]].

In general, if w = aub then the Shirshov standard bracketing gives [w] = [a[uc]d], where b = cd. Now, c = c1 ••• ct, each ci is an LS-word, and c1 ^ ••• ^ ct in the lex ordering (Shirshov's factorization theorem). Then we must change the bracketing of [uc]:

[w]u = [a[...[[u][ci]] ...[ct ]]d ]

The main property of [w]u is that [w]u is a monic associative polynomial with the maximal monomial w; hence, [w]u = w.

Actually, Shirshov [207], 1962 needed a 'double' relative bracketing of a regular word with two disjoint LS subwords. Then he implicitly used the following property: every LS subword of c = c1 ••• ct as above is a subword of some ci for 1 < i < t.

Shirshov defined regular (LS) monomials [203], 1958, as follows: (w) = ((u)(v)) is a regular monomial iff:

(1) w is a regular word;

(2) (u) and (v) are regular monomials (then automatically u >- v in the lex ordering);

(3) if (u) = ((u1)(u2)) then u2 ^ v.

Once again, if we formally omit all Lie brackets in Shirshov's paper [207] then essentially the same algorithm and essentially the same CD-lemma (with the same but much simpler proof) yield a linear basis for associative algebra presented by generators and defining relations. The differences are the following:

- no need to use LS monomials and LS words, since the set X* is a linear basis of the free associative algebra k<X);

- the definition of associative composition for monic polynomials f and g,

(f, g)w = fb - ag, w = fb = ag, deg(w) < deg(f) + deg(g),

(f, g)w = f - agb, w = f = agb, w, a, b e X*,

are much simpler than the definition of Lie composition for monic Lie polynomials f and g,

<f, g)w = [fb]f - [ag]g, w = fb = ag, deg(w) < deg(f) + deg(g),

<f, g)w = f - [agb]g, w = f = agb, w, a, b, f, g e X*,

where [fb] f, [ag]g, and [agb]g are the Shirshov special bracketings of the LS words w with fixed LS subwords f and g respectively.

- The definition of elimination of the leading word s of an associative monic polynomial s is straightforward: asb ^ a(rs)b whenever s = s - rs and a, b e X*. However, for Lie polynomials, it is much more involved and uses the Shirshov special bracketing: f ^ f - [agb]g whenever f = agb.

We can formulate the main idea of Shirshov's proof as follows. Consider a complete set S of monic Lie polynomials (all compositions are trivial). If w = aisibi = a2 s2b2, where w, ai, bi e X* and w is an LS word, while s1, s2 e S, then the Lie monomials [a1s1b1]si and [a2s2b2]j2 are equal modulo the smaller Lie monomials in Id(S):

[aisibi]si = [a2s2b2]s2 + ^ a [aisibi]s-, i >2

where ai e k, si e S and [aisibi= aisibi < w. Actually, Shirshov proved a more general result: if (a1s1b1) = a1s[b1 and (a2s2b2) = a2s2b2 with w = a1s[b1 = a2s2b2 then

(aisibi) = (a2s2b2) + ^ ai (aisibi),

where ai e k, si e S and (aisibi) = aisibi < w. Below we call a Lie polynomial (asb) a Lie normal S-word provided that (asb) = asb.

This is precisely where he used the notion of composition and other notions and properties mentioned above.

It is much easier to prove an analogue of this property for associative algebras (as well as commutative associative algebras): given a complete monic set S in k(X) (k[X]), for w = a1s~1b1 = a2s2b2 with ai, bi e X* and s1, s2 e S we have

aisibi = a2s2b2 + ^ a^s^i, i>2

where ai e k, si e S and aisibi < w.

Summarizing, we can say with confidence that the work (Shirshov [207]) implicitly contains the CD-lemma for associative algebras as a simple exercise that requires no new ideas. The first author, Bokut, can confirm that Shirshov clearly understood this and told him that "the case of associative algebras is the same". The lemma was formulated explicitly in Bokut [22], 1976 (with a reference to Shirshov's paper [207]), Bergman [11], 1978, and Mora [171], 1986.

Let us emphasize once again that the CD-Lemma for associative algebras applies to every semigroup P = sgp(X |S>, and in particular to every group, by way of the semigroup algebra kP over a field k. The latter algebra has the same generators and defining relations as P, or kP = k (X |S>. Every composition of the binomials u1 - v\ and u2 - v2 is a binomial u - v. As a result, applying Shirshov's algorithm to a set of semigroup relations S gives rise to a complete set of semigroup relations Sc. The Sc-irreducible words in X constitute the set of normal forms of the elements of P.

Before we go any further, let us give some well-known examples of algebra, group, and semigroup presentations by generators and defining relations together with linear

bases, normal forms, and GS bases for them (if known). Consider a field k and a commutative ring or commutative k-algebra K.

- The Grassman algebra over K is

K(X\x2 = 0, XiXj + XjXi = 0, i > j).

The set of defining relations is a GS basis with respect to the deg-lex ordering. A K-basis is

X ■ ■ ■ Xin \Xij e X, j = 1,...,n, ii < ••• < in, n > 0}.

- The Clifford algebra over K is

K(X\XiXj + XjXi = aij, 1 < i, j < n),

where (aij) is an n x n symmetric matrix over K. The set of defining relations is a GS basis with respect to the deg-lex ordering. A K-basis is

{Xii ••• Xin \Xij e X, j = 1,...,n, n > 0, ii < ••• < in}.

- The universal enveloping algebra of a Lie algebra L is

Uk (L) = k{x\XiXj - XjXi = ^ afjXk, i > j.

If L is a free K -module with a well-ordered K -basis

X = {Xi\i e I}, [XiXj] = ^akijXk, i > j, i, j e I,

then the set of defining relations is a GS basis of UK (L). The PBW theorem follows: UK (L) is a free K-module with a K-basis,

{Xi1 ••• Xin \ ii <•••< in, it e I, t = 1,...,n, n > 0}.

- Kandri-Rody and Weispfenning [122] invented an important class of (noncommu-tative polynomial) 'algebras of solvable type', which includes universal enveloping algebras. An algebra of solvable type is

R = k(X\iij = XiXj - XjXi - Pij, i > j, Pij < XiXj),

and the compositions (sij, Sjk)w = 0 modulo (S, w), where w = XiXjXk with i > j > k. Here pij is a noncommutative polynomial with all terms less than XiXj. They created a theory of GS bases for every algebra of this class; thus, they found a linear basis of every quotient of R.

- A general presentation Uk (L) = k(X|S(-)> of a universal enveloping algebra over a field k, where L = Lie(X|S) with S c Lie(X) c k(X> and S(-) is S as a set of associative polynomials. PBW theorem in a Shirshov's form. The following conditions are equivalent:

(i) the set S is a Lie GS basis;

(ii) the set S(-) is a GS basis for k(X>;

(iii) a linear basis for Uk(L) consists of words u1u2 ■■■un, where ui are S-irreducible LS words with u1 ^ u2 ^ ■■■ ^ un (in the lex-ordering), see [56,57];

(iv) a linear basis for L consists of the S-irreducible LS Lie monomials [u] in X;

(v) a linear basis for Uk (L) consists of the polynomials u = [u1 ]■■■ [un], where u 1 ^ ■■■ ^ un in the lex ordering, n > 0, and each [ui ] is an S-irreducible non-associative LS word in X.

- Free Lie algebras LieK(X) over K. Hall, Shirshov, and Lyndon provided different linear K -bases for a free Lie algebra (the Hall-Shirshov series of bases, in particular, the Hall basis, the Lyndon-Shirshov basis, the basis compatible with the free solvable (polynilpotent) Lie algebra) [194], see also [15]. Two anticom-mutative GS bases of LieK(X) were found in [34,37], which yields the Hall and Lyndon-Shirshov linear bases respectively.

- The Lie k-algebras presented by Chevalley generators and defining relations of types An, Bn, Cn, Dn, G2, F4, E6, E7, and E8. Serre's theorem provides linear bases and multiplication tables for these algebras (they are finite dimensional simple Lie algebras over k). Lie GS bases for these algebras are found in [49-51].

- The Coxeter group

W = sgp(S|s2 = 1, mij(si, sj) = mji(sj, si)>

for a given Coxeter matrix M = (mij). Tits [210] (see also [14]) algorithmically solved the word problem for Coxeter groups. Finite Coxeter groups are presented by 'finite' Coxeter matrices An, Bn, Dn, G2, F4, E6, E7, E8, H3, and H4. Coxeter's theorem provides normal forms and Cayley tables (these are finite groups generated by reflections). GS bases for finite Coxeter groups are found in [58].

- The Iwahory-Hecke (Hecke) algebras H over K differ from the group algebras

2 1/2 K (W) of Coxeter groups in that instead of sf = 1 there are relations (si - q/ )(si +

q1/2) = 0 or (si - qi )(si + 1) = 0, where qi are units of K. Two K-bases for H are known; one is natural, and the other is the Kazhdan-Lusztig canonical basis [155]. The GS bases for the Iwahory-Hecke algebras are known for the finite Coxeter matrices. A deep connection of the Iwahory-Hecke algebras of type An and braid groups (as well as link invariants) was found by Jones [116].

- Affine Kac-Moody algebras [117]. The Kac-Gabber theorem provides linear bases for these algebras under the symmetrizability condition on the Cartan matrix. Using this result, Poroshenko found the GS bases of these algebras [178-180].

- Borcherds-Kac-Moody algebras [61-63,117]. GS bases are not known.

- Quantum enveloping algebras (Drinfeld, Jimbo). Lusztig's theorem [154] provides linear canonical bases of these algebras. Different approaches were developed by

Ringel [191,192], Green [110], and Kharchenko [131-135]. GS bases of quantum enveloping algebras are unknown except for the case An, see [55,86,195,220].

- Koszul algebras. The quadratic algebras with a basis of standard monomials, called PBW-algebras, are always Koszul (Priddy [184]), but not conversely. In different terminology, PBW-algebras are algebras with quadratic GS bases. See [177].

- Elliptic algebras (Feigin, Odesskii) These are associative algebras presented by n generators and n(n - 1)/2 homogeneous quadratic relations for which the dimensions of the graded components are the same as for the polynomial algebra in n variables. The first example of this type was Sklyanin algebra (1982) generated by x1, x2, and x3 with the defining relations [x3, x2] = x2, [x2, x1] = x|, and [x1; x3] = x|. See [175]. GS bases are not known.

- Leavitt path algebras. GS bases for these algebras are found in Alahmedi et al. [2] and applied by the same authors to determine the structure of the Leavitt path algebras of polynomial growth in [3].

- Artin braid group Brn. The Markov-Artin theorem provides the normal form and semi-direct structure of the group in the Burau generators. Other normal forms of Brn were obtained by Garside, Birman-Ko-Lee, and Adjan-Thurston. GS bases for Brn in the Artin-Burau, Artin-Garside, Birman-Ko-Lee, and Adjan-Thurston generators were found in [23-25,89] respectively.

- Artin-Tits groups. They differ from Coxeter groups in the absence of the relations sf = 1. Normal forms are known in the spherical case, see Brieskorn, Saito [64]. GS bases are not known except for braid groups (the Artin-Tits groups of type An).

- The groups of Novikov-Boon type (Novikov [173], Boon [60], Collins [97], Kalorkoti [118-121]) with unsolvable word or conjugacy problem. They are groups with standard bases (standard normal forms or standard GS bases), see [16-18,77].

- Adjan's [1] and Rabin's [187] constructions of groups with unsolvable isomorphism problem and Markov properties. A GS basis is known for Adjan's construction [26].

- Markov's [161] and Post's [183] semigroups with unsolvable word problem. The GS basis of Post's semigroup is found in [223].

- Markov's construction of semigroups with unsolvable isomorphism problem and Markov properties. The GS basis for the construction is not known.

- Plactic monoids. A theorem due to Richardson, Schensted, and Knuth provides a normal form of the elements of these monoids (seeLothaire [151]). New approaches to plactic monoids via GS bases in the alphabets of row and column generators are found in [29].

- The groups of quotients of the multiplicative semigroups of power series rings with topological quadratic relations of the type k((x, y, z, t \xy = zt)) embeddable (without the zero element) into groups but in general not embeddable into division algebras (settling a problem of Malcev). The relative standard normal forms of these groups found in [19,20] are the reduced words for what was later called a relative GS basis [59].

To date, the method of GS bases has been adapted, in particular, to the following

classes of linear universal algebras, as well as for operads, categories, and semirings.

Unless stated otherwise, we consider all linear algebras over a field k. Following

the terminology of Higgins and Kurosh, we mean by a ((differential) associative)

algebra a linear space ((differential) associative algebra) with a set of multi-linear operations

- Associative algebras, Shirshov [207], Bokut [22], Bergman [11];

- Associative algebras over a commutative algebra, Mikhalev and Zolotykh [170];

- Associative T-algebras, where V is a group, Bokut and Shum [59];

- Lie algebras, Shirshov [207];

- Lie algebras over a commutative algebra, Bokut et al. [31];

- Lie p-algebras over k with char k = p, Mikhalev [166];

- Lie superalgebras, Mikhalev [165,167];

- Metabelian Lie algebras, Chen and Chen [75];

- Quiver (path) algebras, Farkas et al. [101];

- Tensor products of associative algebras, Bokut et al. [30];

- Associative differential algebras, Chen et al. [76];

- Associative (n-)conformal algebras over k with char k = 0, Bokut et al. [45], Bokut et al. [43];

- Dialgebras, Bokut et al. [38];

- Pre-Lie (Vinberg-Koszul-Gerstenhaber, right (left) symmetric) algebras, Bokut et al. [35],

- Associative Rota-Baxter algebras over k with char k = 0, Bokut et al. [32];

- L-algebras, Bokut et al. [33];

- Associative ^-algebras, Bokut et al. [41];

- Associative differential ^-algebras, Qiu and Chen [185];

- ^-algebras, Bokut et al. [33];

- Differential Rota-Baxter commutative associative algebras, Guo et al. [111];

- Semirings, Bokut et al. [40];

- Modules over an associative algebra, Golod [108], Green [109], Kang and Lee [123,124], Chibrikov [90];

- Small categories, Bokut et al. [36];

- Non-associative algebras, Shirshov [206];

- Non-associative algebras over a commutative algebra, Chen et al. [81];

- Commutative non-associative algebras, Shirshov [206];

- Anti-commutative non-associative algebras, Shirshov [206];

- Symmetric operads, Dotsenko and Khoroshkin [98].

At the heart of the GS method for a class of linear algebras lies a CD-lemma for a free object of the class. For the cases above, the free objects are the free associative algebra k(X>, the doubly free associative k[Y]-algebra k[Y](X>, the free Lie algebra Lie(X), and the doubly free Lie k[Y]-algebra Liek[Y](X). For the tensor product of two associative algebras we need to use the tensor product of two free algebras, k(X> ® k(Y>. We can view every semiring as a double semigroup with two associative products ■ and o. So, the CD-lemma for semirings is the CD-lemma for the semiring algebra of the free semiring Rig(X). The CD-lemma for modules is the CD-lemma for the doubly free module Modk(Y> (X), a free module over a free associative algebra. The CD-lemma for small categories is the CD-lemma for the 'free partial k-algebra' kC(X> generated by an oriented graph X (a sequence z1z2 ■■■ zn, where zi e X, is a

partial word in X iff it is a path; a partial polynomial is a linear combination of partial words with the same source and target).

All CD-lemmas have essentially the same statement. Consider a class V of linear universal algebras, a free algebra V( X ) in V, and a well-ordered k-basis of terms N ( X ) of V(X). A subset S c V(X) is called a GS basis if every composition of the elements of S is trivial (vanishes upon the elimination of the leading terms s for s e S). Then the following conditions are equivalent:

(i) S is a GS basis.

(ii) If f e Id(S) then the leading term f contains the subterm s for some s e S.

(iii) The set of S-irreducible terms is a linear basis for the V-algebra V (X | S> generated by X with defining relations S.

In some cases ((n-) conformal algebras, dialgebras), conditions (i) and (ii) are not equivalent. To be more precise, in those cases we have (i) ^ (ii) ^ (iii).

Typical compositions are compositions of intersection and inclusion. Shirshov [206,207] avoided inclusion composition. He suggested instead that a GS basis must be minimal (the leading words do not contain each other as subwords). In some cases, new compositions must be defined, for example, the composition of left (right) multiplication. Also, sometimes we need to combine all these compositions. We present here a new approach to the definition of a composition, based on the concept of the least common multiple lcm(u, v) of two terms u and v.

In some cases (Lie algebras, (n-) conformal algebras) the 'leading' term f of a polynomial f e V(X) lies outside V(X). For Lie algebras, we have f e k(X>, for (n-) conformal algebras f belongs to an '^-semigroup'.

Almost all CD-lemmas require the new notion of a 'normal S-term'. A term (asb) in {X, Œ}, where s e S, with only one occurrence of s is called a normal S-term whenever (asb) = (a(s)b). Given S c k(X>, every S-word (that is, an S-term) is a normal S-word. Given S c Lie(X), every Lie S-monomial (Lie S-term) is a linear combination of normal Lie S-terms (Shirshov [207]).

One of the two key lemmas asserts that if S is complete under compositions of multiplication then every element of the ideal generated by S is a linear combination of normal S-terms. Another key lemma says that if S is a GS basis and the leading words of two normal S-terms are the same then these terms are the same modulo lower normal S-terms. As we mentioned above, Shirshov proved these results [207] for Lie(X) (there are no compositions of multiplication for Lie and associative algebras).

This survey continues our surveys with Kolesnikov, Fong, Ke, and Shum [27,28, 42,46,52,53], Ufnarovski's survey [213], and the book of the first named author and Kukin [54].

The paper is organized as follows. Section 2 is for associative algebras, Sect. 3 is for semigroups and groups, Sect. 4 is for Lie algebras, and the short Sect. 5 is for Œ-algebras and operads.7

To conclude this introduction, we give some information about the work of Shir-shov; for more on this, see the book [209]. Shirshov (1921-1981) was a famous Russian

7 The first definitions of the symmetric operad were given by Kurosh's student Artamonov under the name 'clone of multilinear operations' in 1969, see Kurosh [144] and Artamonov [4], cf. Lambek (1969) [146] and May (1972) [162].

mathematician. His name is associated with notions and results on the Grobner-Shirshov bases, the Composition-Diamond lemma, the Shirshov-Witt theorem, the Lazard-Shirshov elimination, the Shirshov height theorem, Lyndon-Shirshov words, Lyndon-Shirshov basis (in a free Lie algebra), the Hall-Shirshov series of bases, the Cohn-Shirshov theorem for Jordan algebras, Shirshov's theorem on the Kurosh problem, and the Shirshov factorization theorem. Shirshov's ideas were used by his students Efim Zelmanov to solve the restricted Burnside problem and Aleksander Kemer to solve the Specht problem.

1.1 Digression on the history of Lyndon-Shirshov bases and Lyndon-Shirshov words

Lyndon [156], 1954, defined standard words, which are the same as Shirshov's regular words [203], 1958. Unfortunately, the papers (Lyndon [156]) and (Chen et al. [72], 1958) were practically unknown before 1983. As a result, at that time almost all authors (except four who used the names Shirshov and Chen-Fox-Lyndon, see below) refer to the basis and words as Shirshov regular basis and words, cf. for instance [8,9,96,188,212,224]. To the best of our knowledge, none of the authors mentioned Lyndon's paper [156] as a source of 'Lyndon words' before 1983(!).

In the following papers the authors mentioned both (Chen et al. [72]) and (Shirshov [203]) as a source of 'Lyndon-Shirshov basis' and 'Lyndon-Shirshov words':

- Schutzenberger and Sherman [196], 1963;

- Schutzenberger [197], 1965;

- Viennot [217], 1978;

- Michel [163], 1975; [164], 1976.

The authors of [196] thank Cohn for pointing out Shirshov's paper [203]. They also formulate Shirshov's factorization theorem [203]. They mention [72,203] as a source of 'LS words'. Schutzenberger also mentions [197] Shirshov's factorization theorem, but in this case he attributes it to both Chen et al. [72] and Shirshov [203]. Actually, he cites [72] by mistake, as that result is absent from the paper, see Berstel and Perrin [12].8

Starting with the book of Lothaire, Combinatorics on words ([151], 1983), some authors called the words and basis 'Lyndon words' and 'Lyndon basis'; for instance, see Reutenauer, Free Lie algebras ([190], 1993).

2 Grobner-Shirshov bases for associative algebras

In this section we give a proof of Shirshov's CD-lemma for associative algebras and Buchberger's theorem for commutative algebras. Also, we give the Eisenbud-Peeva-

8 From [12]: "A famous theorem concerning Lyndon words asserts that any word w can be factorized in a unique way as a non-increasing product of Lyndon words, i.e. written w = X1X2 ■■■xn with X1 > X2 > ••• > xn. This theorem has imprecise origin. It is usually credited to Chen et al., following the paper of Schutzenberger [197] in which it appears as an example of factorization of free monoids. Actually, as pointed out to one of us by Knuth in 2004, the reference [72] does not contain explicitly this statement."

Sturmfels lifting theorem, the CD-lemmas for modules (following Kang and Lee [124] and Chibrikov [90]), the PBW theorem and the PBW theorem in Shirshov's form, the CD-lemma for categories, the CD-lemma for associative algebras over commutative algebras and the Rosso-Yamane theorem for Uq (An).

2.1 Composition-Diamond lemma for associative algebras

Let k be a field, k (X) be the free associative algebra over k generated by X and X* be the free monoid generated by X, where the empty word is the identity, denoted by 1. Suppose that X* is a well-ordered set. Take f e k (X) with the leading word f and f = a f - rf, where 0 = a e k and rj < f. We call f monic if a = 1.

A well-ordering > on X* is called a monomial ordering whenever it is compatible with the multiplication of words, that is, for all u ,v e X* we have

u > v ^ w1uw2 > w1 vw2, for all w1, w2 e X*.

A standard example of monomial ordering on X* is the deg-lex ordering, in which two words are compared first by the degree and then lexicographically, where X is a well-ordered set.

Fix a monomial ordering < on X* and take two monic polynomials f and g in k(X). There are two kinds of compositions:

(i) If w is a word such that w = fb = ag for some a, b e X* with \ f \ + \g \ > \w\ then the polynomial (f, g)w = fb - ag is called the intersection composition of f and g with respect to w.

(ii) If w = f = agb for some a, b e X* then the polynomial (f, g)w = f - agb is called the inclusion composition of f and g with respect to w.

Then (f, g)w < w and (f, g)w lies in the ideal Id{ f, g} of k(X) generated by f and g.

In the composition (f, g)w, we call w an ambiguity (or the least common multiple lcm( f, g), see below).

Consider S c k(X) such that very s e S is monic. Take h e k(X) and w e X*. Then h is called trivial modulo (S, w), denoted by

h = 0 mod (S, w),

if h = X aiaisibi, where ai e k, at, bi e X*, and si e S with atltbi < w. The elements asb, a, b e X*, and s e S are called S-words. A monic set S c k( X) is called a GS basis in k( X) with respect to the monomial ordering < if every composition of polynomials in S is trivial modulo S and the corresponding w.

A set S is called a minimal GS basis in k (X) if S is a GS basis in k(X) avoiding inclusion compositions; that is, given f, g e S with f = g, we have f = agb for all a, b e X*.

Irr(S) = {u e X*\u = asb, s e S, a, b e X*}.

The elements of Irr(S) are called S-irreducible or S-reduced.

A GS basis S in k(X) is reduced provided that supp(s) c Irr(S\{s}) for every s e S, where supp(s) = {u1; u 2 ,...,un} whenever s = 'Yn=1 aiUi with 0 = ai e k and ui e X*. In other words, each ui is an S\{s}-irreducible word.

The following lemma is key for proving the CD-lemma for associative algebras.

Lemma 1 If S is a GS basis ink (X) and w = a1sib1 = a2s2b2, where a1, b1; a2, b2 e X* and s1, s2 e S, then a1s1b1 = a2s2b2 mod (S, w).

Proof There are three cases to consider.

Case 1 Assume that the subwords s1 and s2 of w are disjoint, say, \a2\ > \a1\ + \s 1\. Then, a2 = a1s1c and b1 = cs2b2 for some c e X*, and so w1 = a1s1cs2b2. Now,

aisibi - a2s2b2 = aisics2b2 - aisics2b2

= aisic(s2 - s2)b2 + ai(si -s i)cs2b2.

Since sj - s2 < s2 and s1 - si < s1, we conclude that

aisibi - a2s2b2 = ^aiuisivi + ^Pjujs2vj i j

with ai, fij e k and S-words uis1vi and ujs2vj satisfying uis1vi, ujs2vj < w.

Case 2 Assume that the subword s1 of w contains s2 as a subword. Then s1 = as2b with a2 = a1a and b2 = bb1, that is, w = a1as2bb1 for some S-word as2b. We have

a1s1b1 - a2s2b2 = a1s1b1 - a1as2bb1 = a1(s1 - as2b)b1 = a1(s1, s2)sib1.

The triviality of compositions implies that a1s1b1 = a2s2b2 mod (S, w).

Case 3 Assume that the subwords s1 and s2 of w have a nonempty intersection. We may assume that a2 = a1a and b1 = bb2 with w = s1b = as2 and \w\ < \s 1\ + \s2\. Then, as in Case 2, we have a1s1b1 = a2s2b2 mod (S, w). □

Lemma 2 Consider a set S c k(X) ofmonic polynomials. For every f e k (X) we have

f aiui + X fjajsjbj

ui < f ajsjbj < f

where ai, fj e k, ui e Irr( S), and ajsjbj are S-words. So, Irr (S) is a set of linear generators of the algebra k( X\S).

Proof Induct on f . □

Theorem 1 (The CD-lemma for associative algebras) Choose a monomial ordering < on X*. Consider a monic set S c k (X) and the ideal Id (S) ofk (X) generated by S. The following statements are equivalent:

(i) S is a Grobner-Shirshov basis in k (X).

(ii) f e Id(S) ^ f = asb for some s e S and a, b e X*.

(iii) Irr(S) = {u e X *\u = asb, s e S, a, b e X *} is a linear basis of the algebra k( X\S).

Proof (i)^(ii). Assume that S is a GS basis and take 0 = f e Id(S). Then, we have f = Xn=1 aiaisibi where at e k, at, bi e X*, and si e S. Suppose that wi = aisibi satisfy

w1 = w2 = ••• = wi > wl+1 >••• .

Induct on w1 and l to show that f = asb for some s e S and a, b e X*. To be more precise, induct on (w1, l) with the lex ordering of the pairs.

If l = 1 then f = a1s1b1 = a{s\b1 and hence the claim holds. Assume that l > 2. Then w1 — a1sb — a2sjb2. Lemma 1 implies that a1s1b1 = a2sjb2 mod (S, w1). If a1 + a2 = 0 or l > 2 then the claim follows by induction on l. For the case a1 + a2 = 0 and l = 2, induct on w1. Thus, (ii) holds.

(ii)^(iii). By Lemma 2, Irr(S) generates k(X\S) as a linear space. Suppose that aiui = 0 in k(X\S), where 0 = ai e k and ui e Irr(S). It means that Xi aiui e

Id(S) in k(X). Then £i aiui = uj e Irr(S) for some j, which contradicts (ii).

(iii)^(i). Given f, g e S, Lemma 2 and (iii) yield (f, g)w = 0 mod (S, w). Therefore, S is a GS basis. □

A new exposition of the proof of Theorem 1 (CD-lemma for associative algebras). Let us start with the concepts of non-unique common multiple and least common multiple oftwo words u, v e X * .A common multiple cm (u, v) means that cm(u, v) = a1ub1 = a2vb2 for some ai, bi e X*. Then lcm(u, v) means that some cm(u, v) contains some lcm(u, v) as a subword: cm(u, v) = c ■ lcm(u, v) ■ d with c, d e X*, where u and v are the same subwords in both sides. To be precise,

lcm(u, v) e {ucv, c e X* (a trivial lcm(u, v)); u = avb, a, b e X* (an inclusion lcm(u, v)); ub = av, a, b e X*, \ub\ < \u\ + \v\ (an intersection lcm(u, v))}.

Define the general composition (f, g)lcm( j g) of monic polynomials f, g e k( X) as

(f, g)lcm(f,~g) = lcm(f , g)\f^f - lcm(f , g)lg^g.

The only difference with the previous definition of composition is that we include the case of trivial lcm( f , g). However, in this case the composition is trivial,

(f, g)jcg = 0 mod ({f, g}, fcg).

It is clear that if a1 fb1 = a2gb2 then, up to the ordering of f and g,

a1 fb1 - a2gb2 = c ■ (f, g)lcm(f,-g) ■ d■

This implies Lemma 1. The main claim (i)^(ii) of Theorem 1 follows from Lemma 1.

Shirshov algorithm. If a monic subset S c k (X) is not a GS basis then we can add to S all nontrivial compositions, making them monic. Iterating this process, we eventually obtain a GS basis Sc that contains S and generates the same ideal, Id(Sc) = Id(S). This Sc is called the GS completion of S. Using the reduction algorithm (elimination of the leading words of polynomials), we may obtain a minimal GS basis Sc or a reduced GS basis.

The following theorem gives a linear basis for the ideal Id(S) provided that S c k( X) is a GS basis.

Theorem 2 If S c k (X) is a Grobner-Shirshov basis then, given u e X*\Irr(S), by Lemma 2 there existsH e kIrr(S) with iU < u (ifu = 0) such that u - ¡m e Id(S) and the set {u - u\u e X *\Irr(S)} is a linear basis for the ideal Id (S) ofk (X).

Proof Take 0 = f e Id(S). Then by the CD-lemma for associative algebras, f = aisib1 = u1 for some s1 e S and a1, b1 e X*, which implies that f = u1 e X*\Irr(S). Put f1 = f - a1(u 1 - Mi), where a1 is the coefficient of the leading term of f and uj < u1 or u1 = 0. Then f1 e Id(S) and f1 < f. By induction on f, the set {u - u\u e X*\Irr(S)} generates Id(S) as a linear space. It is clear that {u - u\u e X*\Irr(S)} is a linearly independent set. □

Theorem 3 Choose a monomial ordering > on X*. For every ideal I ofk(X) there exists a unique reduced Grobner-Shirshov basis Sfor I.

Proof Clearly, a Grobner-Shirshov basis S c k(X) for the ideal I = Id(S) exists; for example, we may take S = I .By Theorem 1, we may assume that the leading terms of the elements of S are distinct. Given g e S, put

Ag = {f e S\ f = g and f = agb for some a, b e X*}

and S1 = S\ UgsS Ag.

For every f e Id(S) we show that there exists an s1 e S1 such that f = asib for some a, b e X*.

In fact, Theorem 1 implies that f = a'hb' for some a', b' e X* and h e S. Suppose that h e S\S1. Then we have h e UgeS Ag, say, h e Ag. Therefore, h = g and h = agb for some a, b e X*. We claim that h > g. Otherwise, h < g. It follows that h = agb > ahb and so we have the infinite descending chain

h > ahb > a2hb2 > a3hb3 > ...,

which contradicts the assumption that > is a well ordering.

Suppose that g e Si. Then, by the argument above, there exists g1 e S such that g e Ag1 and g > gi. Since > is a well ordering, there must exist s1 e S1 such that f = a{s\b1 for some a1, b1 e X*.

Put f1 = f - a1a1s1b1, where a1 is the coefficient of the leading term of f. Then f1 e Id(S) and f >J\.

By induction on f, we know that f e Id(S1), and hence I = Id(S1). Moreover, Theorem 1 implies that S1 is clearly a minimal GS basis for the ideal Id(S). Assume that S is a minimal GS basis for I.

For every s e S we have s = s' + s", where supp(s') c Irr(S\{s}) and s" e Id(S\{s}). Since S is a minimal GS basis, it follows that s = s' for every s e S.

We claim that S2 = {s'|s e S} is a reduced GS basis for I. In fact, it is clear_that S2 c Id(S) = I .By Theorem 1, for every f e Id(S) we have f = a{s\b1 = a1s1 b1 for some a1, b1 e X*.

Take two reduced GS bases S and R for the ideal I. By Theorem 1, for every s e S,

1 = arb, r = cs[d

for some a, b, c, d e X*, r e R, and s1 e S, and hence s = acsidb. Since s e supp(s) c Irr(S\{s}), we have s = s1. It follows that a = b = c = d = 1, and so s = r.

If s = r then 0 = s - r e I = Id(S) = Id(R). By Theorem 1, s—r = a1rlb1 = c1s2d1 for some a1, b1, c1, d1 e X* with rj", s2 < s = r. This means that s2 e S\{s} and r1 e R\{r}. Noting that s - r e supp(s) U supp(r), we have either s - r e supp(s) or s - r e supp(r). If s - r e supp(s) then s - r e Irr(S\{s}), which contradicts s - r = c1s2d1;ifs - r e supp(r) then s - r e Irr(R\{r}),which

contradicts s - r = a1?Tb1. This shows that s = r, and then S c R. Similarly, R c S.

Remark 1 In fact, a reduced GS basis is unique (up to the ordering) in all possible cases below.

Remark 2 Both associative and Lie CD-lemmas are valid when we replace the base field k by an arbitrary commutative ring K with identity because we assume that all GS bases consist of monic polynomials. For example, consider a Lie algebra L over K which is a free K-module with a well-ordered K-basis {a; |/ e I}. With the deg-lex ordering on {a; |j e I}*, the universal enveloping associative algebra UK(L) has a (monic) GS basis

jiaiaj - aja; = ^ atijat > j, i, j e IJ,

where aij e K and [a*, aj] = X a\jat in L, and the CD-lemma for associative algebras over K implies that L c UK (L) and

a ••• ain | i 1 <•••< in, n > 0, *1, ...,in e I}

is a K-basis for UK (L).

In fact, for the same reason, all CD-lemmas in this survey are valid if we replace the base field k by an arbitrary commutative ring K with identity. If this is the case then claim (iii) in the CD-lemma should read: K(X\S) is a free K-module with a K-basis Irr(S). But in the general case, Shirshov's algorithm fails: if S is a monic set then S', the set obtained by adding to S all non-trivial compositions, is not a monic set in general, and the algorithm may stop with no result.

2.2 Grobner bases for commutative algebras and their lifting to Grobner-Shirshov bases

Consider the free commutative associative algebra k[X]. Given a well ordering < on X = {xi\i e I},

[ X ] = {xi1 ...Xit \ i i <••• < it, ii,...,it e I, t > 0}

is a linear basis for k[X].

Choose a monomial ordering < on [X]. Take two monic polynomials f and g in k[ X ] such that w = lcm( f, g) = fa = gb for some a, b e [X ] with \ f \ + \g \ > \w\ (so, f andgarenotcoprimein [X]).Then (f, g)w = fa-gbiscalledthes-polynomial of f and g.

A monic subset S c k[X] is called a Grobner basis with respect to the monomial ordering < whenever all s-polynomials of two arbitrary polynomials in S are trivial modulo S and corresponding w.

An argument similar to the proof of the CD-lemma for associative algebras justifies the following theorem due to Buchberger.

Theorem 4 (Buchberger Theorem) Choose a monomial ordering < on [X]. Consider a monic set S c k[X] and the ideal Id(S) ofk[X] generated by S. The following statements are equivalent:

(i) S is a Grobner basis in k[X].

(ii) f e Id(S) ^ f = sa for some s e S and a e [X].

(iii) Irr(S) = {u e [X ]\u = sa, s e S, a e [X ]} is a linear basis for the algebra k[ X \S] = k [ X ]/Id(S).

Proof Denote by lcm(u, v) be the usual (unique) least common multiple of two commutative words u,v e [X]:

lcm(u, v) e {uv (the triviallcm(u, v)); au = bv, a, b e [X], \au\ < \u\ + \v\ (the nontriviallcm(u, v))}.

If cm(u, v) = a1u = a2v is a common multiple of u and v then cm(u, v) = b ■ lcm(u, v).

The s-polynomial of two monic polynomials f and g is

(f, g )lcm( f ,g) = lcm( f, g)\f^ f - lcm( f, g)lg^g.

An analogue of Lemma 1 is valid for k[X] because if a1s1 = a2s2 for two monic polynomials s1 and s2 then

aisi - a2s2 = b ■ (si, s2)lcm(si,s2). Lemma 1 implies the main claim (i)^(ii) of Buchberger's theorem. □

Theorem 5 Given an ideal I ofk[X] and a monomial ordering < on [X], there exists a unique reduced Grobner basis Sfor I. Moreover, ifX is finite then so is S.

Eisenbud et al. [99] constructed a GS basis in k (X) by lifting a commutative Grobner basis for k[X] and adding all commutators. Write X = {x1, x2,..., xn} and put

Si = {hij = xtxj - xjxt \ i > j} c k(X).

Consider the natural map y : k(X) ^ k[X] carrying xi to xi and the lexicographic splitting of y , which is defined as the k-linear map

8 : k [ X ] ^ k (X), xi1 x;2 •••xir ^ xi1 xi2 •••xir if ii < i2 •••< ir.

Given u e [X], we express it as u = xJ x22 ••• x"", where h > 0, using an arbitrary monomial ordering on [X].

Following [99], define an ordering on X* using the ordering x1 < x2 < ••• < xn as follows: given u,v e X*, put

u > v if y(u) > Y(v) in [X] or (y(u) = y(v) and u >iex v).

It is easy to check that this is a monomial ordering on X* and 8(s) = 8(s) for every s e k[X]. Moreover, v > 8(u) for every v e y-1(m).

Consider an arbitrary ideal L of k[X] generated by monomials. Given m =

xii xif ' * * xir

e L , ii < i2 ••• < ir, denote by Ul(m) the set of all monomials u e [xii+1,xir-1] such that neither uxi2 ••• xir nor uxii ■ ■ ■ xir-1 lie in L.

Theorem 6 ([99]) Consider the orderings on [X] and X* defined above. If S is a minimal Grobner basis ink[X] then S' = {8 (us)\s e S, u e UL (s)}U S1 is a minimal Grobner-Shirshov basis in k (X), where L is the monomial ideal ofk [X] generated by S.

Jointly with Yongshan Chen [30], we generalized this result to lifting a GS basis S c k[Y] ® k(X), see Mikhalev and Zolotykh [170], to a GS basis of Id(S, [yt, yj] for all (i, j))of k(Y) ®k(X).

Recall that for a prime number p the Gauss ordering on the natural numbers is described as s <p t whenever Q ^ 0 mod p. Let <0 = < be the usual ordering on the natural numbers. A monomial ideal L of k[X] is called p-Borel-fixed whenever it satisfies the following condition: for each monomial generator m of L, if m is divisible by xj but no higher power of xj then (xi /xj )sm e L for all i < j and s <p t. Thus, we have the following Eisenbud-Peeva-Sturmfels lifting theorem.

Theorem 7 ([99]) Given an ideal I of k[X], take L = Id( f, f e I) and J = Y-1(I) C k<X).

(i) If L is 0-Borel-fixed then a minimal Grobner-Shirshov basis of J is obtained by applying 8 to a minimal Grobner basis of I and adding commutators.

(ii) IfL is p-Borel-fixedfor some p then J has a finite Grobner-Shirshov basis.

Proof Assume that L is p-Borel-fixed for some p. Take a generator m = xi1 xi2 ••• xir of L, where xi1 < xi2 < ••• < xir, and suppose that x\ is the highest power of xir dividing m. Since t <p t, it follows that x\m/x\ e L for l < ir . This implies that xfm/xir e L for l < ir, and hence, every monomial in UL(m) satisfies degxi (u) < t for i1 < l < ir. Thus, UL (m) is a finite set, and the result follows from Theorem 6. In particular, if p = 0 then UL (m) = 1. □

In characteristic p > 0 observe that if the field k is infinite then after a generic change of variables L is p-Borel-fixed. Then Theorems 6 and 7 imply

Corollary 1 ([99]) Consider an infinite field k and an ideal I c k[ X ]. After a general linear change of variables, the ideal y-1(I) in k <X) has a finite Grobner-Shirshov basis.

2.3 Composition-Diamond lemma for modules

Consider S, T c k<X) and f, g e k<X). Kang and Lee define [123] the composition of f and g as follows.

Definition 1 ([123,127])

(a) If there exist a, b e X * such that w = fa = bg with |w| < | f | + |g| then the intersection composition is defined as (f, g)w = fa - bg.

(b) If there exist a, b e X * such that w = afb = g then the inclusion composition is defined as (f, g)w = afb - g.

(c) The composition (f, g)w is called right-justified whenever w = f = ag for some a e X*.

If f - g = Z atatstbt + X Pjcjtj,where at ,Pj e k, at, bi, cj e X*, s* e S, and tj e T with aisibi < w and cjtj < w for all i and j, then we call f - g trivial with respect to S and T and write f = g mod (S, T; w).

Definition 2 ([123,124]) A pair (S, T) ofmonic subsets of k<X) is called a GSpair if S is closed under composition, T is closed under right-justified composition with respect to S, and given f e S, g e T, and w e X* such that if (f, g)w is defined, we have (f, g)w = 0 mod (S, T; w). In this case, say that (S, T) is a GS pair for the A-module aM = a k<X)/(k<X)T + Id(S)), where A = k<X|S).

Theorem 8 (Kang and Lee [123,124], the CD-lemma for cyclic modules) Consider a pair (S, T) of monic subsets of k< X), the associative algebra A = k< X |S) definedbyS, and the left cyclic module AM = A k< X)/(k< X)T + Id(S)) definedby (S, T).Suppose that (S, T) is a Grobner-Shirshov pair for the A-module AM and p e k <X )T+Id(S). Then p = asb or p = ct , where a, b, c e X*, s e S, and t e T.

Applications of Theorem 8 appeared in [125-127].

Take two sets X and Y and consider the free left k(X)-module Modk(X)(Y) with k(X)-basis Y. Then Modk(X)(Y) = ©ysYk(X)y is called a double-free module. We now define the GS basis in Modk(X) (Y). Choose a monomial ordering < on X*, and a well-ordering < on Y. Put X*Y = {uy\u e X*, y e Y} and define an ordering < on X* Y as follows: for any w1 = u 1 y1, w2 = u2y2 e X*Y,

w1 < w2 ^ u 1 < u2 or u 1 = u2, y1 < y2

Given S c Modk(X) (Y) with all s e S monic, define composition in S to be only inclusion composition, which means that f = ag for some a e X*, where f, g e S. If (f, g) f = f - ag = X atatst, where at e k, at e X*, si e S, and at^i < f, then this composition is called trivial modulo (S, f).

Theorem 9 (Chibrikov [90], see also [78], the CD-lemma for modules) Consider a non-empty set S c modk(X) (Y) with all s e S monic and choose an ordering < on X* Y as before. The following statements are equivalent:

(i) S is a Grobner-Shirshov basis in Modk(X) (Y).

(ii) If 0 = f e k( X )S then f = as for some a e X * and s e S.

(iii) Irr(S) = {w e X *Y \w = as, a e X *, s e S} is a linear basis for the quotient Modk( x ) (Y\S) =Modk(X) (Y)/k( X )S.

Outline of the proof. Take u e X* Y and express it as u = uXyu with uX e X* and yu e Y. Put

cm(u, v) = aXu = bXv, lcm(u, v) = u = dXv,

where yu = yv. Up to the order of u and v, we have cm(u, v) = c ■ lcm(u, v). The composition of two monic elements f, g e Modk(X) (Y) is

(f, g)\lcm(f,~g) = lcm(f , g)\/Wf - lcm(f , g)\g^g.

If a1s1 = a2s2 for monic s1 and s2 then a1s1 - a2s2 = c ■ (s1, s2)lcm(si,i2). This gives an analogue of Lemma 1 for modules and the implication (i)^(ii) of Theorem 9.

Given S c k(X), put A = k(X\S). We can regard every left A-module AM as a k(X)-module in a natural way: fm := (f + Id(S))m for f e k(X) and m e M. Observe that A M is an epimorphic image of some free A-module. Assume now that aM = Mod a(Y \T) = Mod a (Y)/AT, where T c Mod a (Y). Put

Ti = {X fiyi e Modk(x)(Y(fi + Id(S))yt e t}

and ^ = SX* Y U T1. Then AM = mod k(X)(Y\as k(X)-modules.

Theorem 10 Given a submodule I of Modk(X)(Y) and a monomial ordering < on X* Y as above, there exists a unique reduced Grobner-Shirshov basis Sfor I.

Corollary 2 (Cohn) Every left ideal I ofk <X) is a free leftk <X) -module.

Proof Take a reduced Grobner-Shirshov basis S of I as a k<X)-submodule of the cyclic k<X)-module. Then I is a free left k<X)-module with a k<X)-basis S. □

As an application of the CD-lemma for modules, we give GS bases for the Verma modules over the Lie algebras of coefficients of free Lie conformal algebras. We find linear bases for these modules.

Let B be a set of symbols. Take the constant locality function N : B x B ^ Z+; that is, N (a, b) = N for all a, b e B. Put X = {b(n)| b e B, n e Z} and consider the Lie algebra L = Lie(X|S) over a field k of characteristic 0 generated by X with the relations

S = j X(-1)s (N)[b(n - s)a(m + s)] = 0| a, b e B, m, n e zj .

For every b e B, put b = ^nb(n)z-n-1 e L[[z, z-1]]. It is well-known that these elements generate a free Lie conformal algebra C with data (B, N) (see [194]). Moreover, the coefficient algebra of C is just L.

Suppose that B is linearly ordered. Define an ordering on X as

a(m) < b(n) ^ m < n or (m = n and a < b).

We use the deg-lex ordering on X*. It is clear that the leading term of each polynomial in S is b(n)a(m) with

n - m > N or (n - m = N and (b > a or (b = a and N is odd))).

The following lemma is essentially from [194].

Lemma 3 ([78]) With the deg-lex ordering on X*, the set S is a GS basis in Lie(X).

Corollary 3 ([78]) A linear basis of the universal enveloping algebra U = U(L) of L consists of the monomials

a1(n1)a2(n2) •••ak(nk) with ai e B and ni e Z such that for every 1 < i < k we have

IN - 1 if ai > ai+1 or a = ai+1 and N is odd) N otherwise.

An L-module M is called restricted if for all a e C and v e M there is some integer T such that a(n)v = 0 for n > T.

An L-module M is called a highest weight module whenever it is generated over L by a single element m e M satisfying L+m = 0, where L + is the subspace of L generated by {a (n) |a e C, n > 0}. In this case m is called a highest weight vector.

Let us now construct a universal highest weight module V over L, which is often called the Verma module. Take the trivial 1-dimensional L+-module kIv generated by Iv; hence, a(n)Iv = 0 for all a e B, n > 0. Clearly,

V = IndL+ kIv = U(L) ®u(L+) kIv = U(L)/U(L)L + .

Then V has the structure of the highest weight module over L with the action given by multiplication on U(L)/U(L)L + and a highest weight vector I e U(L). In addition,

V = U (L )/U (L )L + is the universal enveloping vertex algebra of C and the embedding

V : C ^ V is given by a ^ a(-1)I (see also [194]).

Theorem 11 ([78]) With the above notions, a linear basis ofV consists of the elements

ai(ni)a2(n2) • --ak(nk), at e B, m e Z satisfying the condition in Corollary 3 and nk < 0. Proof Clearly, as k(X}-modules, we have

u V = u (U (L )IU ( L )L+) = Modk( x} ( 11 S(-) X * I, a(n) I, n > 0} =k(x} ( 11 S'},

where S' = {S(-)X* I, a(n)I, n > 0}. In order to show that S' is a Grôbner-Shirshov basis, we only need to verify that w = b(n)a(m)I, where m > 0. Take

f = 2](-l)sQ(b(n - s)a(m + s) - a(m + s)b(n - s))I and g = a(m)I.

Then (f, g)w = f - b(n)a(m)I = 0 mod (S', w) since n - m > N, m + s > 0, n - s > 0, and 0 < s < N .It follows that S' is a Grôbner-Shirshov basis. Now, the result follows from the CD-lemma for modules. □

2.4 Composition-Diamond lemma for categories Denote by X an oriented multi-graph. A path

an ^ an-\ ai ^ a0, n > 0,

in X with edges xn,..., x2, x1 is a partial word u = x1 x2 ••• xn on X with source an and target a0. Denote by C (X) the free category generated by X (the set of all partial words (paths) on X with partial multiplication, the free 'partial path monoid' on X). A well-ordering on C (X) is called monomial whenever it is compatible with partial multiplication.

A polynomial f e kC(X) is a linear combination of partial words with the same source and target. Then kC(X) is the partial path algebra on X (the free associative partial path algebra generated by X).

Given S c kC(X), denote by Id(S) the minimal subset of kC(X) that includes S and is closed under the partial operations of addition and multiplication. The elements of Id(S) are of the form £ atatstbi with ai e k, at, bi e C(X), and si e S, and all S-words have the same source and target.

Both inclusion and intersection compositions are possible.

With these differences, the statement and proof of the CD-lemma are the same as for the free associative algebra.

Theorem 12 ([36], the CD-lemma for categories) Consider a nonempty set S c kC(X) ofmonic polynomials and a monomial ordering < on C (X). Denote by Id(S) the ideal ofkC (X) generated by S. The following statements are equivalent:

(i) The set S is a Grobner-Shirshov basis in kC(X).

(ii) f e Id(S) ^ f = asbfor some s e S and a, b e C(X).

(iii) the set Irr (S) = {u e C (X )\u = asb a, b e C (X), s e S} is a linear basis for kC(X)/Id(S), which is denoted by kC (X \S).

Outline of the proof.

Define w = lcm (u, v), u,v e C (X) and the general composition (f, g)w for f, g e kC(X) and w = lcm( f, g) by the same formulas as above. Under the conditions of the analogue of Lemma 1, we again have a1s1b1 - a2s2b2 = c(s1, s2)wd = 0 mod (S, w), where w = lcm(s1, s2) and c, d e C(X). This implies the analogue of Lemma 1 and the main assertion (i)^(ii) of Theorem 12.

Let us present some applications of CD-lemma for categories.

For each non-negative integer p, denote by [p] the set {0, 1, 2,..., p} of integers in their usual ordering. A (weakly) monotonic map / : [q] ^ [p] is a function from [q] to [p] such that i < j implies /(i) < /(j). The objects [p] with weakly monotonic maps as morphisms constitute the category A called the simplex category. It is convenient to use two special families of monotonic maps,

V(X) = {[p]\ p e Z +U{0}},

E(X) = {ep : [p - 1]^ [p], n]q :[q + 1]^[q]\ p > 0, 0 < i < p, 0 < j < q} .

e'q : [q - 1]^ [q], n'q :[q + 1]^ [q]

defined for i = 0, 1,... q (and for q > 0 in the case of el) by

n'q (j )

eq (j)

j if i > j;

j + 1 if i < j,

j if i > j-

j - 1 if j > i.

Take the oriented multi-graph X = (V(X), E(X)) with

Consider the relation S c C (X) x C (X) consisting of:

fq+1,q : 4+1eq-1 = for j > i'

gqq+1 : 4 n'q +1 = nlq n}q +1 for j > i;

hq-l,q : nq-1eq =

4-i nqfor j > î,

1q-1 for î = j Or î = j + 1,

î-1 ni-2 fOr i > j + 1

q-Vi q

This yields a presentation A = C(X |S) of the simplex category A. Order now C(X) as follows.

Firstly, for nlp, nq ^ Wp Ip > 0, 0 < i < p} put nlp > Vq iff p > q or (p = q and i < j).

Secondly, for

u = np1 n% • • • nX e {np Ip > 0, 0 < i < p}*

(these are all possible words on {np |p > 0, 0 < i < p}, including the empty word 1v, where v e Ob(X)), define

wt (u) = (n, np ,nt\, ••• ,np1 )•

Then, for u,v e {nlp |p > 0, 0 < i < p}* put u > v iff wt(u) > wt(v) lexicographically.

Thirdly, for e1p, eJq e {ep, |p e Z+, 0 < i < p}, put e1p > e]q iff p > q or (p = q and i < j).

Finally, for u = v0elpi1 v1etj22 • • • elpnvn e C(X), where n > 0, and vj e {np|p >

0, 0 < i < p}* put wt(u) = (n, v0, v1,...,vn ,elpi1,..., efin). Then for every u,v e C (X),

u v ^ wt(u) > wt(v) lexicographically. It is easy to check that is a monomial ordering on C (X). Then we have

Theorem 13 ([36]) For X and S defined above, with the ordering on C(X), the set S is a Grobner-Shirshov basis for the simplex partial path algebra kC (X | S).

Corollary 4 ([157]) Every morphism /x : [q] ^ [p] of the simplex category has a unique expression of the form

ei1 elm n j1 njn

ep ... ep-m+1 nq-n . . . nq-1

with p > i1 > ••• > im > 0, 0 < j1 < ••• < jn < q, and q - n + m = p.

The cyclic category is defined by generators and relations as follows, see [104]. Take the oriented (multi) graph Y = (V(Y), E(Y)) with V (Y) = {[p]| p e Z +U{0}} and

E(Y) = {ep : [p - 1]^ [p], nJq :[q + 1]^ [q], tq : [q]

^ [q]| p > 0, 0 < i < p, 0 < j < q}.

Consider the relation S c C (Y) x C (Y) consisting of:

fq+1,q : 4+1eq-1 = eq+ 1^ for j > 1

gq,q+1: nq nq+1 = nq nq+1 for j >i;

e'q-1n]q-2 for j > i:

hq-1,q : nq-1 eq =

1q-1 for i = j or i = j + 1,

nq-2 for i >j+^

P1 P2 P3

tqelq = elq 1tq-1 for i = 1,---,q;

tq nq = nlq-1tq+1 for i = 1,...,q; tq+1 — 1

tq = 1q •

The category C(Y|S) is called the cyclic category and denoted by A. Define an ordering on C(Y) as follows.

Firstly, for tp, t}q e {tq |q > 0}* put (tp)i > (tq)j iff i > j or (i = j and p > q).

Secondly, for nip, nJq e {np |p > 0, 0 < i < p} put nip > nJq iff p > q or (p = q and i < j ). Thirdly, for

u = w0nlp1 W1n% ■ ■ ■ wn-1npn wn e {tq, nip ^, p > 0, 0 < i < p}*, where wi e {tq |q > 0}*, put

wt(u) = (n, W0, W1,..., Wn, npn, np-\,..., nlp1).

Then for every u,v e {tq, nip |q, p > 0, 0 < i < p}* put u > v iff wt(u) > wt(v) lexicographically.

Fourthly, for elp,eJq e {ep, |p e Z +, 0 < i < p}, elp > e]q iff p > q or (p = q and i < j ).

Finally, for u = v0elpi1 v1 elp2 ■ ■ ■ elpnn vn e C(Y) and vj e {tq, nip |q, p > 0, 0 < i <

p}* define wt(u) = (n, v0, v1,..., vn, ep,..., elpn).

Then for every u,v e C(Y) put u v ^ wt(u) < wt(v) lexicographically. It is also easy to verify that is a monomial ordering on C(Y) which extends . Then we have

Theorem 14 ([36]) Consider Y and S defined as the above. Put p4 : tqe°q = e| and P5 : tqn0 = nftq2+1. Then

(1) With the ordering >-2 on C(Y), the set S U {p4, p5} is a Grobner-Shirshov basis for the cyclic category C(Y |S).

(2) Every morphism / :[q [p] of the cyclic category A = C (Y | S) has a unique expression of the form

ei1 eim n j1 njn tk

ep ... ep-m+1 nq-n . . . nq-1 q

with p > i1 > ••• > im > 0, 0 < j1 < ••• < jn < q, 0 < k < q, and q - n + m = p.

2.5 Composition-Diamond lemma for associative algebras over commutative algebras

Given two well-ordered sets X and Y, put

N = [X]Y* = {u = uXuYe [X] and uY e Y*}

and denote by kN the k-space spanned by N. Define the multiplication of words as

u = uXuY, v = vXvY e N ^ uv = uXvXuYvY e N.

This makes kN an algebra isomorphic to the tensor product k[X] ® k(Y), called a'double free associative algebra'. It is a free object in the category of all associative algebras over all commutative algebras (over k): every associative algebra KA over a commutative algebra K is isomorphic to k[X] ® k(Y)/Id(S) as a k-algebra and a k[ X ]-algebra.

Choose a monomial ordering > on N. The following definitions of compositions and the GS basis are taken from [170].

Take two monic polynomials f and g in k[X]® k(Y) and denote by L the least common multiple of f X and gX.

1. Inclusion. Assume that gY isasubwordof fY ,say,fY = cgYd forsome c, d e Y *. If fY = gY then f X > gX and if gY = 1 then we set c = 1. Put w = LfY =

LcgYd. Define the composition C1(f, g, c)w = -L^f - g^cgd.

2. Overlap. Assume that a non-empty beginning of gY is a non-empty ending of f Y, say, f Y = cc0, gY = c0 d, and fYd = cgY for some c, d, c0 e Y * and c0 = 1. Put w = LfYd = LcgY. Define the composition C2 (f, g, c0)w = -Lrfd - gL^cg.

3. External. Take a (possibly empty) associative word c0 e Y*. In the case that the greatest common divisor of f X and g X is non-empty and both f Y and gY are non-empty, put w = LfYc0 gY and define the composition C3( f, g, c0 )w =

fXfc0gY - jrfYc0g.

A monic subset S of k[X] ® k(Y} is called a GS basis whenever all compositions of elements of S, say (f, g)w, are trivial modulo (S, w):

(f, g)w = ^atatstbi, i

where ai, bi e N, si e S, ai e k, and aisibi < w for all i.

Theorem 15 (Mikhalev and Zolotykh [170,228], the CD-lemma for associative algebras over commutative algebras) Consider a monic subset S c k[X] ® k(Y} and a monomial ordering < on N. The following statements are equivalent:

(i) The set S is a Grobner-Shirshov basis in k[X] ® k(Y}.

(ii) For every element f e Id(S), the monomial f contains s as its subwordfor some s e S.

(iii) The set Irr (S) = {w e N |w = asb, a, b e N, s e S} is a linear basis for the quotient k [ X ]® k (Y}.

Outline of the proof. For

w = lcm(u, v) = lcm(uX, vX) lcm(uY, vY) the general composition is

X X X X X X

(s1, s2)w = (lcm(uX, vX)/uX)w|a^s1 - (lcm(uX, vX)/vX)w|v^s2,

where s1, s2 e k[X](Y} are k-monic with u = s 1 and v = s2. Moreover, (s1, s2)w = 0 mod ({s1, s2}, w) whenever w = uXvXuYcYvY with cY e Y*, that is, w is a trivial least common multiple relative to both X-words and Y-words. This implies the analog of Lemma 1 and the claim (i)^(ii) in Theorem 15. We apply this lemma in Sect. 4.3.

2.6 PBW-theorem for Lie algebras

Consider a Lie algebra (L, []) over a field k with a well-ordered linear basis X = {xi |i e I} and multiplication table S = {[xixj] = [|xixj |]|i > j, i, j e I}, where for every i, j e I we write [|xixj |] = £taijxt with aij e k. Then U(L) = k(X|S(-)} is

called the universal enveloping associative algebra of L, where S(-) = {xixj - xjxi = [|xixj |]|i > j, i, j e I}.

Theorem 16 (PBW Theorem) In the above notation and with the deg-lex ordering on X *, the set S(-) is a Grobner-Shirshov basis ofk( X}. Then by the CD-lemma for associative algebras, the set Irr (S(-)) consists of the elements

xi1 ...xin with i1 <•••< in, i1,...,in e I, n > 0,

and constitutes a linear basis of U(L).

Theorem 17 (The PBW Theorem in Shirshov's form) Consider L = Lie(X|S) with S c Lie( X) c k( X) and U (L) = k (X |S(-)>. The following statements are equivalent.

(i) For the deg-lex ordering, S is a GS basis ofLie(X).

(ii) For the deg-lex ordering, S(-) is a GS basis ofk(X).

(iii) A linear basis ofU (L) consists of the words u = u 1 ••• un, where u 1 ^ ••• ^ un in the lex ordering, n > 0,andeveryui isanS(-) -irreducible associative Lyndon-Shirshov word in X.

(iv) A linear basis ofL is the set of all S-irreducible Lyndon-Shirshov Lie monomials [u] in X.

(v) A linear basis of U (L) consists of the polynomials u = [u1]--- [un], where u1 < ••• ^ un in the lex ordering, n > 0, and every [ui ] is an S-irreducible non-associative Lyndon-Shirshov word in X.

The PBW theorem, Theorem 33, the CD-lemmas for associative and Lie algebras, Shirshov's factorization theorem, and property (VIII) of Sect. 4.2 imply that every LS-subword of u is a subword of some ui.

Makar-Limanov gave [158] an interesting form of the PBW theorem for a finite dimensional Lie algebra.

2.7 Drinfeld-Jimbo algebra Uq(A), Kac-Moody enveloping algebra U(A), and the PBW basis of Uq (AN)

Take an integral symmetrizable N x N Cartan matrix A = (aj). Hence, an = 2, aij < 0 for i = j, and there exists a diagonal matrix D with diagonal entries di, which are nonzero integers, such that the product DA is symmetric. Fix a nonzero element q of k with q4di = 1 for all i. Then the Drinfeld-Jimbo quantum enveloping algebra is

Uq (A) = k (X U H U Y | S+ U K U T U S~

X = {Xi}, H = {¿f1}, Y = [yt}

Z(-1)M1 - jXi1

x1 1 "XjxV, where i = j, t = q2di

v=0 1—a

1 —v j y1

^0 N ' t

y1 1 "yjyV, where i = j, t = q2di

v=0 ¿j

K = [hihj-hjhi, hihi 1 -1, h, 1hi - 1, Xjhf1 - qT1diaij^'Xj

j — j i i i

T1 y..h±l

hf1 yj - q*1 yjh*1},

h2 - h-2

T = \ Xiyj - yjXi - &Hq2di - q-2di

m-i+1 _ ti-m-1

tm-i+i -1, ti -t

(for m > n > 0), (for n = 0 or m = n).

Theorem 18 ([55]) For every symmetrizable Cartan matrix A, with the deg-lex ordering on {X U H U Y }*, the set S+c U T U K U S-c is a Grobner-Shirshov basis of the Drinfeld-Jimbo algebra Uq (A), where S+c and S-c are the Shirshov completions of

S+ and S-.

Corollary 5 (Rosso [195], Yamane [220]) For every symmetrizable Cartan matrix A we have the triangular decomposition

Uq (A) = Uq+(A) ® k[H] ® U-(A)

with U+(A) = k(X|S+} and U-(A) = k(Y|S-}.

Similar results are valid for the Kac-Moody Lie algebras g( A) and their universal enveloping algebras

U(A) = k(X U H U Y|S+ U H U K U S-},

where S+, S- are the same as for Uq (A),

K = {hjhj - hjhi, xjhi - htxj + dta^xt, htyt - ytht + dtatjyj},

and T = {xtyj - yjxt - Stjht}.

Theorem 19 ([55]) For every symmetrizable Cartan matrix A, the set S+c U T U K U S-c is a Grobner-Shirshov basis of the universal enveloping algebra U (A) of the Kac-Moody Lie algebra g(A).

The PBW theorem in Shirshov's form implies

Corollary 6 (Kac [117]) For every symmetrizable Cartan matrix A, we have the triangular decomposition

U(A) = U+(A) ® k[H]®U-(A), g(A) = g+(A) ©k[H]®g-(A).

_Poroshenko[179,180] found GS bases for the Kac-Moody algebras of types An,

Bn, A, and An. He used the available linear bases of the algebras [117]. Consider now

2 -1 0 • •• 0

-1 2 -1 • •• 0

A = AN = 0 -1 2 • •• 0

0 0 0 • •• 2

and assume that q8 = 1. Introduce new variables, defined by Jimbo (see [220]), which generate Uq(An):

X = {xtj, 1 < i < j < N + 1},

x xi j = i + 1 ,

xij — 1 i

[ qxt, j-1 xj-1, j - q 1 xj-1, jxt, j-1 j > i + 1.

Order the set X as follows: xmn > xij ^^ (m, n) >iex (i, j). Recall from Yamane [220] the notation

C1 = {((i, j), (m, n))|i = m < j < n}, C2 = {((i, j), (m, n))|i < m < n < j}, C3 = {((i, j), (m, n))|i < m < j = n}, C4 = {((i, j), (m, n))|i < m < j < n}, C5 = {((i, j), (m, n))|i < j = m < n}, C6 = {((i, j), (m, n))|i < j < m < n}.

Consider the set S+ consisting of Jimbo's relations:

xmnxij - q xijxmn ((i, j), (m, n)) e C1 U C3,

xmnxij - xijxmn ((i, j), (m, n)) e C2 U C6,

xmnxij - xijxmn + (q2 - q-2)xinxmj ((i, j), (m, n)) e C4,

xmnxij - q xijxmn + qxin ((i, j),(m, n)) e C5.

It is easy to see that U+(AN) = k(X| A+}.

A direct proof [86] shows that S+ is a GS basis for k(il^} = U+(An) [55]. The proof is different from the argument of Bokut and Malcolmson [55]. This yields

Theorem 20_ ([55]) In the above notation and with the deg-lex ordering on {X U H U Y}*, the set S+ U T U K U S- is a Grobner-Shirshov basis of

Uq (An ) = k(Y U H U Y|Y+ U T U K U S-}.

Corollary 7 ([195,220]) For q8 = 1, a linear basis of Uq(An) consists of

ym1U1 • • • ymlHlhi • • • h Nxhj1 • • • xikjk

with (m1, n1) <•••< (ml, nl), (i1, j1) <•••< (ik, jk), k, l > 0 andst e Z. 3 Grobner-Shirshov bases for groups and semigroups

In this section we apply the method of GS bases for braid groups in different sets of generators, Chinese monoids, free inverse semigroups, and plactic monoids in two sets of generators (row words and column words).

Given a set X consider S c X* x X* the congruence p(S) on X* generated by S, the quotient semigroup

A = sgp( X |S) = X */p(S),

and the semigroup algebra k(X*/p(S)). Identifying the set {u = v|(u, v) e S} with S, it is easy to see that

a : k(X|S> ^ k(X*/p(S)), ^am + Id(S) ^ ^aiWi

is an algebra isomorphism.

The Shirshov completion Sc of S consists of semigroup relations, Sc = {ui -vj, i e I}. Then Irr(Sc) is a linear basis of k(X|S), and so a(Irr(Sc)) is a linear basis of k(X*/p(S)). This shows that Irr(Sc) consists precisely of the normal forms of the elements of the semigroup sgp(X |S).

Therefore, in order to find the normal forms of the semigroup sgp(X | S), it suffices to find a GS basis Sc in k (X | S). In particular, consider a group G = gp(X |S), where S = {(u, vi) e F(X) x F(X)|i e I} and F(X) is the free group on a set X. Then G has a presentation

G = sgp(X U X-1|S, xsx-s = 1, s = ±1, x e X), X n X-1 = 0

as a semigroup.

3.1 Grobner-Shirshov bases for braid groups

Consider the Artin braid group Bn of type An-1 (Artin [5]). We have

Bn = gp(01,...,0n | aj ai = ai aj (j -1 > i), a+1ai a+1 = ai ai+1 a, 1 < i < n - 1).

3.1.1 Braid groups in the Artin-Burau generators

Assume that X = YUZ with Y* and Z well-ordered and that the ordering on Y* is monomial. Then every word in X has the form u = u0z1u1 ■ ■ ■ zkuk, where k > 0, e Y*, and e Z. Define the inverse weight of the word u e X* as

inwt(u) = (k, uk, zku1, Z1, u 0)

and the inverse weight lexicographic ordering as

u > v ^ inwt(u) > inwt(v).

Call this ordering the inverse tower ordering for short. Clearly, it is a monomial ordering on X*.

When X = YU Z, Y = TU U, and Y * is endowed with the inverse tower ordering, define the inverse tower ordering on X* with respect to the presentation X = (TUU)UZ. In general, for

X = (■■ ■ (X (n)U X (n-1))U ■ ■ ■ )U X(0)

with X(n)-words equipped with a monomial ordering we can define the inverse tower ordering of X-words.

Introduce a new set of generators for the braid group Bn, called the Artin-Burau generators. Put

2 2 _i _i

Si,i+i = oi , Si,j+1 = Oj ■■■at+i<Ji oi+1 ■■■Oj , 1 < i < j < n - 1;

oi,j+1 = a-1 ■ ■ ■a1 < i < j < n - 1; aii = 1, {a, b} = b-1ab. Form the sets

sj = {Si, j, s-1, 1 < i, j < n} and S-1 = {of1, ■ ■■a"-1!}. Then the set

S = Sn U Sn-1 U^^^U S2 U S-1

generates Bn as a semigroup. Order now the alphabet S as

Sn < Sn-1 < ■■■ < S2 < H

si, j < si, j < s

< Sj-1

a, 1 < ct0 1 <

'n-ll-

Order Sn-words by the deg-inlex ordering; that is, first compare words by length and then by the inverse lexicographic ordering starting from their last letters. Then we use the inverse tower ordering of S-words.

Lemma 4 (Artin [6], Markov [160]) The following Artin-Markov relations hold in the braid group Bn:

_ „ s

ak su-

ai si,i+1

ai- 1Ss ,j =

ai Ss ,j -

aj- 1si, j

aj- Ss ,j -

J „-u

i,i+1a1 a-1 si-1, j ai-1,

<j-1a A

stj - isi\j+1, sj,j+1}aj1,

(1) (2)

where S = ±1;

(8) (9)

(10) (11) (12)

where i < j < k < l and e = ±1;

Oj 1 O— 1 = O— S1 forj < k — 1

Oj, j+1Ok, j+1 = Ok, j+1CTj-1, j for j < k.

(16) (17)

where j < i < k < l or i < k < j < l, and e, S = ±1.

Theorem 21 ([25]) The Artin-Markov relations (1)—(13) form a Grobner—Shirshov basis of the braid group Bn in terms of the Artin—Burau generators with respect to the inverse tower ordering of words.

It is claimed in [25] that some compositions are trivial. Processing all compositions explicitly, [82] supported the claim.

Corollary 8 (Markov-Ivanovskii [6]) The following words are normal forms of the braid group Bn:

where all fj for 2 < j < n are free irreducible words in {sij, i < j}. 3.1.2 Braid groups in the Artin-Garside generators

The Artin-Garside generators of the braid group Bn+1 are ai, 1 < i < n, A, A-1 (Garside [103] 1969), where A = A1 ■■■ An with Aj = a1 ••• a.

Putting A-1 < A < a1 < ■■■ < an, order {A-1, A,a1,..., an}* by the deg-lex ordering.

Denote by V (j, i), W (j, i),... for j < i positive words in the letters

aj, aj+1, ...,ai, assuming that V (i + 1, i) = 1, W (i + 1, i) = 1,____

Given V = V(1, i), for 1 < k < n - i denote by V(k) the result of shifting the indices of all letters in V by k: a1 ^ ak+1,. ..,ai ^ ak+i, and put V' = V(1). Define aij = ai ai-1 ...aj for j < i - 1, while aii = ai and aii+1 = 1.

fn fn — 1 • • • f2 Oinn Oin-1n-1 ■ ■ ■ °i22,

Theorem 22 ([23,47]) A Grobner-Shirshov basis S of Bn+1 in the Artin-Garside generators consists of the following relations:

at+1OiV(1, i - 1)W(j, i)oi+1 j = atoi+1OiV(1, i - 1)oijW(j, i)', Osak = akas for s - k > 2,

a1 V10201 V2 ■ ■ ■ Vn-1an ■■■ 01 = &V(n-1) V(n-2) ■ ■ ■ V^y aiA = Aan-1+1 for 1 < l < n, aiA-1 = A-1an-l+1 for 1 < l < n, AA-1 = 1, A-1 A = 1,

where 1 < i < n - 1 and 1 < j < i + 1; moreover, W(j, i) begins with ai unless it is empty, and Vi = Vi (1, i).

There are corollaries.

Corollary 9 The S-irreducible normal form of each word of Bn+1 coincides with its Garside normal form [103].

Corollary 10 (Garside [103]) The semigroup B++1 of positive braids can be embedded into a group.

3.1.3 Braid groups in the Birman-Ko-Lee generators

Recall that the Birman-Ko-Lee generators ats of the braid group Bn are

ats = (at-1at-2 ...as+1 )as (as+11 ■■■ at-12at-11) and we have the presentation

Bn = gp(ats, n > t > s > IWtsarq = arqats, (t - r)(t - q)(s - r)(s - q)> 0, atsasr = atrats = asratr, n > t > s > r > 1>.

Denote by 8 the Garside word, 8 = ann-1an-1n-2 ■■■ a21. Define the order as 8-1 < 8 < ats < arq iff (t, s) < (r, q) lexicographically. Use the deg-lex ordering on {8-1, 8, ats, n > t > s > 1}*. Instead of aij, we write simply (i, j) or (j, i). We also set

(tm, tm — 1, . ..,t1) = (tm, tm-1)(tm-1, tm-2) . . . (t2, t1)-

where tj = tj+1, 1 < j < m - 1. In this notation, we can write the defining relations of Bn as

(t3, t2, t\) = (t2, t1, t3) = (t1, t3, t2) for t3 > t2 > tx, (k, l)(i, j) = (i, j)(k, l) for k > l, i > j, k > i,

where either k > i > j > l or k > l > i > j.

Denote by V[t2,t1], where n > t2 > t1 > 1, a positive word in (k, l) satisfying t2 > k > l > t1. We can use any capital Latin letter with indices instead of V, and appropriate indices (for instance, t3 and t0 with t3 > t0) instead of t2 and t1. Use also the following equalities in Bn:

V[t2-1,tl](t2, t1) = (t2, t1 )V['t2-1,t1]

for t2 > t1, where V['t2-1,t1 ] = (V[t2-1,t1])l(k,l)^(k,l), if l=t1; (k,t1)^(t2,k)\

W[t2-1,tl](t1, t0) = (t1, t0)W[*2-1,t1]

for t2 > t1 > t0, where W[J2-1,t1] = (W[t2-1,t1])l(k,lif l=t1; (k,t1)^(k,t0).

Theorem 23 ([24]) A Grobner-Shirshov basis of the braid group Bn in the Birman-Ko-Lee generators consists of the following relations:

(k, l)(i, j) = (i, j)(k, l) fork > l > i > j,

(k, l)V[j-1,1](i, j) = (i, j)(k, l)V[j-1,1] fork > i > j > l,

(t3, t2)(t2, t1) = (t2, t1)(t3, t1),

(t3, t1) V[t2-1,1](t3, t2) = (t2, t1)(t3, t1)V[t2-1,1],

(t, s)V[t2-1,1](t2, t1)W[t3-1,t1](t3, t1) = (t3, t2)(t, s)V[t2-1,1](t2, t1)W[/t3-1,t1], (t3, s)V[t2-1,1](t2, t1 )W[t3-1,t1](t3, t1) = (t2, s)(t3, s)V[t2-1,1](t2, t1)W[/t3-1,t1], (2, 1)V2[2,1](3, 1)... V„-1[„-1,1](n, 1) = 5V2[2,1]... Vn-1[„-1,1], (t, s)5 = 5(t + 1, s + 1), (t, s)5-1 = 5-1(t-1, s-1) witht ± 1, s ± 1 (mod n), 55-1 = 1, 5-15 = 1,

where V[k,i] means, as above, a word in (i, j) satisfying k > i > j > l,t > t3, and t2 > s.

There are two corollaries.

Corollary 11 (Birmanetal. [13]) The semigroup B+ of positive braids in the Birman-Ko-Lee generators embeds into a group.

Corollary 12 (Birman et al. [13]) The S-irreducible normal form of a word in Bn in the Birman-Ko-Lee generators coincides with the Birman-Ko-Lee-Garside normal form 5k A, where A e B+.

3.1.4 Braid groups in the Adjan-Thurston generators The symmetric group Sn+1 has the presentation Sn+1 = gp(s1, ...,sn l s2 = 1, sjsi = sisj (j - 1 > i), si+1sisi+1 = sisi+1si >.

Bokut and Shiao [58] found the normal form for Sn+1 in the following statement: the set N = {s1i1 s2i2 ■ ■ ■ snin | ij < j + 1} is a Grobner-Shirshov normal form for Sn+1 in the generators si = (i, i +1) relative to the deg-lex ordering, where sji = sjsj-1 ■■■ s; for j > i and sjj+1 = 1.

Take a e Sn+1 with the normal form a = s1i1 s2i2 ■ ■ ■ snin e N. Define the length of a as |a| = l(s1i1 s2i2 ■ ■ ■ snin) and write a ± ft whenever |aft| = |a| + ft1. Moreover,

every a e N has a unique expression a = sl t sl t ■ ■ ■ s¥i with all sl ,i; = 1. The

12 t j j

number t is called the breadth of a. Now put

Bn+1 = gp{r(a), a e Sn+A{1}|r(a)r(ft) = r(Oft), a ± ft>,

where r (a) stands for a letter with index a.

Then for the braid group with n generators we have Bn+1 = B'n+1. Indeed, define

9 : Bn+1 ^ Bn+1, ai ^ r (si), 9': Bn+1 ^ Bn+1, r (a) ^ a|s,■

These mappings are homomorphism satisfying 99' = lB/+ and 9'9 = lBn+1. Hence,

Bn+1 = gp{r(a), a e Sn+1\{1} | r(a)r(ft) = r(^ft), a ± ft>.

Put X = {r(a), a e Sn+1 \{1}}. These generators of Bn+1 are called the Adjan-Thurston generators.

Then the positive braid semigroup generated by X is

B++1 = sgp{X | r(a)r(ft) = r(aft), a ± ft>.

Assume that s1 < s2 < ■■■ < sn. Define r (a) < r (ft) if and only if |a| > ft | or |a| = ft1 and a <lex ft. Clearly, this is a well-ordering on X. We will use the deg-lex ordering on X*.

Theorem 24 ([89]) The Grobner-Shirshov basis ofB++1 in the Adjan-Thurston generator X relative to the deg-lex ordering on X* consists of the relations

r (a)r (ft) = r (aft) for a ± ft; r (a)r (fty) = r (aft)r (y) for a ± ft ± y.

Theorem 25 ([89]) The Grobner-Shirshov basis ofBn+1 in the Adjan-Thurston generator X with respect to the deg-lex ordering on X* consists of the relations

(1) r (a)r (ft) = r (aft^ for a ± ft,

(2) r (a)r (ftY) = r (aft)r (y) for a ± ft ± y,

(3) r (a)A£ = Asr (a') for a = a|si ^+1-;,

(4) r (aft)r (y fi) = Ar(a')r(fi) for a ± ft ± y ± fi with r(fty) = A,

(5) ASA-S =1.

Corollary 13 (Adjan-Thurston) The normal forms for Bn+1 are Akr (a1) ■ ■ ■ r (as) for k e Z, where r (OT) ••• r (a) is minimal in the deg-lex ordering.

3.2 Grobner-Shirshov basis for the Chinese monoid

The Chinese monoid CH(X, <) over a well-ordered set (X, <) has the presentation CH(X) = sgp(X|S>, where X = {xi |i e I} and S consists of the relations

XiXjXk = xixkxj = xjxixk for i > j > k, xi xj xj = x j xi x j , xi xi x j = xi xj xi for i > j.

Theorem 26 ([85]) With the deg-lex ordering on X *, the following relations (1)-(5) constitute a Grobner-Shirshov basis of the Chinese monoid CH (X):

(1) xixj xk xjxi xk,

(2) xi xkxj xj xi xk,

(3) xi x j xj - x j xi xj ,

(4) xi xi x j - xi x j xi ,

(5) xi x j xi xk - xi xkxi x j ,

where , xj, xk e X and i > j > k.

Denote by A the set consistsing of the words on X of the form un = w1w2 ■■■wn with n > 0, where

w1 = x111

W2 = (x2 x1)t21 x222

W3 = (x3 x1)t31 (x3 x2)t32 xf

Wn = (xnx1)tn1 (xnx2)tn2 ■ ■ ■ (xnxn-1)tn(n-1)x'nm

for e X with x1 < x2 < ■■■ < xn, and all exponents are non-negative.

Corollary 14 ([71]) This A is a set of normal forms of elements of the Chinese monoid CH( X).

Consider a semigroup S. An element s e S is called an inverse of t e S whenever sts = s and tst = t. An inverse semigroup is a semigroup in which every element t has a unique inverse, denoted by t-1.

Given a set X, put X-1 = {x-1|x e X}. On assuming that X n X-1 = 0, denote X U X-1 by Y. Define the formal inverses of the elements of Y* as

1-1 = 1, (x-1)-1 = x (x e X),

(y1 y2 ■ ■ ■ yn)-1 = y- ■ ■ ■ y-1 y-1 (Уl, У2,..., yn e Y).

It is well known that

FI(X) = sgp(Y| aa-1a = a, aa-1bb-1 = bb-1 aa-1, a, b e Y*>

is the free inverse semigroup (with identity) generated by X.

Introduce the notions of a formal idempotent, a (prime) canonical idempotent, and an ordered (prime) canonical idempotent in Y*. Assume that < is a well-ordering on Y.

(i) The empty word 1 is an idempotent.

(ii) If h is an idempotent and x e Y then x-1 hx is both an idempotent and a prime idempotent.

(iii) If e1, e2,..., em, where m > 1, are prime idempotents then e = e1e2 ■■■ em is an idempotent.

(iv) An idempotent w e Y* is called canonical whenever w avoids subwords of the form x-1exfx-1, where x e Y, both e and f are idempotents.

(v) A canonical idempotent w e Y* is called ordered if every subword e = e1e2 ■■■em of w with m > 2 and ei being idempotents satisfies fir(e1) < fir(e2) < ■ ■■ < fir(em), where fir(u) is the first letter of u e Y*.

Theorem 27 ([44]) Denote by S the subset ofk(Y> consisting two kinds ofpolynomi-als:

- ef - fe, where e and f are ordered prime canonical idempotents with ef > fe;

- x-le'xf'x-1 - f'x-le', where x e Y, x-1 e'x, and xf 'x-1 are ordered prime canonical idempotents.

Then, with the deg-lex ordering on Y *, the set S is a Grober-Shirshov basis of the free inverse semigroup sgp(Y |S>.

Theorem 28 ([44]) The normal forms of elements of the free inverse semigroup sgp(Y |S> are

uoe1u1 ■ ■ ■ emum e Y*,

where m > 0, u1,..., um-1 = 1 and u0u1 ■■■ um avoids subwords of the form yy-1 for y e Y, while e1,...,em are ordered canonical idempotents such that the first (respectively last) letter of ei, for 1 < i < m is not equal to the first (respectively last) letter of (respectively -1).

The above normal form is analogous to the semi-normal forms of Poliakova and Schein [176], 2005.

3.4 Approaches to plactic monoids via Grobner-Shirshov bases in row and column generators

Consider the set X = {x1,..., xn} of n elements with the ordering x1 < ■■■ < xn. Schutzenberger called Pn = sgp(X|T> a plactic monoid (see also Lothaire [153],

Chapter 5), where T consists of the Knuth relations

XiXkXj = XkXiXj for xi < Xj < Xk, XjXiXk = XjXkXi for xi < Xj < Xk.

A nondecreasing word R e X* is called a row and a strictly decreasing word C e X* is called a column; for example, x1 x1 x3x5x5x5x6 is a row and x6x4x2x1 is a column.

For two rows R, S e A* say that R dominates S whenever | R| < | S| and every letter of R is greater than the corresponding letter of S, where | R| is the length of R.

A (semistandard) Young tableau on A (see [152]) is a word w = R1R2 ■■■ Rt in U* such that R; dominates R;+1 for all i = 1,..., t - 1. For example,

X4 X5 X5 X6 ■ X2 X2 X3 X3 X5 X7 ■ X1X1X1X2 X4 X4 X4

is a Young tableau.

Cain et al. [69] use the Schensted-Knuth normal form (the set of (semistandard) Young tableaux) to prove that the multiplication table of column words, uv = u'v', forms a finite GS basis of the finitely generated plactic monoid. Here the Young tableaux u V is the output of the column Schensted algorithm applied to uv, but u'v' is not made explicit.

In this section we give new explicit formulas for the multiplication tables of row and column words. In addition, we give independent proofs that the resulting sets of relations are GS bases in row and column generators respectively. This yields two new approaches to plactic monoids via their GS bases.

3.4.1 Plactic monoids in the row generators

Consider the plactic monoid Pn = sgp{X|T>, where X = {1, 2,..., n} with 1 < 2 < ■■■ < n. Denote by N the set of non-negative integers. It is convenient to express the rows R e X * as R = (r1, r2,..., rn), where ri for i = 1, 2,...,n is the number of occurrences of the letter i. For example, R = 111225 = (3, 2, 0, 0, 1, 0,..., 0). Denote by U the set of all rows in X* and order U* as follows. Given R =

(r1, r2,..., rn) e U, define the length |R| = r1 +-----+ rn of R in X*.

Firstly, order U: for every R, S e U, put R < S if and only if |R| < |S| or | R| = |S| and (r1, r2,..., rn) > (s1, s2,..., sn) lexicographically. Clearly, this is a well-ordering on U. Then, use the deg-lex ordering on U*.

Lemma5 ([29]) Take $= (&1, ...,&n) e U. For 1 < p < nput$ p = X ,

where & (w;, z;, w[, and z[, see below) stands for a lowercase symbol, and $p ( Wp, Zp, W'p, and Z'p, see below) for the corresponding uppercase symbol. Take W = (w1, w2,..., wn) and Z = (z1, z2,..., zn) in U. Put W' = (w'1, w'2,..., w'n) and Z' = (z1, z2,..., z'n), where w1 = 0, w'p = min(Zp-1 - W'p-1, wp), z'q = wq + zq - w'q (*) for n > p > 2 and n > q > 1. Then W ■ Z = W' ■ Z' in Pn = sgp(X|T> and W' ■ Z' is a Young tableau on X, which could have only one row, that is, Z' = (0, 0,..., 0). Moreover, Pn = sgp( X |T > = sgp(U |T>, where T = {W ■ Z = W' ■ Z' | W, Z e V}. We should emphasize that (*) gives explicitly the product of two rows obtained by the Schensted row algorithm. Jointly with our students Weiping Chen and Jing Li we proved [29], independently of Knuth's normal form theorem [137], that T is a GS basis of the plactic monoid algebra in row generators with respect to the deg-lex ordering. In particular, this yields a new proof of Knuth's theorem. 3.4.2 Plactic monoids in the column generators Consider the plactic monoid Pn = sgp(X^>, where X = {1, 2,..., n} with 1 < 2 < ■■■ < n. Every Young tableaux is a product of columns. For example, 4,556 ■ 223,357 ■ 1,112,444 = (421)(521)(531)(632)(54)(74)(4) is a Young tableau. Given a column C e X*, denote by a the number of occurrences of the letter i in C. Then a e {0, 1} for i = 1, 2,...,n. We write C = (c1; c2; ...; cn). For example, C = 6,421 = (1; 1; 0; 1; 0; 1; 0; ...; 0). Put V = {C | C is a column in X*}. For R = (r 1; r2;...; rn) e V define wt(R) = (| R|, r1,...,rn).Order V as follows: for R, S e V ,put R < S ifandonlyifwt(R) > wt(S) lexicographically. Then, use the deg-lex ordering on V*. For$ = (01;...; ) e V, put $p = ^p=1 , 1 < p < n, where$ stands for some lowercase symbol defined above and \$ stands for the corresponding uppercase symbol.

Lemma 6 ([29]) Take W = (w1; w2;...; w„), Z = (z1; z2;...; z„) e V. Define W' = (w1; w2; ...; w„) and Z' = (z1; z'2; ...; z„), where

z'1 = min(w1, z1), z'p = min(Wp - Zp-1, zp), w'q = wq + zq - z!q (**)

for n > p > 2 and n > q > 1. Then W', Z' e V and W ■ Z = W' ■ Z' in P„ = sgp(X|T>, and W' ■ Z' is a Young tableau on X. Moreover,

Pn = sgp( X |T > = sgp(V |A>,

where A = {W ■ Z = W' ■ Z' | W, Z e V}.

Equation (**) gives explicitly the product of two columns obtained by the Schensted column algorithm.

Jointly with our students Weiping Chen and Jing Li we proved [29], independently of Knuth's normal form theorem [137], that A is a GS basis of the plactic monoid algebra in column generators with respect to the deg-lex ordering. In particular, this yields another new proof of Knuth's theorem. Previously Cain, Gray, and Malheiro [69] established the same result using Knuth's theorem, and they did not find A explicitly.

Remark All results of [29] are valid for every plactic monoid, not necessarily finitely generated.

4 Grobner-Shirshov bases for Lie algebras

In this section we first give a different approach to the LS basis and the Hall basis of a free Lie algebra by using Shirshov's CD-lemma for anti-commutative algebras. Then, using the LS basis, we construct the classical theory of GS bases for Lie algebras over a field. Finally, we mention GS bases for Lie algebras over a commutative algebra and give some applications.

4.1 Lyndon-Shirshov basis and Lyndon-Shirshov words in anti-commutative algebras

A linear space A equipped with a bilinear product x ■ y is called an anti-commutative algebra if it satisfies the identity x2 = 0, and so x ■ y = -y ■ x for every x, y e A.

Take a well-ordered set X and denote by X** the set of all non-associative words. Define three orderings >lex, >deg-lex, and >n-deg-lex (non-associative deg-lex) on X**. For (u), (v) e X** put

- (u) = ((ui)(u2)) ^lex (v) = ((v1)(v2)) (here (u2) or (v2) is empty when |(u)| = 1 or | (v) | = 1) iff one of the following holds:

(a) u1u2 > v1v2 in the lex ordering;

(b) u1u2 = v1v2 and (u1) >lex (v1);

(c) u1u2 = v1v2, (u1) = (v1), and (u2) >lex (v2)';

- (u) = ((u 1 )(u2)) >deg-lex (v) = ((v1 )(v2)) iff one of the following holds:

(a) u1u2 > v1v2 in the deg-lex ordering;

(b) u1u2 = v1v2 and (ui) >deg-lex (vO;

(c) u1u2 = v1v2, (u1) = (v1), and W) >deg-lex fa)';

- (u) >n-deg-lex (v) iff one of the following holds:

(a) Ku^ > |(v)|;

(b) if |(u)| = |(v)|, (u) = ((u1)(u2)), and (v) = ((v1)(v2)) then (u1) >n-deg-lex (v1) or ((u1) = (v1) and (u2) >n-deg-lex v)).

Define regular words (u) e X** by induction on |(u)|:

(i) xi e X is a regular word.

(ii) (u) = ((u1)(u2)) is regular if both (u1) and (u2) are regular and (u1) >-lex (u2).

Denote (u) by [u] whenever (u) is regular.

The set N (X) of all regular words on X constitutes a linear basis of the free anti-commutative algebra AC(X) on X.

The following result gives an alternative approach to the definition of LS words as the radicals of associative supports u of the normal words [u].

Theorem 29 ([37]) Supposethat [u] is a regular word ofthe anti-commutative algebra AC (X).Thenu = vm,where v is a Lyndon-Shirshov word in X andm > 1.Moreover, the set of associative supports ofthe words in N (X) includes the set of all Lyndon-Shirshov words in X.

Fix an ordering >deg-lex on X** and choose monic polynomials f and g in AC(X). If there exist a, b e X * such that [w] = [f ] = [a[g ]b] then the inclusion composition of f and g is defined as (f, g)[w] = f - [a[g]b].

A monic subset S of AC(X) is called a GS basis in AC(X) if every inclusion composition (f, g)[w] in S is trivial modulo (S, [w]).

Theorem 30 (Shirshov's CD-lemma for anti-commutative algebras, cf. [206]) Consider a nonempty set S c AC (X) of monic polynomials with the ordering >deg-lex on X **. The following statements are equivalent:

(i) The set S is a Grobner-Shirshov basis in AC (X).

(ii) If f e Id(S) then [f ] = [a[s]b] for some s e Sand a,b e X*, where [asb] is a normal S-word.

(iii) The set

Irr(S) = {[u] e N(X)| [u] = [a[s]b] a, b e X*, s e S and [asb]is a normalS-word}

is a linear basis ofthe algebra AC (X |S) = AC (X)/Id(S). Define the subset S1 the free anti-commutative algebra AC(X) as

S1 = {([u][v])[w] - ([u][w])[v] - [u]([v][w]) |

[u], [v], [w] e N(X) and [u] ^lex [v] ^lex [w]}.

It is easy to prove that the free Lie algebra admits a presentation as an anti-commutative algebra: Lie(X) = AC(X)/Id(S1).

The next result gives an alternating approach to the definition of the LS basis of a free Lie algebra Lie(X) as a set of irreducible non-associative words for an anti-commutative GS basis in AC(X).

Theorem 31 ([37]) Under the ordering >deg-lex, the subset S1 of AC (X) is an anti-commutative Grobner-Shirshov basis in AC (X). Then Irr (S1) is the set of all non-associative LS words in X. So, the LS monomials constitute a linear basis of the free Lie algebra Lie(X).

Theorem 32 ([34])Define S2 by analogy with Si, but using >n-deg-lex insteadof >lex. Then with the ordering >n-deg-lex the subset S2 of AC (X) is also an anti-commutative GS basis. The set Irr (S2) amounts to the set of all Hall words in X and forms a linear basis of a free Lie algebra Lie( X).

4.2 Composition-Diamond lemma for Lie algebras over a field

We start with some concepts and results from the literature concerning the theory of GS bases for the free Lie algebra Lie(X) generated by X over a field k.

Take a well-ordered set X — (x; |i e I} with Xi > xt whenever i > t, for all i, t e I. Given u — Xii X^2 • •• Xim e X*, define the length (or degree) of u to be m and denote it by |u | — m or deg(u) — m, put fir(u) — Xj1, and introduce

xp — min(u) — min(Xi1,x;2, ••• ,Xim},

X'(u) — (x/ — XiXp • •• xp |i > p, j > 0}. j

Order the new alphabet X'(u) as follows:

xj1 > x/2 ^ ii > i2 or ii — i2 and /2 > ji.

Assuming that

u — Xri Xp ••• Xp ••• Xrt Xp • • • Xp,

where ri > p, define the Shirshov elimination

u' — xmi • • • xm e (X'(u))*.

We use two linear orderings on X*:

(i) the lex ordering (or lex-antideg ordering): i ^ v if v — i and, by induction, if u — xiui and v — Xj vi then u >- v if and only if Xi > xj or Xi — Xj and ui ^ vi;

(ii) the deg-lex ordering: u > v if |u| > |v| or |u| — |v| and u >- v.

Remark In commutative algebras, the lex ordering is understood to be the lex-deg ordering with the condition v > i for v — i.

We cite some useful properties of ALSWs and NLSWs (see below) following Shirshov [203,204,207], see also [209]. Property (X) was given by Shirshov [204] and Chen et al. [72]. Property (VIII) was implicitly used in Shirshov [207], see also Chibrikov [94].

We regard Lie(X) as the Lie subalgebra of the free associative algebra k(X) generated by X with the Lie bracket [u, v] — uv-vu.Below weprove that Lie(X) is the free Lie algebra generated by X for every commutative ring k (Shirshov [203]). For a field,

this follows from the PBW theorem because the free Lie algebra Lie(X) = Lie(X|0) has the universal enveloping associative algebra k(X> = k (X |0>.

Given f e k(X>, denote by f the leading word of f with respect to the deg-lex ordering and write f = a ff- rf with a f e k.

Definition 3 ([156,203]) Refer to w e X*\{1} as an associative Lyndon-Shirshov word, or ALSW for short, whenever

(Vu, v e X*, u, v = 1)w = uv ^ w > vu.

Denote the set of all ALSWs on X by ALSW(X).

Associative Lyndon-Shirshov words enjoy the following properties (Lyndon [156], Chen et al. [72], Shirshov [203,204]).

(I) Put xp = min(uv). If fir(u) = xp and fir(v) = xp then

u >- v (in the lex ordering on X*) & u' > v' (in the lex ordering on (X'uv)*).

(II) (Shirshov's key property of ALSWs) A word u is an ALSW in X* if and only if u' is an ALSW in (X'(u))*.

Properties (I) and (II) enable us to prove the properties of ALSWs and NLSWs (see below) by induction on length.

(III) (down-to-up bracketing) u e ALSW(X) ^ (3k) |u(k)| (k) = 1, where

u(k) = (u')(k-1) and (X(u))(k) = (X'(u)){k-1). In the process u ^ u' ^ u" ^ ■■ ■ we use the algorithm of joining the minimal letters of u, u'... to the previous words.

(IV) If u,v e ALSW(X) then uv e ALSW(X) ^ u >■ v.

(V) w e ALSW(X) ^ (for every u, v e X*\{1} and w = uv ^ w >- v).

(VI) If w e ALSW(X) then an arbitrary proper prefix of w cannot be a suffix of w and wxp e ALSW(X) if xp = min(w).

(VII) (Shirshov's factorization theorem) Every associative word w can be uniquely represented as w = c1c2 . ..c„, where c1,..., c„ e ALSW(X) and c1 ^ c2 ^ ■■■ ^ c„ .

Actually, if we apply to w the algorithm of joining the minimal letter to the previous word using the Lie product, w ^ w' ^ w" , then after finitely many steps we obtain w(k) = [c1][c2 ]... [c„ ], with c1 ^ c2 ^ ■■■ ^ c„, and w = c1c2 ...c„ would be the required factorization (see an example in the Introduction).

(VIII) If an associative word w is represented as in (VII) and v is a LS subword of w then v is a subword of one of the words c1, c2,..., c„.

(IX) If u1u2 and u2u3 are ALSWs then so is u1u2u3 provided that u2 = 1.

(X) If w = uv is an ALSW and v is its longest proper ALSW ending, then u is an ALSW as well (Chen et al. [72], Shirshov [204]).

Definition 4 (down-to-up bracketing of ALSW, Shirshov [203]) For an ALSW w, there is the down-to-up bracketing w ^ w' ^ w" w(k) = [w], where each time

we join the minimal letter of the previous word using Lie multiplication. To be more

precise, we use the induction [w] = [w'] . .

xi ^[[x!xp ]-x£ 1

Definition 5 (up-to-down bracketing of ALSW, Shirshov [204], Chen et al. [72]) For an ALSW w, we define the up-to-down Lie bracketing [[w]] by the induction [[w]] — [[[u]][[v]]], where w — uv as in (X).

(XI) If w e ALSW(X) then [w] — [[w]].

(XII) Shirshov's definition of aNLSW (non-associative LS word) (w) below is the same as [w] and [[w]]; that is, (w) — [w] — [[w]]. Chen et al. [72] used [[w]].

Definition 6 (Shirshov [203]) A non-associative word (w) in X is a NLSW if

(i) w is an ALSW;

(ii) if (w) — ((u)(v)) then both (u) and (v) are NLSWs (then (IV) implies that

u >- v);

(iii) if (w) — (((ui)(u2))(v)) then u2 ^ v. Denote the set of all NLSWs on X by NLSW(X).

(XIII) If u e ALSW(X) and [u] e NLSW(X) then H — u in k(X).

(XIV) The set NLSW(X) is linearly independent in Lie(X) c k(X) for every commutative ring k.

(XV) NLSW(X) is a set of linear generators in every Lie algebra generated by X over an arbitrary commutative ring k.

(XVI) Lie(X) c k(X) is the free Lie algebra over the commutative ring k with the k-basis NLSW(X).

(XVII) (Shirshov's special bracketing [203]) Consider w — aub with w, u e ALSW(X). Then

(i) [w] — [a[uc]d], where b — cd and possibly c — i.

(ii) Express c in the form c — ci c2 ...cn, where ci,..., cn e ALSW(X) and ci ^ c2 ^ ••• ^ cn. Replacing [uc] by [... [[u][ci]]... [cn]], we obtain the word

[w]u — [a[...[[[u][ci]][c2 ]] ...[c„ ]]d]

which is called the Shirshov special bracketing of w relative to u.

(iii) [w]u — a[u]b + Xi aiai[u]bi in k(X) with ai e k and ai, bi e X* satisfying aiubi < aub, and hence [w]u — w.

Outline of the proof. Put xp — min(w). Then w' — a'(uxm)'(bi)' in (X(w)')*, where b — xmbi and uxm is an ALSW. Claim (i) follows from (II) by induction on length. The same applies to claim (iii).

(XVIII) (Shirshov's Lie elimination of the leading word) Take two monic Lie polynomials f and s with f — asb for some a, b e X*. Then fi — f - [asb]s is a Lie polynomial with smaller leading word, and so fi < f.

(XIX) (Shirshov's double special bracketing) Assume that w — aubvc with w, u,v e ALSW(X). Then there exists a bracketing [w]u,v such that [w]u,v — [a[u]b[v]c]„,„ and [w]„,„ — w.

More precisely, [w]u,v — [a[up]uq[vr]vs] if [w] — [a[up]q[vr]s], and

[w]u,v — [a[...[... [[u][ci]]... [c;]v]... [c„]]p]

if [w] = [a[uc]p], where c = c1 ...cn is the Shirshov factorization of c and v is a subword of ci .In both cases [w]u,v = a[u ]b[v]d + X aiai [u]bi [v]di in k (X), where aiubivdi < w.

(XX) (Shirshov's algorithm for recognizing Lie polynomials, cf. the Dynkin-Specht-Wever and Friedrich algorithms). Take s e Lie(X) c k(X). Then s is an ALSW and s1 = s - as [s] is a Lie polynomial with a smaller maximal word (in the

deg-lex ordering), s1 < s, where s = as[s] +____Then s2 = s1 - asi [s 1], s2 < si.

Consequently, s e Lie(X) if and only if after finitely many steps we obtain

sm+1 = s - as [s]- a~s1 [s 1]-----a~sm [sm] = 0.

Here k can be an arbitrary commutative ring.

Definition 7 Consider S c Lie(X) with all s e S monic. Take a, b e X* and s e S. If asb is an ALSW then we call [asb]s = [asb]s |[s a special normal S-word (or a special normal s-word), where [asb]s is defined in (XVII) (ii). A Lie S-word (asb) is called a normal S-word whenever (asb) = asb. Every special normal s-word is a normal s-word by (XVII) (iii).

For f, g e S there are two kinds of Lie compositions:

(i) If w = f = agb for some a, b e X* then the polynomial (f, g)w = f - [agb]g is called the inclusion composition of f and g with respect to w.

(ii) If w is a word satisfying w = fb = ag for some a, b e X* with deg(f) + deg(g) > deg(w) then the polynomial (f, g)w = [fb]f - [ag]g is called the intersection composition of f and g with respect to w, and w is an ALSW by (IX).

Given a Lie polynomial h and w e X*, say that h is trivial modulo (S, w) and write h =Lie 0 mod(S, w) whenever h = ^i ai (aisibi), where each ai e k, (aisibi) is a normal S-word and aisibi < w.

A set S is called a GS basis in Lie( X) if every composition (f, g)w of polynomials f and g in S is trivial modulo S and w.

(XXI) If s e Lie(X) is monic and (asb) is a normal S-word then (asb) = asb + aiaisbi, where aisbi < asb.

A proof of (XXI) follows from the CD-lemma for associative algebras since {s} is an associative GS basis by (IV).

(XXII) Given two monic Lie polynomials f and g, we have

(f, g)w - (f, g)w =ass 0 mod ({f, g}, w).

Proof If (f, g)w and (f, g)w are intersection compositions, where w = fb = ag, then (XIII) and (XVII) yield

(f, g)w = [fb]f - [ag]g = fb + Zaiaifbi - ag - Pjajgbj,

where aifbi, ajgbj < fb = ag = w. Hence,

<f, g)w - (f, g)w =ass 0 mod ({f, g}, w). In the case of inclusion compositions we arrive at the same conclusion. □

Theorem 33 (PBW Theorem in Shirshov's form [56,57], see Theorem 17) A nonempty set S c Lie( X) c k< X) of monic Lie polynomials is a Grobner-Shirshov basis in Lie( X) if and only if S is a Grobner-Shirshov basis in k< X).

Proof Observe that, by definition, for any f, g e S the composition lies in Lie(X) if and only if it lies k<X).

Assume that S is a GS basis in Lie(X). Then we can express every composition < f, g)w as < f, g)w = Zi1 a (atsibt), where (atstbt) are normal S-wordsand atstbt < w. By (XXI), we have < f, g)w = XI PjCjsjdj with Cjs jdj < w. Therefore, (XXII) yields (f, g)w =ass 0 mod (S, w). Thus, S is a GS basis in k<X).

Conversely, assume that S is a GS basis in k<X). Then the CD-lemma for associative algebras implies that < f, g)w = asb < w for some a, b e X* and s e S. Then h = <f, g)w - a[asb]s e Idass(S) is a Lie polynomial and h < <f, g)w. Induction on < f, g)w yields <f, g)w =Lie 0 mod (S,w). □

Theorem 34 (The CD-lemma for Lie algebras over a field) Consider a nonempty set S c Lie(X) c k<X) of monic Lie polynomials and denote by Id(S) the ideal of Lie( X) generated by S. The following statements are equivalent:

(i) The set S is a Grobner-Shirshov basis in Lie(X).

(ii) If f e Id(S) then f = asb for some s e S and a, b e X*.

(iii) The set

Irr(S) = {[u] e NLSW(X) | u = asb, s e S, a, b e X*}

is a linear basis for Lie( X |S).

Proof (i)^(ii). Denote by Idass(S) and IdLie(S) the ideals of k<X) and Lie(X) generated by S respectively. Since IdLie(S) c Idass(S), Theorem 33 and the CD-lemma for associative algebras imply the claim.

(ii)^(iii). Suppose that Xai[Ui] = 0 in Lie(X|S) with [ui] e Irr(S) and u1 > u2 > •••, that is, X ai [ui] e IdLie(S). Then all ai must vanish. Otherwise we may assume that a1 = 0. Then £ ai [ui] = u1 and (ii) implies that [u1] e Irr(S), which is a contradiction. On the other hand, by the next property (XXIII), Irr(S) generates Lie(X|S) as a linear space.

(iii)^(i). This part follows from (XXIII). □

The next property is similar to Lemma 2.

(XXIII) Given S c Lie(X), we can express every f e Lie(X) as

f = ai [ui ] + X P j [aj sj bj ]sj

with ai, Pj e k, [ui] e Irr(S) satisfying [ui] < f, and [ajsjbj]s-. are special normal S-word satisfying [ajsjbj]s-. < f.

(XXIV) Given a normal s-word (asb), take w = asb. Then (asb) = [asb]s mod (s, w).Itfollowsthath =Lie 0 mod (S, w) is equivalent to h = X i ai [aisibi ]s-., where [aisibi]s-. are special normal S-words with aisibi < w.

Proof Observe that for every monic Lie polynomial s, the set {s} is a GS basis in Lie(X). Then (XVIII) and the CD-lemma for Lie algebras yield (asb) = [asb]s mod (s, w). □

Summary of the proof of Theorem 34.

Given two ALSWs u and v, define the ALSW-lcm(u, v) (or lcm(u, v) for short) as follows:

w = lcm(u, v) e {aucvb (an ALSW), a, b, c e X* (a triviallcm);

u = avb, a, b e X* (an inclusion lcm); ub = av, a, b e X*, deg(ub) < deg(u) + deg(v) (an intersection lcm)}.

Denote by [w]u,v the Shirshov double special bracketing of w in the case that w is a trivial lcm(u, v), by [w]u and [w]v the Shrishov special bracketings of w if w is an inclusion or intersection lcm respectively. Then we can define a general Lie composition for monic Lie polynomials f and g with f = u and g = v as

(f, g)w = [w]u,v f - [w]u,v

if w is a trivial lcm(u, v) (it is 0 mod ({f, g}, w)), and (f, g)w = [w]„ |[a]^ f - [w]v

if w is an inclusion or intersection lcm(u, v).

If S c Lie( X) c k( X) is a Lie GS basis then S is an associative GS basis. This follows from property (XVII) (iii) and justifies the claim (i)^(ii) of Theorem 34.

Shirshov's original proof of (i)^(ii) in Theorem 34, (see [207,209]), rests on an analogue of Lemma 1 for Lie algebras.

Lemma 7 ([207,209]) If (a1s1b1), (a2s2b2) are normal S-words with equal leading associative words, w = a1s~1b1 = a2s2b2, then they are equal mod (S, w), that is, (a1 s1 b1) - (a2s2b2) = 0 mod (S, w).

Outline of the proof. We have w1 = cwd and w = lcm(s~1, s2). Shirshov's (double) special bracketing lemma yields

[wJw = [c[[w]d1]d2] = c[w]d + aiai [w]b,

with ai wbi < w1. The ALSW w includes u = s1 and v = s2 as subwords, and so there is a bracketing {w} e {[w]u,v, [w]u, [w]v} such that

[a1s1b1] = [c{w}|[„]^s1 d], [a2s2b2] = [c{w}|[„]^s2 d]

are normal s1- and s2- words with the same leading associative word w1. Then

[aisibi] - [a2s2b2] = [c(si, s2)wd] = 0 mod (S, wi).

Now it is enough to prove that two normal Lie s-words with the same leading associative words, say w1, are equal mod (s, w1):

f = (asb) - [asb] =Lie 0 mod (s, w1) provided that f < w1.

Since f e Idass (s), we have f = c1sd1 by the CD-lemma for associative algebras with one Lie polynomial relation s. Then f - a[c1sd1]s is a Lie polynomial with the leading associative word smaller than w1. Induction on w1 finishes the proof.

4.2.1 Grobner-Shirshov basis for the Drinfeld-Kohno Lie algebra

In this section we give a GS basis for the Drinfeld-Kohno Lie algebra Ln.

Definition 8 Fix an integer n > 2. The Drinfeld-Kohno Lie algebra Ln over Z is defined by generators tij = tji for distinct indices 1 < i, j < n - 1 satisfying the relations [tijtki] = 0 and [tij(tik + tjk)] = 0 for distinct i, j, k, and l.

Therefore, we have the presentation Ln = Liez(T|S), where T = {j 1 < i < j < n - 1} and S consists of the following relations:

[tijtkl] = 0 if k < i < j, k < l, l = i, j; (18) [tjktij] + [tiktij] =0 if i < j < k; (19)

[tjktik] - [tiktij] = 0 ifi < j < k. (20)

Order T by setting tij < tkl if either i < k or i = k and j < l. Let < be the deg-lex ordering on T *.

Theorem 35 ([80]) With S = {(18), (19), (20)} as before and the deg-lex ordering < on T*, the set S is a Grobner-Shirshov basis of Ln.

Corollary 15 The Drinfeld-Kohno Lie algebra Ln is a free Z-module with Z-basis un-2 NLSW(Ti), where T = {tij | i < j < n - 1} for i = 1,...,n - 2.

Corollary 16 ([100]) The Drinfeld-Kohno Lie algebra Ln is an iterated semidirect product of free Lie algebras Ai generated by Ti = {tij | i < j < n - 1}, for i = 1,...,n - 2.

4.2.2 Kukin's example of a Lie algebra with undecidable word problem

Markov [161], Post [182], Turing [211], Novikov [173], and Boone [60] constructed finitely presented semigroups and groups with undecidable word problem. For groups this also follows from Higman's theorem [115] asserting that every recursively presented group embeds into a finitely presented group. A weak analogue of Higman's

theorem for Lie algebras was proved in [21], which was enough for the existence of a finitely presented Lie algebra with undecidable word problem. In this section we give Kukin's construction [142] of a Lie algebra AP for every semigroup P such that if P has undecidable word problem then so does AP.

Given a semigroup P = sgp(x, y|ui = vi, i e I}, consider the Lie algebra

AP = Lie(x, x, y, y, z|S)

with S consisting of the relations

(1) [Xx] = 0, [Xy] = 0, [yx] = 0, [yy] = 0;

(2) [xz] = -[zx], [yz] = -[zy];

(3) LzuiJ = LzviJ, i e I.

Here, LzuJ stands for the left normed bracketing.

Put x > y > z > x > y and denote by > the deg-lex ordering on the set {x, y, x, y, z}*. Denote by p the congruence on {x, y}* generated by {(ui, vi), i e I}. Put

(3') LzuJ = LzvJ, (u, v) e p with u > v.

Lemma 8 ([80]) In this notation, the set S1 = {(1), (2), (3')} is a GS basis in Lie(x, y, x, y, z).

Proof For every u e {x, y}*, we can show that Lzu\ = zu by induction on \u\. All possible compositions in Si are the intersection compositions of (2) and (3'), and the inclusion compositions of (3') and (3').

For (2) a (3'), we take f = [Xz] + [zx] and g = \_zu\ - \zv\. Therefore, w = Xzu with (u, v) e p and u > v. We have

([Xz] + [zx], Lzu\ - Lzv\>w = [fu]f - [Xg]g = L([Xz] + [zx])u\ - [X(Lzu\ - Lzv\)] = Lzxu\ + \_XzvJ = \zxu\ - Lzxv\ = 0 mod (S1, w).

For (3') a (3'), we use w = zu1 = zu2e, where e e {x, y}* and (ui, vi) e p with ui > vi for i = 1, 2. We have

(Lzui\ - Lzvi\, Lzu2\ - Lzv2\>w = (Lzui\ - Lzvi\) - L(Lzu2\ - Lzv2\)e\ = LLzv2\e\ - Lzvi\ = Lzv2e\ - Lzvi\ = 0 mod (Si, w).

Thus, Si = {(i), (2), (3')} is a GS basis in Lie(X, y, x, y, z). □

Corollary 17 (Kukin [i42]) For u, v e {x, y}* we have

u = v in the semigroup P ^ Lzu\ = Lzv\in the Lie algebra AP.

Proof Assume that u = v in the semigroup P. Without loss of generality we may assume that u = auib and v = avib for some a, b e {x, y}* and (ui, vi) e p. For every r e {x, y} relations (1) yield [Xr] = 0; consequently, LzxcJ = LfeX]cJ = [|_zcJX] and LzycJ = [LzcJy^] for every c e {x, y}*. This implies that in AP we have

LzuJ = LzauibJ = LLzauiJbJ = LLzuilTJbJ = LzuiVbJ = LzvitTbJ = LzavibJ = LzvJ,

where for every xii xi2 ■■■ xin e{x, y}* we put

Xi1 xi2 ••• xin := XinXin-1 ''' Xi1 , Xi1 Xi2 ''' Xin := Xi1 Xi2 ''' Xin ■

Moreover, (3') holds in AP.

Suppose that LzuJ = LzuJ in the Lie algebra AP. Then both LzuJ and LzuJ have the same normal form in AP. Since S1 is a GS basis in AP, we can reduce both LzuJ and LzuJ to the same normal form LzcJ for some c e {x, y}* using only relations (3'). This implies that u = c = u in P. □

By the corollary, if the semigroup P has undecidable word problem then so does the Lie algebra A P.

4.3 Composition-Diamond lemma for Lie algebras over commutative algebras

For a well-ordered set X = {xi |i e I}, consider the free Lie algebra Lie(X) c k(X) with the Lie bracket [u, u] = uu - uu.

Given a well-ordered set Y = {yj | j e J}, the free commutative monoid [Y] generated by Y is a linear basis of k[Y]. Regard

Liek[Y ](X) = k [Y ]® Lie( X)

as a Lie subalgebra of the free associative algebra k[Y](X) = k[Y] ® k(X) generated by X over the polynomial algebra k[Y], equipped with the Lie bracket [u,u] = uu -uu. Then NLSW(X) constitutes a k[Y]-basis of Liek[Y](X). Put [Y]X* = {/31ft e [Y], t e X*}. For u = 31 e [Y]X*, put uX = t and uY = 3.

Denote the deg-lex orderings on [Y ] and X * by >Y and >X .Define an ordering > on [Y]X* as follows: for u, u e [Y]X*, put

X X X X Y Y

u > u if (uX >X uX) or (uX = uX and uY >Y uY).

We can express every element f e Liek[Y](X) as f = X ai3[ui], where ai e k, Pi e [Y], and [ui] e NSLW(X).

Then f = ^ aipi [ui] = X gj (Y)[uj], where gj (Y) e k[Y] are polynomials in the k-algebra k[Y](X). The leading word f of f in k[Y](X) is of the form p1u1 with 31 e [Y] and u1 e ALSW(X). The polynomial f is called monic (or k-monic) if the

coefficient of f is equal to 1, that is, a\ = 1. The notion of k[Y]-monic polynomials is introduced similarly: a1 = 1 and j = 1.

Recall that every ALSW w admits a unique bracketing such that [w] is a NLSW.

Consider a monic subset S c Liek[Y](X). Given a non-associative word (u) on X with a fixed occurrence of some xi and s e S, call (u)xian S-word. Define |u| to be the s-length of (u)xi. Every S-word is of the form (asb) with a, b e X* and s e S. If asXb e ALSW(X) then we have the special bracketing [asXb]sx of asXb relative to sX. Refer to [asb]s = [asXb]sX|[sXas a special normal s-word (or special normal S-word).

An S-word (u) = (asb) is a normal s-word, denoted by |_u_|s, whenever (asb) = asXb. The following condition is sufficient.

(i) The s-length of (u) is 1, that is, (u) = s;

(ii) if LuJs is a normal S-word of s-length k and [v] e NLSW(X) satisfies |v| = l

then [v]LuJs whenever v > LuJs and LuJs[v] whenever v < LuJs are normal

S-words of s-length k + l.

Take two monic polynomials f and g in Liek[Y](X) and put L = lcm( fY, gY).

There are four kinds of compositions.

C\: Inclusion composition. If f X = agXb for some a, b e X*, then

Cx(f, g)w = fYf - ^Y [agb]g, where w = LfX = LagXb. f Y g

C2: Intersection composition. If f X = aa0 and gX = a0b with a, b, a0 = 1 then

C2< f, g)w = [ fb] f - L [ag]g, where w = LfXb = LagX. f Y gY

C3: External composition. If gcd(fY, gY) = 1 then for all a, b, c e X * satisfying w = LafXbgXc e Ta = {jt j e [Y], t e ALSW(X)}

we have

C3 < f, g)w = L[afbgXc] f - -^Y[afXbgc]g. f Y gY

C4: Multiplication composition. If fY = 1 then for every special normal f -word

[af b]f with a, b e X* we have

C4<f)w = [afXb][afb]f, where w = afXbafb.

Given a k-monic subset S c Liek[Y](X) and w e [Y]X*, which is not necessarily in TA, an element h e Liek[Y](X) is called trivial modulo (S, w) if h can be expressed as a k[Y]-linear combination of normal S-words with leading words smaller than w.

The set S is a Grobner-Shirshov basis in Liek[Y] (X) if all possible compositions in S are trivial.

Theorem 36 ([31], the CD-lemma for Lie algebras over commutative algebras) Consider a nonempty set S c Liek[Y](X) ofmonic polynomials and denote by Id(S) the ideal ofLiek[Y ] (X) generated by S. The following statements are equivalent:

(i) The set S is a Grobner-Shirshov basis in Liek[Y](X).

(ii) If f e Id(S) then f = asb e TA for some s e Sand a, b e [Y ] X *.

(iii) The set Irr (S) = {[u] | [u] e TN, u = asb, for s e S and a, b e [Y]X*} is a linear basis for Li ek[Y ](X |S) = (Liek[Y ](X))/Id(S).

Here Ta = {P1 | P e [Y], t e ALSW(X)} and Tn = {P[t] | P e [Y], [t] e NLSW( X)}.

Outline of the proof.

Take u, u e [Y] ALSW(X) and write u = uYuX and u = uYuX. Define the ALSW-lcm(u, u) (or lcm(u, u) for short) as w = wYwX = lcm(uY, uY) lcm(uX, uX), where

lcm(uX, uX) e {auXcuXb (an ALSW), a, b, c e X*;

uX = auXb, a, b e X*; uXb = auX, a, b e X*, deg(uXb)<deg(uX)+deg(uX)}.

Six lcm(u, u) are possible:

(i) (Y-trivial, X-trivial) (a trivial lcm(u, u));

(ii) (Y-trivial, X-inclusion);

(iii) (Y-trivial, X-intersection);

(iv) (Y-nontrivial, X-trivial);

(v) (Y-nontrivial, X-inclusion);

(vi) (Y-nontrivial, X-intersection).

In accordance with lcm(u, u), six general compositions are possible. Denote by [wX]ux ,vx the Shirshov double special bracketing of wX whenever wX is a X-trivial lcm(uX, uX), by [wX]ux and [wX]vx the Shirshov special bracketings of wX whenever wX is a lcm of X-inclusion or X-intersection respectively.

Define general Lie compositions for k-monic Lie polynomials f and g with f = u and g = u as

Y Y Y X Y Y Y X

(f, g)w = (lcm(uY, uY )/uY )[wX]uX vX |[u]i—> f - (lcm(uY, uY )/uY )[wX]uX^x |[v]^g,

Y Y Y X Y Y Y X

(f, g)w = (lcm(uY, uY )/uY )[wX]u|[u]—f - (lcm(uY, uY )/uY )[wX]u|[v]—g.

Lemma 9 ([31]) The general composition (f, g)w of k-monic Lie polynomials f and g with f = u and g = u, where w is a (Y-trivial, X-trivial) lcm(u, u), is 0 mod ({f, g}, w).

Proof By (XIX), we have

Y X Y X

(f, g)w = v 1 [afb[u ]d] - u [a[u ]bgd] = [afb[u]d] - [aubgd]

= [af b([u] - g)d] - [a([u] - f )bgd] = 0 mod ({f, g}, w).

The proof is complete. □

A Lie GS basis S c Liek[Y](X) c k[Y]<X) need not be an associative GS basis because the PBW-theorem is not valid for Lie algebras over a commutative algebra (Shirshov [201]). Therefore, the argument for Liek(X) above (see Sect. 4.2) fails for Liek[Y ](X).

Moreover, Shirshov's original proof of the CD-lemma fails because the singleton {s} e Li ek[Y] (X) is not a GS basis in general. The reason is that there exists a nontrivial composition (s, s)w of type (Y-nontrivial, X-trivial).

There is another obstacle. For Li ek (X), every s-word is a linear combination of normal s-words. For Liek[Y] (X) this is not the case. Hence, we must use a multiplication composition [uX] f such that f = u = uYuX.

Lemma 10 ([31]) If every multiplication composition [sX ]s, s e S, is trivial modulo (S, w = [uX]u), where u = s, then every S-word is a linear combination of normal S-words.

In our paper with Yongshan Chen [31], we use the following definition of triviality of a polynomial f modulo (S, w):

f = 0 mod (S,w) ^ f = 22atef [aXstbX],

where [ax[si X]bx] is the Shirshov special bracketing of the ALSW a^siXb^X with an ALSW stX.

The previous definition of triviality modulo (S, w) is equivalent to the usual definition by Lemma 11, which is key in the proof of the CD-lemma for Lie algebras over a commutative algebra.

Lemma 11 ([31]) Given a monic set S with trivial multiplication compositions, take a normal s-word (asb) and a special normal s-word [asb] with the same leading monomial w = asb. Then they are equal modulo (s, w).

Lemmas 10 and 11 imply

Lemma 12 ([31]) Given a monic set S with trivial multiplication compositions, every element of the ideal generated by S is a linear combination of special normal S-words.

On the other hand, (XVII) and (XIX) imply the following analogue of Lemma 1 for Liek[Y] (X).

Lemma 13 ([31]) Given two k-monic special normal S-words e\ [aiXsibiX] and e\ [a2Xs2b2X] with the same leading associative word w1, their difference is equal to [a(s1, s2)wb], where w = lcm(s~1, s2), w1 = awb, and [a(s1, s2)wb] = [w1]w|[w]^(s1,s2)w. Hence, if S is a GS basis then the previous special normal Swords are equal modulo (S, w1).

Now the claim (i)^(ii) of the CD-lemma for Liek[Y](X) follows.

For every Lie algebra L = LieK (X |S) over the commutative algebra K = k |Y | R],

U(L) = K(X|S(-)> = k[Y](X|S(-), RX),

where S(-) is just S with all commutators [uv]4 replaced with uv - vu, is the universal enveloping associative algebra of L.

A Lie algebra L over a commutative algebra K is called special whenever it embeds into its universal enveloping associative algebra. Otherwise it is called non-special.

Shirshov (1953) and Cartier (1958) gave classical examples of non-special Lie algebras over commutative algebras over GF(2), justified using ad hoc methods. Cohn (1963) suggested another non-special Lie algebra over a commutative algebra over a field of positive characteristic.

Example 1 (Shirshov (1953)) Take k = GF(2) and

K = k[yi, i = 0, 1, 2, 3|yoyi = yi (i = 0, 1, 2, 3), yty} = 0 (i, j = 0)].

Consider L = LieK(xt, 1 < i < 131Si, S2), where

Sx = {[X2X1] = X11, [X3X1] = X13, [X3X2] = X12,

[X5X3] = [X6X2] = [X8X1] = X10, [XiXj] = 0 (i > j)}; S2 = {y0 Xi = Xi (i = 1, 2,..., 13),

y1 X1 = X4, y1 X2 = X5, y1 X3 = X6, y1 X12 = X10, y2X1 = X5, y2X2 = X7, y2X3 = X8, y2X13 = X10, y3 X1 = X6, y3 X2 = X8, y3 X3 = x9, y3 X11 = x10.

y 1 Xk = 0 (k = 4, 5,..., 11, 13),

y2 Xi = 0 (t = 4, 5,..., 12),

y3 xi = 0 (l = 4, 5,..., 10, 12, 13)}.

Then L = LieK(X|S1, S2) = Liek[Y](X|S1, S2, RX) and

S = S1 U S2 U RX U {y1 X2 = y2X1, y1 X3 = y3X1, y2X3 = y3X2}

is a GS basis in Liek[Y] (X), which implies that x10 belongs to the linear basis of L by Theorem 36, that is, x10 = 0 in L.

On the other hand, the universal enveloping algebra of L has the presentation

Uk(L) = K(X|S(-), S2> = k[Y](X|S(-), S2, RX).

However, the GS completion (see Mikhalev and Zolotykh [170]) of s(-) U S2 U RX in k[Y](X> is

SC = S(-) U S2 U RX U {y1 X2 = y2X1, y1 X3 = y3X1, y2X3 = y3X2, X10 = 0}.

Thus, L is not special. Example 2 (Cartier [70]) Take k = GF(2) and

K = k[yi, y2, ys\yf = 0, i = 1, 2, 3].

Consider L = LieK (xij, 1 < i < j < 3\S), where

S = {[XiiXjj] = Xji (i > j), [XijXkl] = 0, y3X33 = y2X22 + yiXii}.

Then L is not special over K.

Proof The set S' = SU{yfXki = 0 (Vi, k, l)} U Si is a GS basis in Liek[Y](X), where

Si = {y3 X23 = yi Xi2, y3 Xi3 = y2Xi2, y2X23 = yi Xi3, y3 y2X22 = y3 yi Xii, y3 yi Xi2 = 0, y3 y2 Xi2 = 0, y3 y2 yi Xii = 0, y2 yi Xi3 = 0}.

Then, y2yixi2 e Irr(S') and so y2yixi2 = 0 in L. However, in

Uk(L) = K<X\S(-)> = k[Y]<X\S(-), yfxki = 0 (Vi, k, l))

we have

0 = yfXf3 = (y2X22 + yi Xii)2 = yfXf2 + yfXfi + y2 yi[X22, Xn] = y2 yi Xi2. Thus, L Uk (L).

Conjecture (Cohn [95]) Take the algebra K = k[yi, y2, y3\yf = 0, i = i, 2, 3] of truncated polynomials over a field k of characteristic p > 0. The algebra

L p = LieK (Xi, Xf, X3 \ y3 X3 = y2X2 + yi xi),

called Cohn's Lie algebra, is not special. In UK (Lp) we have

0 = (y3 X3 )p = (y2 X2)p + A p (y2 Xf, yi Xi) + (yi Xi)p = A p (y2 X2, yi Xi),

where Ap is a Jacobson-Zassenhaus Lie polynomial. Cohn conjectured that Ap (y2X2, yi x1) = 0inLp. To prove this, we must know a GS basis of Lp up to degree p in X. We found it for p = 2, 3, 5. For example, A2 = [y2x2, yix1] = y2yi[X2x1] and a GS basis of L2 up to degree 2 in X is

y3 X3 = y2 X2 + yi X1, y2Xj = 0 (1 < i, j < 3), y3 y2X2 = y3 yi Xi, y3 y2 yi Xi = 0, y2[X3X2] = yi[X3Xi], y3yi [X2Xi] = 0, y2yi [X3Xi] = 0.

Therefore, y2y1[X2X1] e Irr(SC).

Similar though much longer computations show that A3 = 0 in L3 and A5 = 0 in L5. Thus, we have

Theorem 37 ([31]) Cohn's Lie algebras L2, L3, and L5 are non-special.

Theorem 38 ([31]) Given a commutative k-algebra K = k[Y|R], if S is a Grobner-ShirshovbasisinLiek[Y ](X) such that every s e Sisk[Y ]-monicthen L = LieK (X |S) is special.

Corollary 18 ([31]) Every Lie K-algebra LK = LieK (X | f) with one monic defining relation f = 0 is special.

Theorem 39 ([31]) Suppose that S is a finite homogeneous subset ofLiek (X). Then the word problem ofLieK (X |S) is solvable for every finitely generated commutative k-algebra K.

Theorem 40 ([31]) Every finitely or countably generated Lie K-algebra embeds into a two-generated Lie K-algebra, where K is an arbitrary commutative k-algebra.

5 Grobner-Shirshov bases for fl-algebras and operads

5.1 CD-lemmas for ^-algebras

Some new CD-lemmas for ^-algebras have appeared: for associative conformal algebras [45] and n-conformal algebras [43], for the tensor product of free algebras [30], for metabelian Lie algebras [75], for associative ^-algebras [41], for color Lie superal-gebras and Lie p-superalgebras [165,166], for Lie superalgebras [167], for associative differential algebras [76], for associative Rota-Baxter algebras [32], for L-algebras [33], for dialgebras [38], for pre-Lie algebras [35], for semirings [40], for commutative integro-differential algebras [102], for difference-differential modules and difference-differential dimension polynomials [225], for ^-differential associative ^-algebras [185], for commutative associative Rota-Baxter algebras [186], for algebras with differential type operators [111].

Latyshev studied general versions of GS (or standard) bases [147,148]. Let us state the CD-lemma for pre-Lie algebras, see [35]. A non-associative algebra A is called a pre-Lie (or a right-symmetric) algebra if A satisfies the identity (x, y, z) = (x, z, y) for the associator (x, y, z) = (xy)z - x(yz). It is a Lie admissible algebra in the sense that A(-) = (A, [xy] = xy - yx) is a Lie algebra.

Take a well-ordered set X = {Xj |i e I}. Order X** by induction on the lengths of the words (u) and (v):

(i) When |((u)(v))| = 2 put (u) = Xi > (v) = Xj if and only if i > j.

(ii) When |((u)(v))| > 2 put (u) > (v) if and only if one of the following holds:

(a) |(u)| > |(v)|;

(b) if |(u)| = |(v)| with (u) = ({ui)(u2)) and (v) = ((v1)(v2)) then (u1) > (v1) or (u1) = (v1) and (u2) > (v2).

We now quote the definition of good words (see [198]) by induction on length:

(1) x is a good word for any x e X;

(2) a non-associative word ((v)(w)) is called a good word if

(a) both (v) and (w) are good words and

(b) if (v) = ((v1)(v2)) then (v2) < (w).

Denote (u) by [u] whenever (u) is a good word.

Denote by W the set of all good words in the alphabet X and by RS(X> the free right-symmetric algebra over a field k generated by X. Then W forms a linear basis of RS(X>, see [198]. Kozybaev et al. [141] proved that the deg-lex ordering on W is monomial.

Given a set S c RS(X> of monic polynomials and s e S, an S-word (u)s is called a normal S-word whenever (u)s = (asb) is a good word.

Take f, g e S, [w] e W, and a, b e X *. Then there are two kinds of compositions.

(i) If f = [agb] then (f, g)f = f - [agb] is called the inclusion composition.

(ii) If (f [w]) is not good then f ■ [w] is called the right multiplication composition.

Theorem 41 ([35], the CD-lemma for pre-Lie algebras) Consider a nonempty set S c RS(X> of monic polynomials and the ordering < defined above. The following statements are equivalent:

(i) The set S is a Grobner-Shirshov basis in RS(X>.

(ii) If f e Id(S) then f = [asb] for some s e S and a, b e X*, where [asb] is a normal S-word.

(iii) ThesetIrr(S) = {[u] e W|[u] = [asb] a, b e X*, s e S and [asb] is a normal S-word} is a linear basis of the algebra RS( X |S> = RS( X >/Id(S).

As an application, we have a GS basis for the universal enveloping pre-Lie algebra of a Lie algebra.

Theorem 42 ([35]) Consider a Lie algebra (L, [ ]) with a well-ordered linear basis X = {| i e I}. Write [eiej] = Xm a^jem with am e k. Denote am em by {eiej}. Denote by

U(L) = RS({e;}i| eiej - ejei = {e*ej}, i, j e I> the universal enveloping pre-Lie algebra of L. The set

S = {fij = eiej - ejei - {eiej}, i, j e I and i > j} is a Grobner-Shirshov basis in RS(X>.

Theorems 41 and 42 directly imply the following PBW theorem for Lie algebras and pre-Lie algebras.

Corollary 19 (Segal [198]) A Lie algebra L embeds into its universal enveloping pre-Lie algebra U(L) as a subalgebra of U(L)(-).

Recently the CD-lemmas mentioned above and other combinatorial methods yielded many applications: for groups of Novikov-Boone type [119-121] (see also [16,17,77,118], for Coxeter groups [58,150], for center-by-metabelian Lie algebras [214], for free metanilpotent Lie algebras, Lie algebras and associative algebras [112,168,215,216], for Poisson algebras [159], for quantum Lie algebras and related problems [132,135], for PBW-bases [131,134,158], for extensions of groups and associative algebras [73,74], for (color) Lie (p)-superalgebras [9,48,91,92,105107,169,227,228], for Hecke algebras and Specht modules [125], for representations of Ariki-Koike algebras [126], for the linear algebraic approach to GS bases [127], for HNN groups [87], for certain one-relator groups [88], for embeddings of algebras [39,83], for free partially commutative Lie algebras [84,181], for quantum groups of type Dn, E6, and G2 [174,189,221,222], for calculations of homogeneous GS bases [145], for Picard groups, Weyl groups, and Bruck-Reilly extensions of semigroups [7,128-130,139], for Akivis algebras and pre-Lie algebras [79], for free Sabinin algebras [93].

Following Dotsenko and Khoroshkin ([98], Proposition 3), linear bases for a symmetric operad and a shuffle operad are the same provided both of them are defined by the same generators and defining relations. It means that we need CD-lemma for shuffle operads only (and we define a GS basis for a symmetric operad as a GS basis of the corresponding shuffle operad).

We express the elements of the free shuffle operad using planar trees.

Put V = UVn, where V = [s(n)\i e In} is the set of n-ary operations.

Call a planar tree with n leaves decorated whenever the leaves are labeled by [n] = {1, 2, 3,..., n} for n e N and every vertex is labeled by an element of V.

For an arrow in a decorated tree, let its value be the minimal value of the leaves of the subtree grafted to its end. A decorated tree is called a tree monomial whenever for each its internal vertex the values of the arrows beginning from it increase from the left to the right.

Denote by Fy (n) the set of all tree monomials with n leaves and put T = Un>i#v (n). Given a = a(x\, ...,xn) e Fy (n) and j e Fy (m), define the shuffle composition a oi,a j as

a(x\, . .., xi-1, j(xi, xa(i + 1), •••, xa(i+m-1)X xa(i +m), ■■■, xa(m+n-1)),

which lies in Fy (n + m — 1), where 1 < i < n and the bijection

a :{i + 1,...,m + n — 1}^[i + 1,..., m + n — 1}

is an (m — 1, n — i)-shuffle, that is,

a(i + 1) < a(i + 2) < ••• < a(i + m — 1), a(i + m) < a(i + m + 1) < ••• < a(n + m — 1).

The set T is freely generated by V with the shuffle composition.

Denote by Fy = kT the k-linear space spanned by T. This space with the shuffle compositions oi,a is called the free shuffle operad.

Take a homogeneous subset S of Fy. For s e S, define an S-word u|s as before.

A well ordering > on T is called monomial (admissible) whenever

a > ft ^ u|a > u|ft for any u e T.

Assume that T is equipped with a monomial ordering. Then each S-word is a normal S-word.

For example, the following ordering > on T is monomial, see Proposition 5 of [98].

Every a = a(x1,..., xn) e Fy (n) has a unique expression

a = (path(1),..., path(n), [i1... in]),

where path(r) e V* for 1 < r < n is the unique path from the root to the leaf r and the permutation [i1... in] lists the labels of the leaves of the underlying tree in the order determined by the planar structure, from left to right. In this case define

wt(a) = (n, path(1),..., path(n), [i1... in]).

Assume that V is a well-ordered set and use the deg-lex ordering on V*. Take the order on the permutations in reverse lexicographic order: i > j if and only if i is less than j as numbers.

Now, given a, ft e T, define

a > ft & wt(a) > wt(ft) lexicographically.

An element of Fy is called homogeneous whenever all tree monomials occurring in this element with nonzero coefficients have the same arity degree (but not necessarily the same operation degree).

For two tree monomials a and ft, say that a is divisible by ft whenever there exists a subtree of the underlying tree of a for which the corresponding tree monomial a' is equal to a.

A tree monomial y is called a common multiple of two tree monomials a and ft whenever it is divisible by both a and ft .A common multiple y of two tree monomials a and ft is called a least common multiple and denoted by y = lcm(a, ft) whenever |a| + |ft| > |y|, where |5| = n for 5 e FV(n).

Take two monic homogeneous elements f and g of Fy. If f and g have a least common multiple w then (f, g)w = w^ - wg^g.

Theorem 43 ([98], the CD-lemma for shuffle operads) In the above notation, consider a nonempty set S c Fy of monic homogeneous elements and a monomial ordering < on T. The following statements are equivalent:

(i) The set S is a Grobner-Shirshov basis in Fy.

(ii) If f e Id(S) then f = u|s for some S-word u|s.

(iii) The set Irr(S) = [u e T\u = v\-sforall S-word v\s} is a k-linear basis of Fy / Id(S).

As applications, the authors of [98] calculate Grobner-Shirshov bases for some well-known operads: the operad Lie of Lie algebras, the operad As of associative algebras, and the operad PreLie of pre-Lie algebras.

Acknowledgments Authors thank Pavel Kolesnikov, Dima Piontkovskii, Yongshan Chen and Yu Li for valuable comments and help in writing some parts of the survey. They thank the referee for valuable comments and suggestions.

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