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Grobner-Shirshov bases and their calculation∗

L. A. Bokut†

School of Mathematical Sciences, South China Normal University

Guangzhou 510631, P. R. China

Sobolev Institute of Mathematics, Novosibirsk 630090, Russia

bokut@math.nsc.ru

Yuqun Chen

School of Mathematical Sciences, South China Normal University

Guangzhou 510631, P. R. China

yqchen@scnu.edu.cn

Abstract: In this survey, we formulate the Grobner-Shirshov bases1 theory for associativealgebras and Lie algebras. Some new Composition-Diamond lemmas and applications arementioned.

Key words: Grobner basis, Grobner-Shirshov basis, Composition-Diamond lemma, con-gruence, normal form, Braid group, free semigroup, Chinese monoid, plactic monoid,associative algebra, Lie algebra, Lyndon-Shirshov basis, Lyndon-Shirshov word, PBWtheorem, Ω-algebra, dialgebra, semiring, pre-Lie algebra, Rota-Baxter algebra, category,module.

AMS 2010 Subject Classification: 13P10, 16-xx, 16S15, 16S35, 16W99, 16Y60, 17-xx,17B01, 17B37, 17B66, 17D99, 18Axx, 18D50, 20F05, 20F36, 20Mxx, 20M18.

Notations∗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 Pro-gram on International Cooperation and Innovation, Department of Education, Guangdong Province(2012gjhz0007).

†Supported by RFBR 12-01-00329, LSS–3669.2010.1 and SB RAS Integration grant No.2009.97 (Rus-sia) and Federal Target Grant Scientific and educational personnel of innovation Russia for 2009-2013(government contract No.02.740.11.5191).

1Though Shirshov [181] 1962 was the first with the idea of “Grobner-Shirshov basis” for Lie andnon-commutative polynomial algebras, his paper became practically unknown outside Russia. In themeantime, Buchberger’s “Grobner basis” (Thesis 1965 [58], paper 1970 [59]) for (commutative) polyno-mials became very popular in science. As a result, the first author suggested the name “Grobner-Shirshovbasis” for non-commutative and non-associative polynomials. For (commutative) differential polynomialsan analogous, or better to say closed “basis” is called Ritt-Kolchin characteristic set due to Ritt [167]1950 and Kolchin [121] 1973, rediscovered by Wen-Tsun Wu [193] 1978.

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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: Poincare-Birkhoff-Witt theorem.

X∗: the free monoid generated by the set X .

[X ]: the free commutative monoid generated by the set 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 unit.

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: Grobner-Shirshov completion of S.

Id(S): the ideal generated by the set S.

s: the leading term of the polynomial s.

Irr(S): the set of all monomials which do not contain any subwords s, s ∈ 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 over a commutative algebra K generated by X .

1 Introduction

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

A.I. Shirshov (also A.I. Sirsov) in his pioneering work ([181], 1962) posted the followingfundamental question:

How to find a linear basis of any Lie algebra presented by generators and definingrelations?

He gave an infinite algorithm for a solution of this problem using a new notion ofcomposition (“s-polynomial” in a later terminology of Buchberger [58, 59]) of two Liepolynomials and a new notion of completion of any set of Lie polynomials (to add con-sequently “non-trivial” compositions, critical pair/completion (cpc-) algorithm in a laterKnuth-Bendix [120] and Buchberger [60, 61] terminology).

Shirshov’s algorithm is as follows. Let S be a subset of Lie polynomials in Lie(X) ofthe free algebra k〈X〉 on X over a field k (the algebra of non-commutative polynomialson X over k) . Let S ′ be a set by adding all non-trivial Lie compositions (“Lie s-polynomials”) of elements of S to S (S ⊆ S ′). A problem of triviality of a Lie polynomial

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modulo a finite (or recursive) set S is algorithmically solvable using Shirshov Lie reductionalgorithm from his previous paper [177], 1958. Then one gets in general an infinitesequence of Lie multi-compositions S ⊆ S ′ ⊆ S ′′ ⊆ · · · ⊆ S(n) ⊆ . . . . The union Sc of thesequence has the property that any Lie composition of elements of Sc is trivial relativeto Sc. It is what is now called a Lie GS basis. Then from a new “Composition-Diamondlemma2 for Lie algebras” (Lemma 3 in [181]) it follows that the set Irr(Sc) of all Sc-irreducible (or Sc-reduced) “basic” Lie monomials [u] in X is a linear basis of the Liealgebra Lie(X|S) generated by X with defining relations S. Here “basic” Lie monomialmeans a Lie monomial from a special linear basis of the free Lie algebra Lie(X) ⊂ k〈X〉,so called Lyndon-Shirshov (LS for short) basis (Shirshov [181] and Chen-Fox-Lyndon[64], see below). A LS monomial [u] is called Sc-irreducible (or Sc-reduced) if u, theassociative support of [u], does not contain word s for any s ∈ S, where s is the maximum(in deg-lex order) word of s as associative polynomial. To be more precise, Shirshov usedhis reduction algorithm at each step S, S ′, S ′′, . . . . Then one has a direct system of setsS → S ′ → S ′′ → . . . and Sc = lim−→S(n) is what is now called a “minimal GS basis”. Asthe result, Shirshov algorithm gives a solution of the above problem for Lie algebras.

Shirshov algorithm dealing with the word problem is an infinite algorithm as Knuth-Bendix algorithm [120], 1970 that deals with the identity problem for any variety ofuniversal algebras.3 An initial data for Knuth-Bendix algorithm is defining identities ofa variety. The result of the algorithm, if any, is a “Knuth-Bendix basis” of identities ofthe variety (not a GS basis of defining relations, say, a Lie algebra).

Shirshov algorithm gives linear basis and algorithmic decidability of the word problemfor one relator Lie algebras [181], (recursive) linear basis for Lie algebras with (finite)homogeneous defining relations [181], and linear basis for free product of Lie algebraswith known linear basis [180]. Also he proved the Freiheitssatz (the freeness theorem)for Lie algebras [181] (for any one-relator Lie algebra Lie(X|f), a subalgebra 〈X\xi〉,where xi is involved in f , is a free Lie algebra). Shirshov problem [181] for decidabilityof the word problem for Lie algebras was solved negatively in [15]. More generally, it wasproved [15] that some recursively presented Lie algebras with an unsolvable word problemcan be embedded into finitely presented Lie algebras (with the unsolvable word problem).It is a weak analogy of Higman embedding theorem for groups [101]. A problem [15]whether an analogy of Higman embedding theorem is valid for Lie algebras is still open.For associative algebras the similar problem [15] was solved positively by V.Y. Belyaev[6]. A simple example of a Lie algebra with the unsolvable word problem was given byG.P. Kukin [123].

Following Shirshov’s paper [181] a similar algorithm is valid also for associative algebras.It is since he fixed that Lie(X) is the subspace of the Lie polynomials in the free associativealgebra k〈X〉. Then to define a Lie composition 〈f, g〉w of two Lie polynomials relativeto an associative word w, he defines first the associative composition (non-commutative“s-polynomial”) (f, g)w = fb − ag, a, b ∈ X∗. Then he puts some brackets 〈f, g〉w =[fb]f − [ag]g following his special bracketing lemma in [177]. One may get Sc for anyS ⊂ 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 get that Irr(Sc) is a linear basis of associative

2The name “Composition-Diamond lemma” is a combination name of “Neuman Diamond Lemma”[148], “Shirshov Composition Lemma” [181] and “Bergman Diamond Lemma” [7].

3We use the usual algebraic terminology “the word problem”, “ the identity problem”, see, for example,O.G. Kharlampovich, M.V. Sapir [119].

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algebra k〈X|S〉 generated by X with defining relations S. All proofs are in the samemanner and much easier than in [181].

Even more, the cases of semigroups and groups presented by generators and definingrelations are just special cases of associative algebras via semigroup and group algebras.All in all, Shirshov algorithm gives linear bases and normal forms of elements of any Liealgebra, associative algebra, semigroup or group presented by generators and definingrelations! The algorithm does work in many cases (see below), as opposed to for examplean analogous algorithm for non-associative algebras.

The Grobner bases theory and Buchberger algorithm were initiated by B. Buchberger(Thesis [58] 1965, paper [59] 1970) for commutative-associative algebras. Buchbergeralgorithm is a finite algorithm for finitely generated commutative algebras. It is one ofthe most useful and famous algorithm in modern computer science.

The Shirshov paper [181] was in a spirit of A.G. Kurosh (1908-1972) program to studynon-associative (relatively) free algebras and free products of algebras initiated in Kuroshpaper [124], 1947. In this paper he proved non-associative analogies of Nielsen-Shreierand Kurosh theorems for groups. It took quite a few years to clarify the situation for Liealgebras in Shirshov’s papers [174], 1953 and [180], 1962 closely connected with his GSbases theory. It is important to note that quite unexpectedly the Kurosh program led tothe Shirshov’s theory of GS bases for Lie and associative algebras [181].

A step in the Kurosh program was done by his student A.I. Zhukov in his PhD Thesis[199], 1950 who algorithmically solved the word problem for non-associative algebras. In asense it was a beginning GS bases theory for non-associative algebras. The main differencewith the future Shirshov’s approach is that Zhukov did not use any linear order of non-associative monomials. Instead he chooses any monomial of maximal degree as a “leading”monomial of a polynomial. Also for non-associative algebras there is no “composition ofintersection” (“s-polynomial”). In this sense it cannot be a model for Lie and associativealgebras.4

A.I. Shirshov, also a student of Kurosh, defended his Candidate of Sciences Thesis[173] at Moscow University in 1953. His Thesis together with next followed paper [177],1958 may be viewed as a background of his later method of GS bases. He proved thefree subalgebra theorem for free Lie algebras (now known as Shirshov-Witt theorem,see also Witt [192], 1956) using the elimination process rediscovered by Lazard [128],1960. He used the elimination process later [177], 1958 as a general method to proveproperties of regular (LS) words, including an algorithm of (special) bracketing of a LSword (with a fix LS subword). The last algorithm is of some importance in his GSbases theory for Lie algebras (particular in a definition of the compositions of two Liepolynomials). Also Shirshov proved the free subalgebra theorem for (anti-) commutativenon-associative algebras [176], 1954. He used the paper lately in [182], 1962 for GS basestheory of (commutative, anti-commutative) non-associative algebras. Last but not theleast Shirshov found a series of bases of a free Lie algebra published 10 years later in thefirst issue of the new A.I. Malcev’s Journal “Algebra and Logic” [179], 1962.5

4After PhD Thesis, 1950, A.I. Zhukov moved to now Keldysh Institute of Applied Mathemat-ics (Moscow) and then to an atomic organization that is now known as Chelyabinsk-90 (Ural Re-gion) doing computational mathematics. S.K. Godunov in “Reminiscence about numerical schemes”,arxiv.org/pdf/0810.0649, 2008, mentioned his name in according with a creation of the famous Godunovnumerical method. So, A.I. Zhukov was a forerunner of two important computational methods!

5It must be pointed out that A.I. Malcev (1909-1967) inspired Shirshov’s works very much. Malcev

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LS basis is a particular case of the Shirshov or Hall-Shirshov series of bases (cf. Reutenauer[164], where this series was called “Hall series”). In the definition of his series, Shirshovused the Hall inductive procedure (see Ph. Hall [99], 1933, M. Hall [100], 1950): anon-associative monomial w = ((u)(v)) is a basic monomial if 1) (u), (v) are basic, 2)(u) > (v), 3) if (u) = ((u1)(u2)), then (u2) ≤ (v). But instead of an order by the de-gree function (Hall words), he used any linear order of non-associative monomials suchthat ((u)(v)) > (v). For example, in his Thesis [173], 1953 he used an order by the“content” of monomials (content of, say, a monomial (u) = ((x2x1)((x2x1)x1)) is a vector(x2, x2, x1, x1, x1)). Actually content u of (u) may be viewed as a commutative-associativeword that equals to u in the free commutative semigroup. Two contents are comparedlexicographically (a proper prefix of a content is greater than the content).

If one uses lexicographical order, (u) ≻ (v) if u ≻ v lexicographically, then one will getthe LS basis.6 For example, for alphabets x1, x2, x2 ≻ x1 one has Lyndon-Shirshov basicmonomials by induction:

x2, x1, [x2x1], [x2[x2x1]] = [x2x2x1], [[x2x1]x1] = [x2x1x1], [x2[x2x2x1]] = [x2x2x2x1],[x2[x2x1x1]] = [x2x2x1x1], [[x2x1x1]x1] = [x2x1x1x1], [[x2x1][x2x1x1]] = [x2x1x2x1x1] andso on.

They are exactly all Shirshov regular (LS) Lie monomials and their associative supportsare exactly all Shirshov regular words with 1-1 correspondence between two sets given bythe Shirshov elimination (bracketing) algorithm.

Let us remind that an elementary step of Shirshov elimination algorithm is to joint theminimal letter of a word to previous ones by the bracketing and to continue this processwith lexicographical order of the new alphabet. For example, let x2 ≻ x1. Then one hasinduction bracketing:

x2x1x2x1x1x2x1x1x1x1x2x1x1, [x2x1][x2x1x1][x2x1x1x1][x2x1x1],[x2x1][[x2x1x1][x2x1x1x1]][x2x1x1], [[x2x1][[x2x1x1][x2x1x1x1]]][x2x1x1],[[[x2x1][[x2x1x1][x2x1x1x1]]][x2x1x1]],x2x1x1x1x2x1x1x2x1x2x2x1, [x2x1x1x1][x2x1x1][x2x1]x2[x2x1],[x2x1x1x1][x2x1x1][x2x1][x2[x2x1]], x2x1x1x1 ≺ x2x1x1 ≺ x2x1 ≺ x2x2x1.

By the way, the second series of partial bracketing is an illustration of Shirshov factor-ization theorem [177], 1958 that any word is a non-decreasing product of LS words (it isoften called by a mistake, see [8], as “Lyndon theorem”.)

Shirshov’s [177] the special bracketing of any LS word with a fix LS subword is as follows.Let us give an example of special bracketing of LS word w = x2x2x1x1x2x1x1x1 with LS-

was an official opponent (referee) of his (second) Doctor of Sciences Dissertation, MSU, 1958. L.A. Bokutremembers this event at the Science Council Meeting, Chairman A.N. Kolmogorov, and Malcev’s words“Shirshov’s Dissertation is a brilliant one!”. Malcev and Shirshov worked together at the now SobolevInstitute of Mathematics (Novosibirsk) since 1959 until Malcev’ unexpected death at 1967, and have beenfriends despite the age difference. Malcev was head of the Algebra and Logic department (by the way,L.A. Bokut is a member of the departement since 1960) and Shirshov was the first deputy-director of theInstitute (director was S.L. Sobolev). In those years, Malcev was interested in the theory of algorithmsof mathematical logic and algorithmic problems of model theory. Because of that, Shirshov had anadditional motivation to work on algorithmic problems for Lie algebras. Both Maltsev and Kurosh weredelighted with the Shirshov’s results of [181]. Malcev successfully presented the paper for an award ofthe Presidium of the SB of the RAS (Sobolev and Malcev were only members of the Presidium fromInstitute of mathematics at that time).

6Lyndon-Shirshov basis for the alphabet x1, x2 is different from above Shirshov “content” basis startingwith monomials of degree 7.

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subword u = x2x2x1. The Shirshov standard bracketing is [w] = [x2[[[x2x1]x1][x2x1x1x1]]].The Shirshov special bracketing is [w]u = [[[u]x1][x2x1x1x1]]. In general, if w = aub thenthe Shirshov standard bracketing gives [w] = [a[uc]d], where b = cd. Now, c = c1 · · · ct,each ci is LS-word, c1 · · · ct in lex-order (Shirshov factorization theorem). Then onemust change bracketing of [uc]:

[w]u = [a[. . . [[u]c1]] . . . [ct]]d]

The main property of [w]u is that [w]u is a monic associative polynomial with the maximalmonomial w, i.e. [w]u = w.

Actually, Shirshov [181], 1962 needs to use “double” relative bracketing of a regularword with two separated LS subwords. Then he used implicitly the following property:any LS subword of c = c1 · · · ct as above is a subword of some ci, 1 ≤ i ≤ t.

The Shirshov [177], 1958 definition of regular monomials ia as follows: (w) = ((u)(v))is a regular monomial iff 1) w is a regular word, 2) (u), (v) are regular monomials (thenautomatically u ≻ v in lex order), 3) if (u) = ((u1)(u2)), then u2 v.

Once again, if one formally omits all Lie brackets in Shirshov’s paper [181], essentiallythe same algorithm and essentially the same CD-lemma (with the same but much simplerproof) give a linear basis of an associative algebra presented by generators and definingrelations. The differences are the following:

—No need to use LS monomials and LS words (since the set X∗ is a linear basis of thefree associative algebra k〈X〉);

—The definition of associative compositions for monic polynomials f, g,

(f, g)w = fb − ag, w = f b = ag, deg(w) < deg(f) + deg(g), or (f, g)w = f − agb, w =agb, w, a, b ∈ X∗, are much simpler than the definition of Lie compositions for monic Liepolynomials f, g,

〈f, g〉w = [fb]f − [ag]g, w = f b = ag, deg(w) < deg(f) + deg(g), or

〈f, g〉w = f − [agb]g, w = agb, w, a, b, f , g ∈ X∗,

where [fb]f , [ag]g, [agb]g are special Shirshov’s bracketing of LS words w with fixed LSsubwords f , g correspondingly.

—A definition of the elimination of a leading word s of an associative monic polynomials is straightforward (asb → a(rs)b, if s = s− rs, a, b ∈ X∗). But for Lie polynomials, itis much more involved and uses the special Shirshov’s bracketing.

—The main idea of the Shirshov’s proof may be formulated as follows. Let S be acomplete set of monic Lie polynomials. If w = a1s1b1 = a2s2b2, where w, , ai, bi ∈ X∗, wis a LS word, s1, s2 ∈ S, then Lie monomials [a1s1b1]s1 and [a2s2b2]s2 are equal modulothe less Lie monomials from the Id(S):

[a1s1b1]s1 = [a2s2b2]s2 +∑

i>2

αi[aisibi]si , where each si ∈ S, [aisibi]si = aisibi < w.

Actually Shirshov proved more general result: if (a1s1b1) = a1s1b1, (a2s2b2) = a2s2b2 andw = a1s1b1 = a2s2b2 then

(a1s1b1) = (a2s2b2) +∑

i>2

αi(aisibi), where each si ∈ S, (aisibi) = aisibi < w.

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Later we call a Lie polynomial (asb) as Lie normal S-word if (asb) = asb.

Exactly in this place he used the notion of composition and other notions and propertiesmentioned above.

For associative algebras (and for commutative-associative algebras) it is much easy toprove the analogy of this property: If S is a complete monic set in k〈X〉 (k[X ]) andw = a1s1b1 = a2s2b2, ai, bi ∈ X∗, s1, s2 ∈ S, then

a1s1b1 = a2s2b2 +∑

i>2

αiaisibi, where each si ∈ S, aisibi < w.

Summarizing we can say with confidence that the work Shirshov [181] implicitly containsthe CD-lemma for associative algebras as a simple exercise that does not require new ideas.L.A. Bokut can confirm that Shirshov clearly understood this and told to him, that “thecase of associative algebras is the same”. Explicitly the lemma was formulated in Bokut[16], 1976 (with reference to the Shirshov’s paper [181]), Bergman [7], 1978 and Mora[147], 1986.

Let us stress once again that CD-Lemma for associative algebras can be applied to anysemigroups P = sgp〈X|S〉, in particular to any group, through the semigroup algebrakP over a field k. The last algebra has the same generators and defining relations as P ,kP = k〈X|S〉. Any composition of “twonomials” u1 − v1, u2 − v2 is a twonomial u − v.As the result, Shirshov’s algorithm applying to a set of semigroups relations S gives rise acomplete set of semigroup relations Sc. The Sc-irreducible words in X is a set of normalforms of elements of P .

Before we go any farther, let us give some well known examples of presentations ofalgebras, groups and semigroup.

—Grassman algebra k〈X|x2i = 0, xixj + xjxi = 0, i > j〉.

—Clifford algebra k〈X|xixj+xjxi = aij , 1 ≤ i, j ≤ n〉, where (aij) is an n×n symmetricmatrix over k.

—Universal enveloping algebra U(L) = k〈X|xixj − xjxi =∑

αkijxk, i > j〉, where L is

a Lie algebra with a well-ordered linear basis X = xi|i ∈ I, [xixj ] =∑

αkijxk, i > j.

The PBW theorem provides a linear basis of U(L): xi1xi2 · · ·xin | i1 ≤ i2 ≤ · · · ≤in, it ∈ I, t = 1, . . . , n, n ≥ 0. A. Kandri-Rody and V. Weispfenning [108] inventedan important class of “algebras of polynomial type” that included Clifford and universalenveloping algebras. They created a GS bases theory for this class.

—More generally, let L = Lie(X|S) be a Lie algebra presented by generators anda Lie GS basis S (in deg-lex order). Then the universal enveloping algebra U(L) haspresentation U(L) = k〈X|S(−)〉 , where S(−) is S as the set of associative polynomials.PBW Theorem in a Shirshov form (see Theorem 2.24) says that S(−) is an associative GSbasis and a linear basis of U(L) consists of words u1u2 · · ·un, where ui are S-reduced LSwords, u1 u2 · · · un (in lex-order),7 see [49, 50].

—Free Lie algebras over a commutative algebra. Ph. Hall, M. Hall, A.I. Shirshov, R.Lyndon provide different linear bases of the algebras (Hall basis, Hall-Shirshov series ofbases, Lyndon-Shirshov basis). See also R-basis [10] and recent papers [29, 31].

7For any Lie polynomial s ∈ Lie(X) ⊂ k〈X〉, the maximal associative word s of s is a LS word andthe regular LS monomial [s] is the maximal LS monomial of s (both in deg-lex order).

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—Lie algebras presented by Chevalley generators and defining relations of types An, Bn,Cn, Dn, G2, F3, E6, E7, E8. The Serre theorem provides linear bases and multiplicationtables of the algebras (they are finite dimensional simple Lie algebras over a field k). Seealso [42, 43, 44].

—Finite Coxeter groups. Groups presented by Coxeter generators and defining rela-tions of types An, Bn, Dn, G2, F3, E6, E7, E8. Coxeter theorem provides normal forms andCayley tables (they are finite groups generated by involutions). See also [51].

—Coxeter groups (for any Coxeter matrix). J. Tits [184] algorithmically solved theword problem for any Coxeter group. Normal forms of elements are unknown.

—Affine Kac-Moody algebras [103]. Kac-Gabber theorem provides linear bases of thealgebras under “symmetrizability” condition on the Cartan matrix. See also [154, 155,156].

—Borcherds-Kac-Moody algebras [54, 55, 56, 103].

—Quantum enveloping algebras (V.G. Drinfeld, M. Jimbo). Lusztig theorem [132]provides linear “canonical bases” of the algebras. Different approaches were given byC.M. Ringel [165, 166] and V.K. Kharchenko [114, 115, 116, 117, 118]. See also [48, 76].

—Koszul algebras. Quadratic algebras having a basis of “standard” monomials, calledPBW-algebras, are always Koszul (S. Priddy [159]), but not conversely. In another ter-minology PBW-algebras are algebras with quadratic GS bases. See [153].

—Elliptic algebras. Elliptic algebras (B. Feigin, A. Odesskii) are associative algebraspresented by n generators and n(n− 1)/2 homogeneous quadratic relations for which thedimensions of graded components are the same as for polynomial algebra in n variables.The first example of such algebras was Sklyanin algebra (1982) generated by x1, x2, x3

with 3 defining relations [x3, x2] = x21, [x2, x1] = x2

3, [x1, x3] = x22. See [151].

—Cluster algebras. A cluster algebra (S. Fomin, A. Zelevinski) is a subalgebra of a fieldof rational functions Q(x1, . . . , xn) generated by “cluster” elements. Cluster algebras wereinvented in order to better understand the Lustig canonical bases of quantum envelopingalgebras.

—Hecke (Iwahory-Hecke) algebras. A deep connection of Hecke algebras (of type An)and braid groups (and invariants of links) were found in V.F.R. Jones [102].

—Artin braid group Bn. Markov-Artin theorem provides normal form and a semi-directstructure of the group in Burau generators. Other normal forms of Bn were obtained byGarside, Birman-Ko-Lee and Adjan-Thurston. See also [17, 18].

—Artin-Tits groups. Normal forms are known in the spherical case, see E. Brieskorn,K. Saito [57].

—Novikov-Boon type of groups [53, 87, 104, 105, 106, 107, 149] with the unsolvableword or conjugacy problem. See also [11, 12, 69].

—Adjan-Rabin constructions of groups with unsolvable isomorphism and Markov prop-erties. See also [19, 20].

—Markov-Post semigroups with unsolvable word problem. See also [138].

—Markov construction of semigroups with unsolvable isomorphism and Markov prop-erties.

—Plactic monoids. A theorem due to Richardson, Shensted, Knuth provides normalform of elements of the monoids. See also [25].

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—Multiplicative semigroups of some power series rings with “quadratic defining re-lations” (of the type k〈〈x, y, z, t|xy = zt〉〉) that are embeddable into groups but notembeddable, in general, into division algebras (an answer to Malcev problem). Relativenormal forms of the semigroups were found in [13, 14] between line of what was latercalled “relative GS basis” [52].

Up to date, GS bases method invented in particular for the following classes of linearuniversal algebras as well as for operads, categories and semirings. Unless otherwisestated, all linear algebras are considered over a field k. Following the terminology ofHiggins-Kurosh, by an ((differential) associative) Ω-algebra we mean a linear space (an(differential) associative algebra) with a set of multi-linear operations Ω:

—Associative algebras,

—Associative algebras over a commutative algebra,

—Associative Γ-algebras, where Γ is a group,

—Lie algebras,

—Lie algebras over a commutative algebra ,

—Lie p-algebras over k, chk = p,

—Lie superalgebras,

—Metabelian Lie algebras,

—Quiver algebras,

—Associative (n−)conformal algebras over k, chk = 0,

—Dialgebras,

—Pre-Lie (Vinberg-Koszul-Gerstenhaber) algebras,

—Associative Rota-Baxter algebras over k, chk = 0,

—Associative Ω-algebras,

—Associative differential Ω-algebras,

—Ω-algebras,

—Differential Rota-Baxter commutative associative algebras,

—Tensor products of associative algebras,

—Semirings,

—Modules over an associative algebra,

—Small categories,

—Non-associative algebras,

—Non-associative algebras over a commutative algebra,

—Commutative non-associative algebras,

—Anti-commutative non-associative algebras,

—Symmetric operads.

At the heart of the GS method for a class of linear algebras, there is a CD-lemma fora free object of the the class. For above cases, free objects are a free associative algebrak〈X〉, a “double free” associative k[Y ]-algebra k[Y ]〈X〉, a free Lie algebra Lie(X), a“double free” Lie k[Y ]-algebra Liek[Y ](X). For tensor product of two associative algebras,

9

one needs to use tensor product of two free algebras, k〈X〉 ⊗ k〈Y 〉. Any semiring can beviewed as a “double semigroup” with two associative “products” · and . So, CD-lemmafor semirings is CD-lemma for a “semiring algebra” kRig(X) of the free semiring Rig(X).CD-lemma for modules is CD-lemma for a “double free” module Modk〈Y 〉(X), a freemodule over a free associative algebra. CD-lemma for small categories is CD-lemma for a“free partial k-algebra” kC〈X〉 generated by an oriented graph X (a sequence z1z2 · · · zn,where zi ∈ X , is a partial word in X , iff it is a path; a partial polynomial is a linearcombination of partial words with the same sources and targets). All CD-lemmas areessentially the same wording. Let V be a class of linear universal algebras, V(X) a freealgebra from V, N(X) a well-ordered term basis of V(X). A subset S ∈ V(X) is called aGS basis, if any “composition” of elements of S is “trivial” (goes to zero by “elimination”of leading terms s, s ∈ S). Then the following conditions are equivalent:

(i) S is a GS basis.

(ii) If f ∈ Id(S), then the leading term f contains a sub-term s for some s ∈ S.

(iii) A set of S-irreducible terms is a linear basis of the V-algebra V〈X|S〉, generatedby X with defining relations S.

In some cases ((n−) conformal algebras, dialgebras), conditions (i) and (ii) are notequivalent. To be more precise in that cases (i)⇒ (ii)⇔ (iii).

Typical compositions are compositions of intersection and inclusion. Shirshov [181, 182]avoided the inclusion composition. Instead, he suggested that a GS basis must be reduced(leading words do not contain each other as subwords). In some cases, one needs todefine new compositions, for example, the composition of left (right) multiplication. Also,sometimes one needs to combine all these compositions.

In some cases (Lie algebras, (n−) conformal algebras) the “leading” term f of a poly-nomial f ∈ V(X) does not belong to V(X). For Lie algebras, one has f ∈ k〈X〉, for((n)-) conformal algebras f belongs to an associative Ω-semigroup.

For almost all CD-lemmas one needs to use a new notion of “normal S-term”. A term(asb) in X, Ω, s ∈ S, with only one occurrence of s is called a normal S-term if(asb) = (a(s)b). For any S ⊂ k〈X〉, each S-word (i.e. S-term) is a normal S-word. Forany S ⊂ Lie(X), each Lie S-monomial (Lie S-term) is a linear combination of normal LieS-terms (Shirshov [181]).

One of the two key lemmas asserts that if S is complete under compositions of multi-plications then any elements of the ideal generated by S is a linear combination of normalS-terms. Another key lemma says that if S is a GS basis and the leading words of twonormal S-terms are the same, then these terms are the same modulo lower normal S-terms. As we mentioned above these results were proved in Shirshov [181] for Lie(X)(there are no compositions of multiplications for Lie and associative algebras).

The survey is in line to our with P. Kolesnikov, Y. Fong and W.-F. Ke surveys [21, 22,36, 40, 45, 46].

The paper is organized as follows. Chapter 2 is for associative algebras, chapter 3 forsemigroups and groups, chapter 4 for Lie algebras, and chapter 5 for Ω-algebras and someapplications.

In conclusion, we give some information about the works of Shirshov. See more on thisin the book [183]. A.I. Shirshov (1921-1981) was a famous Russian mathematician. His

10

name associated with notions and results on GS bases, the CD-lemma, the Shirshov-Witttheorem, the Lazard-Shirshov elimination, the Shirshov height theorem, LS words, LSbasis (in free Lie algebras), Hall-Shirshov series of bases, the Cohn-Shirshov theorem, theShirshov theorem on the Kurosh problem, the Shirshov factorization theorem. Shirshov’sideas were used by his student Efim Zelmanov for a solution of the restricted Burnsideproblem.

We thank P.S. Kolesnikov, Yongshan Chen and Yu Li for valuable comments and helpin writing some parts of the survey.

1.1 Degression to the history of Lyndon-Shirshov basis and Lyndon-

Shirshov words

Standard (it is the same as Shirshov regular) words are defined by Lyndon in [133], 1954.

Unfortunately, papers Lyndon [133] and Chen-Fox-Lyndon [64] were practically un-known before 1983. As a result at that time almost all (but four, see below) authors callthe basis and words by Shirshov regular basis and words, cf. for example, [4, 5, 86, 162,186, 197].

To the best of our knowledge, no one of authors mentioned Lyndon’s paper [133] as asource of “Lyndon words” before 1983(!).

As for papers Chen-Fox-Lyndon [64] and Shirshov [177], they were mentioned (before1983) in the following papers as a source of “LS basis” and “LS words”:

—M.P. Schutzenberger, S. Sherman [170]

—M.P. Schutzenberger [171]

—G. Viennot [191]

—J. Michel [139, 140]

In the paper [170], the authors thank P.M. Cohn who let them know Shirshov paper[177]. Also they formulate the Shirshov factorization theorem [177]. They mention [64,177] as a source of “LS words”. In the paper [171], M.P. Schutzenberger also mentionsthe Shirshov factorization theorem but in this case he attributes it to both [64] and [177].Actually, he cites [64] by a mistake, there is no this result in the paper, see Berstel-Perrin[8].8

Starting with the book [130], some authors called the words and the basis as “Lyndonwords” and “Lyndon basis”, see, for example, [164].

2 Grobner-Shirshov bases for associative algebras

In this chapter, we give a proof of Shirshov CD-lemma for associative algebras and for-mulate Buchberger theorem for commutative algebras. Also, we give Eisenbud-Peeva-

8From [8]: “A famous theorem concerning Lyndon words asserts that any word w can be factorizedin 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-Fox-Lyndon, followingthe paper of Schutzenberger [171] in which it appears as an example of factorization of free monoids.Actually, as pointed out to one of us by D. Knuth in 2004, the reference [64] does not contain explicitlythis statement.”Despite of this, the Shirshov factorization theorem is called “Chen-Fox-Lyndon theorem” in Wikipedia

and Enciclopedia of Mathematics, Springer, on line.

11

Sturmfels lifting theorem, CD-lemmas for modules (following S.-J. Kang and K.-H. Lee[110] and E.S. Chibrikov [80]), PBW theorem and PBW theorem in a Shirshov form,CD-lemma for categories, CD-lemma for associative algebras over commutative algebrasand Rosso-Yamane theorem for Uq(An).

2.1 Composition-Diamond lemma for associative algebras

Let k be a field, k〈X〉 the free associative algebra over k generated by X and X∗ the freemonoid generated by X , where the empty word is the identity which is denoted by 1. Fora word w ∈ X∗, we denote the length (degree) of w by |w|. Let X∗ be a well-ordered set.Let f ∈ k〈X〉 with the leading word f and f = αf − rf where 0 6= α ∈ k. We call fmonic if α = 1.

A well order > on X∗ is called monomial if it is compatible with the multiplication ofwords, that is, for u, v ∈ X∗, we have

u > v ⇒ w1uw2 > w1vw2, for all w1, w2 ∈ X∗.

A standard example of monomial order on X∗ is the deg-lex order to compare two wordsfirst by degree and then lexicographically, where X is a well-ordered set.

Let f and g be two monic polynomials in k〈X〉 and < a monomial order on X∗. Then,there are two kinds of compositions:

(i) If w is a word such that w = f b = ag for some a, b ∈ X∗ with |f |+ |g| > |w|, thenthe polynomial (f, g)w = fb − ag is called the intersection composition of f and g withrespect to w.

(ii) If w = f = agb for some a, b ∈ X∗, then the polynomial (f, g)w = f − agb is calledthe inclusion composition of f and g with respect to w.

In the composition (f, g)w, w is called an ambiguity.

Let S ⊂ k〈X〉 such that every s ∈ S is monic. Let h ∈ k〈X〉 and w ∈ X∗. Then h iscalled trivial modulo (S, w), denoted by h ≡ 0 mod(S, w), if h =

∑αiaisibi, where each

αi ∈ k, ai, bi ∈ X∗, si ∈ S and aisibi < w.

Elements asb, a, b ∈ X∗, s ∈ S are called S-words.

A monic set S ⊂ k〈X〉 is called a GS basis in k〈X〉 with respect to the monomial order< if any composition of polynomials in S is trivial modulo S and corresponding w.

A set S is called a minimal GS basis in k〈X〉 if S is a GS basis in k〈X〉 such that thereis no inclusion compositions in S, i.e. for any f, g ∈ S with f 6= g, one has f 6= agb forany a, b ∈ X∗.

DenoteIrr(S) = u ∈ X∗|u 6= asb, s ∈ S, a, b ∈ X∗.

A GS basis S in k〈X〉 is reduced if for any s ∈ S, supp(s) ⊆ Irr(S \ s), wheresupp(s) = u1, u2, . . . , un if s =

∑n

i=1 αiui each 0 6= αi ∈ k, ui ∈ X∗.

The following lemma plays a key role for proving CD-lemma for associative algebras.

Lemma 2.1 If S is a GS basis in k〈X〉 and w = a1s1b1 = a2s2b2, where a1, b1, a2, b2 ∈X∗, s1, s2 ∈ S, then a1s1b1 ≡ a2s2b2 mod(S, w).

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Proof. There are three cases to consider.

Case 1. Suppose that subwords s1 and s2 of w are disjoint, say, |a2| ≥ |a1|+ |s1|. Then,a2 = a1s1c and b1 = cs2b2 for some c ∈ X∗, and so, w1 = a1s1cs2b2. Now,

a1s1b1 − a2s2b2 = a1s1cs2b2 − a1s1cs2b2

= a1s1c(s2 − s2)b2 + a1(s1 − s1)cs2b2.

Since s2 − s2 < s2 and s1 − s1 < s1, we conclude that

a1s1b1 − a2s2b2 =∑

i

αiuis1vi +∑

j

βjujs2vj

for some αi, βj ∈ k, S-words uis1vi and ujs2vj such that uis1vi, uj s2vj < w.

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

a1s1b1 − a2s2b2 = a1s1b1 − a1as2bb1 = a1s1 − as2bb1 = a1(s1, s2)s1b1.

By the triviality of compositions it follows that a1s1b1 ≡ a2s2b2 mod(S, w).

Case 3. s1 and s2 have a nonempty intersection as a subword of w. We may assumethat a2 = a1a, b1 = bb2, w = s1b = as2, |w| < |s1|+ |s2|. Then, similar to the Case 2, wehave a1s1b1 ≡ a2s2b2 mod(S, w).

Lemma 2.2 Let S ⊂ k〈X〉 be a subset of monic polynomials. Then for any f ∈ k〈X〉,

f =∑

ui≤f

αiui +∑

ajsjbj≤f

βjajsjbj

where each αi, βj ∈ k, ui ∈ Irr(S) and ajsjbj an S-word. So, Irr(S) is a set of lineargenerators of the algebra f ∈ k〈X|S〉.

Proof. The result follows by induction on f .

Theorem 2.3 (CD-lemma for associative algebras) Let < be a monomial order onX∗, S ⊂ k〈X〉 a monic set and Id(S) the ideal of k〈X〉 generated by S. Then the followingstatements are equivalent.

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

(ii) f ∈ Id(S)⇒ f = asb for some s ∈ S and a, b ∈ X∗.

(iii) Irr(S) = u ∈ X∗|u 6= asb, s ∈ S, a, b ∈ X∗ is a linear basis of the algebra k〈X|S〉.

Proof. (i) ⇒ (ii). Let S be a GS basis and 0 6= f ∈ Id(S). Then, we have f =∑n

i=1 αiaisibi where each αi ∈ k, ai, bi ∈ X∗, si ∈ S. Let

wi = aisibi, w1 = w2 = · · · = wl > wl+1 ≥ · · · .

We will use the induction on l and w1 to prove that f = asb for some s ∈ S and a, b ∈ X∗.

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If l = 1, then f = a1s1b1 = a1s1b1 and hence the result holds. Assume that l ≥2. Then, we have w1 = a1s1b1 = a2s2b2. It follows from Lemma 2.1 that a1s1b1 ≡a2s2b2 mod(S, w1). If α1 + α2 6= 0 or l > 2, the result follows from the induction on l.For the case α1 + α2 = 0 and l = 2, we use the induction on w1. This shows (ii).

(ii) ⇒ (iii). By Lemma 2.2, Irr(S) generates k〈X|S〉 as a linear space. Suppose that∑i

αiui = 0 in k〈X|S〉, where 0 6= αi ∈ k, ui ∈ Irr(S). It means that∑i

αiui ∈ Id(S) in

k〈X〉. Then∑i

αiui = uj ∈ Irr(S) for some j which contradicts (ii).

(iii) ⇒ (i). For any f, g ∈ S , by using Lemma 2.2 and (iii), we have (f, g)w ≡0 mod(S, w). Therefore, S is a GS basis.

Shirshov algorithm If a subset S ⊂ k〈X〉 is not a GS basis then one can add allnontrivial compositions of polynomials of S to S. Continuing this process repeatedly, wefinally obtain a GS basis Sc that contains S. Such a process is called Shirshov’s algorithm.Sc is called GS completion of S.

The following theorem gives a linear basis of the ideal Id(S) if S ⊂ k〈X〉 is a GS basis.

Theorem 2.4 Suppose that S ⊂ k〈X〉 is a Grobner-Shirshov basis. Then for any u ∈X∗ \ Irr(S), by Lemma 2.2, there exists u ∈ kIrr(S) with u < u (if u 6= 0) such thatu− u ∈ Id(S) and the set u− u|u ∈ X∗ \ Irr(S) is a linear basis of the ideal Id(S) ofk〈X〉.

Proof. Suppose that 0 6= f ∈ Id(S). Then by CD-lemma for associative algebras,f = a1s1b1 = u1 for some s1 ∈ S and a1, b1 ∈ X∗ which implies that f = u1 ∈ X∗\Irr(S).Let f1 = f−α1(u1−u1), where α1 is the coefficient of the leading term of f and u1 < u1 oru1 = 0. Then f1 ∈ Id(S) and f1 < f . By induction on f , the set u− u|u ∈ X∗ \ Irr(S)generates Id(S) as a linear space. It is clear that the set u− u|u ∈ X∗ \ Irr(S) is linearindependent.

Theorem 2.5 Let I be an ideal of k〈X〉 and > a monomial order on X∗. Then thereexists uniquely the reduced Grobner-Shirshov basis S for I.

2.2 Grobner bases for commutative algebras and their lifting to

Grobner-Shirshov bases

Let k[X ] be free commutative associative algebra. Let< be a well order onX = xi|i ∈ I.Then [X ] = xi1 . . . xit |i1 ≤ · · · ≤ it, i1, . . . , it ∈ I, t ≥ 0 is a linear basis of k[X ].

Let < be a monomial order on [X ]. Let f and g be two monic polynomials in k[X ],w = lcm(f , g) = fa = gb for some a, b ∈ [X ] with |f |+ |g| > |w| (so, f , g are not coprimein [X ]). Then the polynomial (f, g)w = fa− gb is called the s-polynomial of f and g.

A monic subset S ⊆ k[X ] is called a Grobner basis with respect to the monomial order< if any s-polynomial of polynomials in S is trivial modulo S.

By a similar proof of the CD-lemma for associative algebras, we have the followingtheorem which is due to B. Buchberger.

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Theorem 2.6 (Buchberger Theorem) Let < be a monomial order on [X ], S ⊂ k[X ] amonic set and Id(S) the ideal of k[X ] generated by S. Then the following statements areequivalent.

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

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

(iii) Irr(S) = u ∈ [X ]|u 6= sa, s ∈ S, a ∈ [X ] is a linear basis of the algebra k[X|S] =k[X ]/Id(S).

Theorem 2.7 Let I be an ideal of k[X ] and < a monomial order on [X ]. Then thereexists uniquely the reduced Grobner basis S for I. Moreover, S is finite if X is a finiteset.

In [89], a GS basis in k〈X〉 is constructed by lifting a commutative Grobner basis ink[X ] and adding all commutators. Let X = x1, x2, . . . , xn and

S1 = hij = xixj − xjxi| i > j ⊂ k〈X〉.

Then, consider the natural map γ : k〈X〉 → k[X ] which maps xi to xi and the lexico-graphic splitting of γ, which is defined as the k-linear map

δ : k[X ]→ k〈X〉, xi1xi2 · · ·xir 7→ xi1xi2 · · ·xir if i1 ≤ i2 · · · ≤ ir.

For any u ∈ [X ], we present u = xl11 x

l22 · · ·x

lnn , where li ≥ 0.

We use any monomial order on [X ].

Following [89], we define an order on X∗ using the order x1 < x2 < · · · < xn as follows:for any u, v ∈ X∗,

u > v if γ(u) > γ(v) in [X ] or (γ(u) = γ(v) and u >lex v).

It is easy to check that this order is monomial on X∗ and δ(s) = δ(s) for any s ∈ k[X ].Moreover, for any v ∈ γ−1(u), v ≥ δ(u).

Let L be any ideal of k[X ] generated by some monomials. If m = xi1xi2 · · ·xir ∈ L, i1 ≤i2 · · · ≤ ir, denote by UL(m) the set of all the monomials u ∈ [xi1+1, · · · , xir−1] such thatneither u m

xi1nor u m

xirlies in L.

Theorem 2.8 ([89]) Let the orders on [X ] and X∗ be defined as above. If S is a minimalGrobner basis in k[X ], then S ′ = δ(us)|s ∈ S, u ∈ UL(s) ∪ S1 is a minimal Grobner-Shirshov basis in k〈X〉, where L is the monomial ideal of k[X ] generated by S.

Proof. One can check that all the possible compositions of the elements of S ′ are trivialmodulo S ′ and corresponding w. Thus S ′ is a GS basis in k〈X〉.

Let L′ be the monomial ideal of k〈X〉 generated by S ′. By Buchberger Theorem, itremains to show that δ(us) is a minimal generator of L′ if and only if u ∈ UL(s). For the“if” direction it is obvious by the definition of UL(s). For the “only if” direction, suppose

15

that δ(us) is a minimal generator of L′, s = xi1xi2 · · ·xir and u contains the variable xj .Then j > i1, since else, taking j minimal, we have δ(us) = xjδ(

uxjs). Similarly j < ir.

Thus u ∈ [xi1+1, · · · , xir−1]. This implies δ(us) = x1δ(us

xi1) = δ(u s

xir)xir . Therefore

neither δ(u sxi1

) nor δ(u sxir

) lies in L′ and hence neither u sxi1

nor u sxir

lies in L.

Recall that for a prime number p the Gauss order on the natural numbers is described

by s ≤p t if

(ts

)6≡ 0 (mod p). Let ≤0=≤ be the usual order on the natural numbers.

A monomial ideal L of k[X ] is called p-Borel-fixed if it satisfies the following condition:For each monomial generator m of L, if m is divisible by xt

j but no higher power of xj ,then (xi/xj)

sm ∈ L for all i < j and s ≤p t.

Thus we have the following Eisenbud-Peeva-Sturmfels lifting theorem.

Theorem 2.9 ([89]) Let I be an ideal of k[X ], L = Id(f , f ∈ I) and J = γ−1(I) ⊂ k〈X〉.

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

(ii) If L is p-Borel-fixed for some p, then J has a finite Grobner-Shirshov basis.

Proof. Suppose that L is p-Borel-fixed for some p. Let m = xi1xi2 · · ·xir be any generatorof L , where xi1 ≤ xi2 ≤ · · · ≤ xir , and let xt

irbe the highest power of xir dividing m.

Since t ≤p t we have xtlm/xt

ir∈ L for each l < ir. This implies xt

lm/xir ∈ L for l < ir, andhence every monomial in UL(m) satisfies degxl

(u) < t for i1 < l < ir. Thus UL(m) is afinite set and the result follows from Theorem 2.8. In particular, if p = 0 then UL(m) = 1.

Note that if the field k is infinite, then after a generic change of variables, L is p-Borel-fixed in characteristic p ≥ 0. Then Theorems 2.8 and 2.9 imply the following result:

Corollary 2.10 ([89]) Let k be an infinite field and I ⊂ k[X ] be an ideal. After a generallinear change of variables, the ideal γ−1(I) in k〈X〉 has a finite Grobner-Shirshov basis.

2.3 Composition-Diamond lemma for modules

Let S, T ⊂ k〈X〉 and f, g ∈ k〈X〉. In Kang-Lee paper [109], the composition of f and gis defined as follows:

Definition 2.11 ([109, 113])

(a) If there exist a, b ∈ X∗ such that w = fa = bg with |w| < |f | + |g|, then thecomposition of intersection is defined to be (f, g)w = fa− bg.

(b) If there exist a, b ∈ X∗ such that w = afb = g, then the composition of inclusionis defined to be (f, g)w = afb− g.

(c) A composition (f, g)w is called right-justified if w = f = ag for some a ∈ X∗.

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If f − g =∑

αiaisibi +∑

βjcjtj , where αi, βj ∈ k, ai, bi, cj ∈ X∗, si ∈ S, tj ∈ Twith aisibi < w and cjtj < w for each i and j, then we say f − g is trivial with respect toS and T and denote it by f ≡ g mod(S, T ;w).

Definition 2.12 ([109, 110]) A pair (S, T ) of subsets of monic elements of k〈X〉 is calleda GS pair if S is closed under the composition, T is closed under the right-justified com-position with respect to S, and for any f ∈ S, g ∈ T and w ∈ X∗ such that if (f, g)w isdefined, we have (f, g)w ≡ 0 mod (S, T ;w). In this case, we say that (S, T ) is a GS pairfor the A-module AM =A k〈X〉/(k〈X〉T + Id(S)), where A = k〈X|S〉.

Theorem 2.13 ([109, 110] CD-lemma for cyclic modules) Let (S, T ) be a pair of subsetsof monic elements in k〈X〉, A = k〈X|S〉 be the associative algebra defined by S, and

AM =A k〈X〉/(k〈X〉T + Id(S)) be the left cyclic module defined by (S, T ). Suppose that(S, T ) is a Grobner-Shirshov pair for the A-module AM and p ∈ k〈X〉T + Id(S). Thenp = asb or p = ct, where a, b, c ∈ X∗, s ∈ S, t ∈ T .

Let X, Y be sets and modk〈X〉〈Y 〉 a free left k〈X〉-module with the basis Y . Thenmodk〈X〉〈Y 〉 = ⊕y∈Y k〈X〉y is called a “double-free” module. We now define the GSbasis in modk〈X〉〈Y 〉. Suppose that < is a monomial order on X∗, < a well order on Yand X∗Y = uy|u ∈ X∗, y ∈ Y . We define an order < on X∗Y as follows: for anyw1 = u1y1, w2 = u2y2 ∈ X∗Y ,

w1 < w2 ⇔ u1 < u2 or u1 = u2, y1 < y2

Let S ⊂ modk〈X〉〈Y 〉 with each s ∈ S monic. Then we define the composition in Sonly the inclusion composition which means f = ag for some a ∈ X∗, where f, g ∈ S.If (f, g)f = f − ag =

∑αiaisi, where αi ∈ k, ai ∈ X∗, si ∈ S and aisi < f , then this

composition is called trivial modulo (S, f).

Applications of Theorem 2.13 were given in [111, 112, 113].

Theorem 2.14 ([80], see also [70] CD-lemma for modules) Let S ⊂ modk〈X〉〈Y 〉 be anon-empty set with each s ∈ S monic and < the order on X∗Y as before. Then thefollowing statements are equivalent:

(i) S is a Grobner-Shirshov basis in modk〈X〉〈Y 〉.

(ii) If 0 6= f ∈ k〈X〉S, then f = as for some a ∈ X∗, s ∈ S.

(iii) Irr(S) = w ∈ X∗Y |w 6= as, a ∈ X∗, s ∈ S is a linear basis for the factormodk〈X〉〈Y |S〉 = modk〈X〉〈Y 〉/k〈X〉S.

Let S ⊂ k〈X〉 and A = k〈X|S〉. Then, for any left A-module AM , we can regard AMas a k〈X〉-module in a natural way: for any f ∈ k〈X〉, m ∈ M , fm := (f + Id(S))m.We note that AM is an epimorphic image of some free A-module. Now, we assumethat AM = modA〈Y |T 〉 = modA〈Y 〉/AT , where T ⊂ modA〈Y 〉. Let T1 =

∑fiyi ∈

modk〈X〉〈Y 〉|∑

(fi + Id(S))yi ∈ T and R = SX∗Y ∪ T1. Then, as k〈X〉-modules, AM =modk〈X〉〈Y |R〉.

17

As an application of CD-lemma for modules, we give GS bases for the Verma modulesover a Lie algebras of coefficients of a free Lie conformal algebras. We find a linear basisfor such a module.

Let B be a set of symbols. Let the locality function N : B × B → Z+ be a constant,i.e. N(a, b) ≡ N for any a, b ∈ B. Let X = b(n)| b ∈ B, n ∈ Z and L = Lie(X|S) bea Lie algebra generated by X with the relation S, where

S = ∑

s

(−1)s(

ns

)[b(n− s)a(m+ s)] = 0| a, b ∈ B, m, n ∈ Z.

For any b ∈ B, let b =∑n

b(n)z−n−1 ∈ L[[z, z−1]]. Then, it is well-known that they

generate a free Lie conformal algebra C with data (B, N) (see [168]). Moreover, thecoefficient algebra of C is just L. Let B be a linearly ordered set. Define an order on Xin the following way:

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

We use the deg-lex order on X∗. Then, it is clear that the leading term of each polynomialin S is b(n)a(m) such that

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

The following lemma is essentially from [168].

Lemma 2.15 ([70]) With the deg-lex order on X∗, S is a GS basis in Lie(X).

Corollary 2.16 ([70]) Let U = U(L) be an universal enveloping algebra of L. Then alinear basis of U consists of monomials

a1(n1)a2(n2) · · ·ak(nk), ai ∈ B, ni ∈ Z

such that for any 1 ≤ i < k,

ni − ni+1 ≤

N − 1 if ai > ai+1 or (ai = ai+1 and N is odd)N otherwise.

An L-module M is called restricted, if for any a ∈ C, v ∈ M there is some integer Tsuch that for any n ≥ T one has a(n)v = 0.

An L-module M is called highest weight module if it is generated over L by a singleelement m ∈ M such that L+m = 0, where L+ is the subspace of L generated bya(n)|a ∈ C, n ≥ 0. In this case m is called the highest weight vector.

Now we build a universal highest weight module V over L, which is often referred toas Verma module. Let kIv be a 1-dimensional trivial L+-module generated by Iv, i.e.a(n)Iv = 0 for all a ∈ B, n ≥ 0. Clearly,

V = IndLL+kIv = U(L)⊗U(L+) kIv ∼= U(L)/U(L)L+.

Then V has a structure highest weight module over L with the action given by themultiplication on U(L)/U(L)L+ and the highest weight vector I ∈ U(L). Also V =U(L)/U(L)L+ is the universal enveloping vertex algebra of C and the embedding ϕ :C → V is given by a 7→ a(−1)I (see also [168]).

18

Theorem 2.17 ([70]) Let the notions be as above. Then a linear basis of V consists ofelements

a1(n1)a2(n2) · · ·ak(nk), ai ∈ B, ni ∈ Z

such that the condition in Corollary 2.16 holds and nk < 0.

Proof: Clearly, as k〈X〉-modules,

UV =U (U(L)/U(L)L+) = modk〈X〉〈I| S(−)X∗I, a(n)I, n ≥ 0〉 =k〈X〉 〈I| S

′〉,

where S ′ = S(−)X∗I, a(n)I, n ≥ 0. In order to prove that S ′ is a Grobner-Shirshovbasis, we only need to check w = b(n)a(m)I, where m ≥ 0. Let

f =∑

s

(−1)s(

ns

)(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, 0 ≤ s ≤ N . It follows that S ′ is a Grobner-Shirshov basis. Now, the result followsfrom the CD-lemma for modules.

2.4 Composition-Diamond lemma for categories

Let X be an oriented multi-graph, C(X) the free category generated by X (a set of allpartial words on X with a partial product, i.e. a free partial monoid on X) and kC(X)the category algebroid (“partial semigroup algebra”) on C(X) (“a free associative partialalgebra”).

The following theorem is proved as CD-lemma for associative algebras.

There are two kinds of compositions: inclusion and intersection.

Theorem 2.18 ([30] CD-lemma for categories) Let S ⊂ kC(X) be a nonempty set ofmonic polynomials and < a monomial order on C(X). Let Id(S) be the ideal of kC(X)generated by S. Then the following statements are equivalent:

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

(ii) f ∈ Id(S)⇒ f = asb for some s ∈ S and a, b ∈ C(X).

(iii) Irr(S) = u ∈ C(X)|u 6= asb a, b ∈ C(X), s ∈ S is a linear basis of the algebroidkC(X)/Id(S) which is denoted by kC(X|S).

We give some applications of CD-lemma for categories.

For each non-negative integer p, let [p] denote the set 0, 1, 2, . . . , p of integers in theirusual order. A (weakly) monotonic map µ : [q]→ [p] is a function on [q] to [p] such thati ≤ j implies µ(i) ≤ µ(j). The objects [p] with morphisms all weakly monotonic mapsµ constitute a category L called simplicial category. It is convenient to use two specialfamilies of monotonic maps

εiq : [q − 1]→ [q], ηiq : [q + 1]→ [q]

19

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

εiq(j) =

j, if i > j,j + 1, if i ≤ j,

ηiq(j) =

j, if i ≥ j,j − 1, if j > i.

Let X = (V (X), E(X)) be the oriented multi-graph, where V (X) = [p] | p ∈ Z+∪0and E(X) = εip : [p − 1] → [p], ηjq : [q + 1] → [q] | p > 0, 0 ≤ i ≤ p, 0 ≤ j ≤ q. LetS ⊆ C(X)× C(X) be the relation set consisting of the following:

fq+1,q : εiq+1ε

j−1q = εjq+1ε

iq, j > i,

gq,q+1

: ηjqηiq+1 = ηiqη

j+1q+1, j ≥ i,

hq−1,q : ηjq−1ε

iq =

εiq−1ηj−1q−2, j > i,1q−1, i = j, i = j + 1,

εi−1q−1η

jq−2, i > j + 1.

Then the simplicial category L has an expression: L = C(X|S).

Now, we order C(X) by the following way.

Firstly, for any ηip, ηjq ∈ η

ip|p ≥ 0, 0 ≤ i ≤ p, ηip > ηjq iff p > q or (p = q and i < j).

Secondly, for each u = ηi1p1ηi2p2· · · ηinpn ∈ η

ip|p ≥ 0, 0 ≤ i ≤ p* (all possible words

on ηip|p ≥ 0, 0 ≤ i ≤ p, including the empty word 1v, v ∈ Ob(X)), let wt(u) =(n, ηinpn, η

in−1pn−1

, · · · , ηi1p1). Then for any u, v ∈ ηip|p ≥ 0, 0 ≤ i ≤ p*, u > v iff wt(u) > wt(v)lexicographically.

Thirdly, for any εip, εjq ∈ ε

ip, |p ∈ Z+, 0 ≤ i ≤ p, εip > εjq iff p > q or (p = q and i < j).

Finally, for each u = v0εi1p1v1ε

i2p2· · · εinpnvn ∈ C(X), n ≥ 0, vj ∈ η

ip|p ≥ 0, 0 ≤ i ≤ p*,

let wt(u) = (n, v0, v1, · · · , vn, εi1p1, · · · , εinpn). Then for any u, v ∈ C(X),

u ≻1 v ⇔ wt(u) > wt(v) lexicographically.

It is easy to check that the ≻1 is a monomial order on C(X). Then we have the followingtheorem.

Theorem 2.19 ([30]) Let X, S be defined as the above. Then with the order ≻1 onC(X), S is a Grobner-Shirshov basis for the simplical category algebroid kC(X|S).

Corollary 2.20 ([134]) In the simplicial category, each morphism µ : [q] → [p] can beuniquely presented as

εi1p ...εimp−m+1η

j1q−n...η

jnq−1,

where p ≥ i1 > ... > im ≥ 0, 0 ≤ j1 < ... < jn < q, and q − n +m = p.

The cyclic category is defined by generators and defining relations as follows, see [92].Let Y = (V (Y ), E(Y )) be an oriented (multi) graph, where V (Y ) = [p] | p ∈ Z+ ∪ 0,

20

and E(Y ) = εip : [p− 1]→ [p], ηjq : [q + 1]→ [q], tq : [q]→ [q]| p > 0, 0 ≤ i ≤ p, 0 ≤ j ≤q. Let S ⊆ C(Y )× C(Y ) be the set consisting of the following:

fq+1,q : εiq+1ε

j−1q = εjq+1ε

iq, j > i,

gq,q+1 : ηjqη

iq+1 = ηiqη

j+1q+1, j ≥ i,

hq−1,q : ηjq−1ε

iq =

εiq−1ηj−1q−2, j > i,1q−1, i = j, i = j + 1,

εi−1q−1η

jq−2, i > j + 1,

ρ1 : tqεiq = εi−1

q tq−1, i = 1, ..., q,

ρ2 : tqηiq = ηi−1

q tq+1, ı = 1, ..., q,

ρ3 : tq+1q = 1q.

The category C(Y |S) is called cyclic category, denoted by Λ.

An order on C(Y ) is defined by the following way.

Firstly, for any tip, tjq ∈ tq|q ≥ 0∗, (tp)

i > (tq)j iff i > j or (i = j and p > q).

Secondly, for any ηip, ηjq ∈ η

ip|p ≥ 0, 0 ≤ i ≤ p, ηip > ηjq iff p > q or (p = q and i < j).

Thirdly, for each u = w0ηi1p1w1η

i2p2· · ·wn−1η

inpnwn ∈ tq, η

ip|q, p ≥ 0, 0 ≤ i ≤ p*,where

wi ∈ tq|q ≥ 0∗, let wt(u) = (n, w0, w1, · · · , wn, ηinpn, ηin−1

pn−1, · · · , ηi1p1). Then for any u, v ∈

tq, ηip|q, p ≥ 0, 0 ≤ i ≤ p*, u > v iff wt(u) > wt(v) lexicographically.

Fourthly, for any εip, εjq ∈ ε

ip, |p ∈ Z+, 0 ≤ i ≤ p, εip > εjq iff p > q or (p = q and

i < j).

Finally, for each u = v0εi1p1v1ε

i2p2· · · εinpnvn ∈ C(Y ), n ≥ 0, vj ∈ tq, η

ip|q, p ≥ 0, 0 ≤ i ≤

p*, let wt(u) = (n, v0, v1, · · · , vn, εi1p1, · · · , εinpn).

Then for any u, v ∈ C(Y ), u ≻2v ⇔ wt(u) < wt(v) lexicographically.

It is also easy to check the order ≻2 is a monomial order on C(Y ), which is an extensionof ≻1 . Then we have the following theorem.

Theorem 2.21 ([30]) Let Y , S be defined as the above. Let ρ4 : tqε0q = εqq, ρ5 : tqη

0q =

ηqqt2q+1. Then

(1) With the order ≻2 on C(Y ), the set S ∪ρ4, ρ5 is a Grobner-Shirshov basis for thecyclic category algebroid kC(Y |S).

(2) For each morphism µ : [q] → [p] in the cyclic category Λ = C(Y |S), µ can beuniquely presented as

εi1p ...εimp−m+1η

j1q−n...η

jnq−1t

kq ,

where 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

Let X and Y be well-ordered sets and

N = [X ]Y ∗ = u = uXuY |uX ∈ [X ] and uY ∈ Y ∗

21

Let kN be a k-space spanned by N . For any u = uXuY , v = vXvY ∈ N , we define themultiplication of the words of N as follows uv = uXvXuY vY ∈ N. Then, kN is exactlythe tensor product algebra k[X ] ⊗ k〈Y 〉, which is “double free associative algebra”. Itis a free object in the category of all associative algebras over all commutative algebras(over k): any associative algebra KA over a commutative algebra K is isomorphic tok[X ]⊗k〈Y 〉/Id(S) as k-algebras and k[X ]-algebras. Let > be any monomial order on N .The following definitions of compositions and the GS basis are taken from [146].

Let f, g be monic polynomials of k[X ]⊗ k〈Y 〉, L the least common multiple of fX andgX.

1. Inclusion. Let gY be a subword of fY , say, fY = cgY d for some c, d ∈ Y ∗. If fY = gY

then fX ≥ gX and if gY = 1 then we set c = 1. Let w = LfY = LcgY d. We define thecomposition C1(f, g, c)w = L

fX f −LgX

cgd.

2. Overlap. Let a non-empty beginning of gY be a non-empty ending of fY , say,fY = cc0, g

Y = c0d, fY d = cgY for some c, d, c0 ∈ Y ∗ and c0 6= 1. Let w = LfY d = LcgY .

We define the composition C2(f, g, c0)w = LfX fd−

LgX

cg.

3. External. Let c0 ∈ Y ∗ be any associative word (possibly empty). In the case thatthe greatest common divisor of fX and gX is non-empty and fY , gY are non-empty, wedefine the composition C3(f, g, c0)w = L

fX fc0gY − L

gXfY c0g, where w = LfY c0g

Y .

Let S be a monic subset of k[X ]⊗ k〈Y 〉. Then S is called a GS basis if for any elementf ∈ Id(S), f contains s as its subword for some s ∈ S.

Theorem 2.22 ([146, 201] CD-lemma for associative algebras over commutative alge-bras) Let S ⊆ k[X ]⊗ k〈Y 〉 with each s ∈ S monic and < a monomial order on N . Thenthe following statements are equivalent:

(i) S is a Grobner-Shirshov basis in k[X ]⊗ k〈Y 〉.

(ii) Any composition of the elements of S, say (f, g)w, is trivial modulo (S, w), i.e.

(f, g)w =∑

i

αiaisibi,

where ai, bi ∈ N, si ∈ S, αi ∈ k and aisibi < w for any i.

(iii) Irr(S) = w ∈ N |w 6= asb, a, b ∈ N, s ∈ S is a linear basis for the factork[X ]⊗ k〈Y 〉.

We will apply this lemma in Chapter 4.3.

2.6 PBW-theorem for Lie algebras

Let (L, [ ]) be a Lie algebra over a field k with a well-ordered linear basis X = xi|i ∈ I.Let S = [xixj ] = xi, xj|i > j, i, j ∈ I be the multiplication table of L, where forany i, j ∈ I, xi, xj = Σtα

tijxt, αt

ij ∈ k. Then U(L) = k〈X|S(−)〉 is called the universal

enveloping associative algebra of L, where S(−) = xixj − xjxi = xi, xj|i > j, i, j ∈ I.

22

Theorem 2.23 (PBW Theorem) Let the notation be as above. Then with the deg-lexorder on X∗, S(−) is a Grobner-Shirshov basis in k〈X〉. Then by CD-lemma for associativealgebras, the set Irr(S(−)) consists of the elements

xi1 . . . xin , i1 ≤ · · · ≤ in, i1, . . . , in ∈ I, n ≥ 0

which forms a linear basis of U(L).

Theorem 2.24 (PBW Theorem in a Shirshov form) Let L = Lie(X|S), S ⊂ Lie(X) ⊂k〈X〉, S a Grobner-Shirshov basis in Lie(X) and U(L) = k〈X|S〉. Then a linear basis ofU(L) consists of words u = u1 · · ·un, where u1 · · · un in lex-order, n ≥ 0, each ui isS-irreducible associative Lyndon-Shirshov word.

It follows from Theorem 4.7, CD-lemma for associative algebras, Shirshov factorizationtheorem and the property (VI) of Chapter 4.2 that any LS-subword of u is a subword ofsome ui.

An interesting form of PBW theorem for a finite dimensional Lie algebra was given by L.Makar-Limanov [135].

2.7 Drinfeld-Jimbo algebra Uq(A), Kac-Moody enveloping alge-

bra U(A) and PBW basis of Uq(AN)

Let A = (aij) be an integral symmetrizable N × N Cartan matrix so that aii = 2,aij ≤ 0 (i 6= j) and there exists a diagonal matrix D with diagonal entries di which arenonzero integers such that the product DA is symmetric. Let q be a nonzero element ofk such that q4di 6= 1 for each i. Then the Drinfeld-Jimbo quantum enveloping algebra is

Uq(A) = k〈X ∪H ∪ Y |S+ ∪K ∪ T ∪ S−〉,

where

X = xi, H = h±1i , Y = yi,

S+ =

1−aij∑

ν=0

(−1)ν(

1− aijν

)

t

x1−aij−ν

i xjxνi , where i 6= j, t = q2di,

S− =

1−aij∑

ν=0

(−1)ν(

1− aijν

)

t

y1−aij−ν

i yjyνi , where i 6= j, t = q2di,

K = hihj − hjhi, hih−1i − 1, h−1

i hi − 1, xjh±1i − q∓1diaijh

±1xj , h±1i yj − q∓1yjh

±1,

T = xiyj − yjxi − δijh2i − h−2

i

q2di − q−2di and

(mn

)

t

=

n∏i=1

tm−i+1−ti−m−1

ti−t−i (for m > n > 0),

1 (for n = 0 or m = n).

Theorem 2.25 ([48]) For any symmetrizable Cartan matrix A, with the deg-lex orderon X ∪H ∪ Y ∗, S+c ∪ T ∪K ∪ S−c is a Grobner-Shirshov basis of the Drinfeld-Jimboalgebra Uq(A), where S+c, S−c are Shirshov completions of S+, S− correspondingly.

23

Corollary 2.26 (Rosso [169], Yamene [194]) For any symmetrizable Cartan matrix A,one has a triangular decomposition

Uq(A) = U+q (A)⊗ k[H ]⊗ U−

q (A),

where U+q (A) = k〈X|S+〉, U−

q (A) = k〈Y |S−〉.

The same kind of results are valid for Kac-Moody Lie algebras g(A) and their universalenveloping algebras

U(A) = k〈X ∪H ∪ Y |S+ ∪H ∪K ∪ S−〉,

where S+, S− are the same as for Uq(A), K = hihj − hjhi, xjhi − hixj + diaijxi, hiyi −yihi + diaijyj, T = xiyj − yjxi − δijhi.

Theorem 2.27 ([48]) For any symmetrizable Cartan matrix A, a Grobner-Shirshov basisof the universal enveloping algebra U(A) of the Kac-Moody Lie algebras g(A) is S+c∪T ∪K ∪ S−c.

From PBW theorem in a Shirshov form it follows that

Corollary 2.28 (Kac [103]) For any symmetrizable Cartan matrix A, one has a trian-gular decompositions

U(A) = U+(A)⊗ k[H ]⊗ U−(A), g(A) = g+(A)⊕ k[H ]⊕ g−(A).

E.N. Poroshenko [155, 156] found GS bases for Kac-Moody algebras of types An, Bn, Cn, Dn.He used known linear bases of the algebras in [103].

Now, let

A = AN =

2 −1 0 · · · 0−1 2 −1 · · · 00 −1 2 · · · 0· · · · ·0 0 0 · · · 2

and q8 6= 1.

We introduce some new variables defined by Jimbo (see [194]) which generate Uq(AN):

X = xij , 1 ≤ i < j ≤ N + 1,

where

xij =

xi j = i+ 1,

qxi,j−1xj−1,j − q−1xj−1,jxi,j−1 j > i+ 1.

We now order the set X in the following way: xmn > xij ⇐⇒ (m,n) >lex (i, j). Let usrecall from Yamane [194] the following 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.

24

Let the set S+ consist of Jimbo relations:

xmnxij − q−2xijxmn ((i, j), (m,n)) ∈ C1 ∪ C3,

xmnxij − xijxmn ((i, j), (m,n)) ∈ C2 ∪ C6,

xmnxij − xijxmn + (q2 − q−2)xinxmj ((i, j), (m,n)) ∈ C4,

xmnxij − q2xijxmn + qxin ((i, j), (m,n)) ∈ C5.

It is easily seen that U+q (AN ) = k〈X|S+〉.

In paper [76], a direct proof is given that S+ is a GS basis for k〈X|S+〉 = U+q (AN)

([48]). The proof is different from one in L.A. Bokut, P. Malcolmson [48]. As a result,one gets

Theorem 2.29 ([48]) Let the notation be as before. Then with the deg-lex order on

X ∪H ∪ Y ∗, S+ ∪ T ∪K ∪ S− is a Grobner-Shirshov basis for Uq(AN) = k〈X ∪H ∪

Y |S+ ∪ T ∪K ∪ S−〉.

Corollary 2.30 ([169, 194]) For q8 6= 1, a linear basis of Uq(An) consists of elements

ym1n1 · · · ymlnlhs11 · · ·h

sNN xi1j1 · · ·xikjk

with (m1, n1) ≤ · · · ≤ (ml, nl), (i1, j1) ≤ · · · ≤ (ik, jk), k, l ≥ 0 and each st ∈ Z.

3 Grobner-Shirshov bases for groups and semigroups

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

Let X be a set, S ⊆ X∗ ×X∗, ρ(S) the congruence on X∗ generated by S,

A = sgp〈X|S〉 = X∗/ρ(S)

the quotient semigroup and k(X∗/ρ(S)) the semigroup algebra. We identify the set u =v|(u, v) ∈ S with S. Then it is easy to see that

σ : k〈X|S〉 → k(X∗/ρ(S)),∑

αiui + Id(S) 7→∑

αiui

is an algebra isomorphism.

Shirshov completion Sc of S consists of “semigroup relations”, Sc = ui − vi, i ∈ I.Then Irr(Sc) is a linear basis of k〈X|S〉 and so σ(Irr(Sc)) is a linear basis of k(X∗/ρ(S)).This shows that Irr(Sc) is exactly normal forms of elements of the semigroup sgp〈X|S〉.

Thus, in order to find normal forms of the semigroup sgp〈X|S〉, it suffices to find a GSbasis Sc in k〈X|S〉. In particular, let G = gp〈X|S〉 be a group, where S = (ui, vi) ∈F (X)× F (X)|i ∈ I and F (X) is the free group on a set X . Then G has a presentationas a semigroup

G = sgp〈X ∪X−1|S, xεx−ε = 1, ε = ±1, x ∈ X〉, X ∩X−1 = ∅.

25

3.1 Grobner-Shirshov bases for braid groups

Let Bn denote the Artin braid group of type An−1 (Artin [2]). Then

Bn = gp〈σ1, . . . , σn | σjσi = σiσj (j − 1 > i), σi+1σiσi+1 = σiσi+1σi, 1 ≤ i ≤ n− 1〉.

3.1.1 Braid groups in Artin-Burau generators

Let X = Y ∪Z, Y ∗ and Z be well-ordered. Suppose that the order on Y ∗ is monomial.Then, any word in X has the form u = u0z1u1 · · · zkuk, where k ≥ 0, ui ∈ Y ∗, zi ∈ Z.Define the inverse weight of the word u ∈ X∗ by

inwt(u) = (k, uk, zk, · · · , u1, z1, u0).

Now we define the inverse weight lexicographical order as follows

u > v ⇔ inwt(u) > inwt(v).

We call the above order the inverse tower order for short. Clearly, this order is a monomialorder on X∗.

In case X = Y ∪Z, Y = T ∪U and Y ∗ is endowed with the inverse tower order, we definethe inverse tower order on X∗ relative to the presentation X = (T ∪U)∪Z. In general, wecan define the inverse tower order of X-words relative to the presentation

X = (· · · (X(n)∪X(n−1))∪ · · · )∪X(0),

where X(n)-words are endowed by a monomial order.

In the braid group Bn, we now introduce a new set of generators which are called theArtin-Burau generators. We set

si,i+1 = σ2i , si,j+1 = σj · · ·σi+1σ

2i σ

−1i+1 · · ·σ

−1j , 1 ≤ i < j ≤ n− 1;

σi,j+1 = σ−1i · · ·σ

−1j , 1 ≤ i ≤ j ≤ n− 1; σii = 1, a, b = b−1ab.

Form the set

Sj = si,j, s−1i,j , 1 ≤ i, j < n and Σ−1 = σ−1

1 , · · ·σ−1n−1.

Then the setS = Sn ∪ Sn−1 ∪ · · · ∪ S2 ∪ Σ−1

generates Bn as a semigroup.

Now we order the alphabet S in the following way:

Sn < Sn−1 < · · · < S2 < Σ−1,

ands−11,j < s1,j < s−1

2,j < · · · < sj−1,j , σ−11 < σ−1

2 < · · ·σ−1n−1.

With the above notation, we now order the S-words by using the inverse tower order,according to the fixed presentation of S as the union of Sj and Σ−1. We order the Sn-words by the deg − inlex order, i.e. we first compare the words by length and then byinverse lexicographical order, starting from their last letters.

26

Lemma 3.1 (Artin [3], Malkov [137]) The following Artin-Markov relations hold in thebraid group Bn.

σ−1k sδi,j = sδi,jσ

−1k , k 6= i− 1, i, j − 1, j (1)

σ−1i sδi,i+1 = sδi,i+1σ

−11 (2)

σ−1i−1s

δi,j = sδi−1,jσ

−1i−1 (3)

σ−1i sδi,j = s

δi+1,j, si,i+1σ

−1i (4)

σ−1j−1s

δi,j = sδi,j−1σ

−1j−1 (5)

σ−1j sδi,j = s

δi,j+1, sj,j+1σ

−1j (6)

where δ = ±1;

s−1j,ks

εk,l = s

εk,l, s

−1j,l s

−1j,k (7)

sj,ksεk,l = s

εk,l, sj,lsk,lsj,k (8)

s−1j,ks

εj,l = s

εj,l, s

−1k,l s

−1j,l s

−1j,k (9)

sj,ksεj,l = s

εj,l, sk,lsj,k (10)

s−1i,ks

εj,l = s

εj,l, sk,lsi,ls

−1k,l s

−1i,l s

−1i,k (11)

si,ksεj,l = s

εj,l, s

−1i,l s

−1k,l si,lsk,lsi,k (12)

where i < j < k < l, ε = ±1;

sδi,ksεj,l = sεj,ls

δi,k (13)

σ−1j σ−1

k = σ−1k σ−1

j , j < k − 1 (14)

σj,j+1σk,j+1 = σk,j+1σj−1,j , j < k (15)

σ−2i = s−1

i,i+l (16)

s±1i,j s

∓1i,j = 1 (17)

where j < i < k < l or i < k < j < l, and ε, δ = ±1.

Theorem 3.2 ([19]) The Artin-Markov relations (1)-(17) form a Grobner-Shirshov basisof the braid group Bn in terms of the Artin-Burau generators relative to the inverse towerorder of words.

In the paper [19], it was claimed that some compositions are trivial. The paper [72]supported the claim and all compositions worked out explicitly.

Corollary 3.3 (Markov-Ivanovskii [3]) The following words are normal forms of the braidBn: fnfn−1 . . . f2σinnσin−1n−1 . . . σi22, where each fj is free irreducible words in sij, i <j, 2 ≤ j ≤ n.

3.1.2 Braid groups in Artin-Garside generators

The Artin-Garside generators of the braid group Bn+1 are σi, 1 ≤ i ≤ n, , −1 (Garside[91] 1969), where = Λ1 · · ·Λn, with Λi = σ1 · · ·σi.

27

Let us order −1 < < σ1 < · · · < σn. We order −1,, σ1, . . . , σn∗ by deg-lex

order.

By V (j, i),W (j, i), . . . , where j ≤ i, we understand positive words in the lettersσj , σj+1, . . . , σi. Also V (i+ 1, i) = 1, W (i+ 1, i) = 1, . . . .

Given V = V (1, i), let V (k), 1 ≤ k ≤ n− i be the result of shifting in V all indices ofall letters by k, σ1 7→ σk+1, . . . , σi 7→ σk+i, and we also use the notation V (1) = V ′. Wewrite σij = σiσi−1 . . . σj , j ≤ i− 1, σii = σi, σii+1 = 1.

Theorem 3.4 ([17]) A Grobner-Shirshov basis of Bn+1 in the Artin-Garside generatorsconsists of the following relations S:

σi+1σiV (1, i− 1)W (j, i)σi+1j = σiσi+1σiV (1, i− 1)σijW (j, i)′,

σsσk = σkσs, s− k ≥ 2,

σ1V1σ2σ1V2 · · ·Vn−1σn · · ·σ1 = V(n−1)1 V

(n−2)2 · · ·V ′

(n−1),

σl−1 = −1σn−l+1, 1 ≤ l ≤ n,

−1 = 1, −1 = 1,

where 1 ≤ i ≤ n − 1, 1 ≤ j ≤ i + 1, W (j, i) begins with σi if it is not empty, andVi = Vi(1, i).

As results, we have the following corollaries.

Corollary 3.5 The S-irreducible normal form of each word of Bn+1 coincides with theGarside normal form in [91] of the word.

Corollary 3.6 (Garside [91]) The semigroup of positive braids B+n+1 can be embedded

into a group.

3.1.3 Braid groups in Birman-Ko-Lee generators

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

σts = (σt−1σt−2 . . . σs+1)σs(σ−1s+1 · · ·σ

−1t−2σ

−1t−1).

Then Bn has an presentation.

Bn = gp〈σts, n ≥ t > s ≥ 1|σtsσrq = σrqσts, (t− r)(t− q)(s− r)(s− q) > 0,

σtsσsr = σtrσts = σsrσtr, n ≥ t > s > r ≥ 1〉.

Denote by δ = σnn−1σn−1n−2 · · ·σ21, the Garside word.

Let us order δ−1 < δ < σts < σrq iff (t, s) < (r, q) lexicographically. We orderδ−1, δ, σts, n ≥ t > s ≥ 1∗ by deg-lex order.

Instead of σij , 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 6= tj+1, 1 ≤ j ≤ m − 1. In this notation,the defining relations of Bn can be written as (t3, t2, t1) = (t2, t1, t3) = (t1, t3, t2), t3 >t2 > t1, (k, l)(i, j) = (i, j)(k, l), k > l, i > j, k > i, where either k > i > j > l, ork > l > i > j.

28

Let us assume the following notation: V[t2,t1], where n ≥ t2 > t1 ≥ 1, is a positive word in(k, l) such that t2 ≥ k > l ≥ t1. We can use any capital Latin letter with indices instead ofV , and any appropriate numbers (for example, t3, t0 such that t3 > t0) instead of t2, t1. Wewill also use the following equalities in Bn: V[t2−1,t1](t2, t1) = (t2, t1)V

′[t2−1,t1]

, t2 > t1, where

V ′[t2−1,t1]

= (V[t2−1,t1])|(k,l)7→(k,l), if l 6=t1; (k,t1)7→(t2,k); W[t2−1,t1](t1, t0) = (t1, t0)W⋆[t2−1,t1]

, t2 >

t1 > t0, where W ⋆[t2−1,t1]

= (W[t2−1,t1])|(k,l)7→(k,l), if l 6=t1; (k,t1)7→(k,t0).

Theorem 3.7 ([18]) 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), k > l > i > j,

(k, l)V[j−1,1](i, j) = (i, j)(k, l)V[j−1,1], k > 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) . . . Vn−1[n−1,1](n, 1) = δV ′2[2,1] . . . V

′n−1[n−1,1],

(t, s)δ = δ(t+ 1, s+ 1), (t, s)δ−1 = δ−1(t− 1, s− 1), t± 1, s± 1 (modn),

δδ−1 = 1, δ−1δ = 1,

where V[k,l] means as before any word in (i, j) such that k ≥ i > j ≥ l, t > t3, t2 > s.

As results, we have the following corollaries.

Corollary 3.8 (Birman-Ko-Lee [9]) The semigroup of positive braids B+n in Birman-Ko-

Lee generators is embeddable into a group.

Corollary 3.9 (Birman-Ko-Lee [9]) The S-irreducible normal form of a word of Bn

in Birman-Ko-Lee generators coincides with the Birman-Ko-Lee-Garside normal formδkA, A ∈ B+

n .

3.1.4 Braid groups in Adyan-Thurston generators

The symmetry group is as follow:

Sn+1 = gp〈s1, . . . , sn | s2i = 1, sjsi = sisj (j − 1 > i), si+1sisi+1 = sisi+1si〉.

L.A. Bokut, L.-S. Shiao [51] found the normal form for Sn+1 in the following theorem.

Theorem 3.10 ([51]) The set N = s1i1s2i2 · · · snin | ij ≤ j + 1 is a Grobner-Shirshovnormal form for Sn+1 in generators si = (i, i + 1) relative to the deg-lex order, wheresji = sjsj−1 · · · si (j ≥ i), sjj+1 = 1.

Let α ∈ Sn+1 and α = s1i1s2i2 · · · snin ∈ N be the normal form of α. Define the lengthof α as |α| = l(s1i1s2i2 · · · snin) and α ⊥ β if |αβ| = |α| + |β|. Moreover, each α ∈ N has

29

a unique expression α = sl1il1

sl2il2· · · s

ltilt

, where each slj ilj

6= 1. Such a t is called the

breath of α.

Now, we let

B′n+1 = gp〈r(α), α ∈ Sn+1 \ 1 | r(α)r(β) = r(αβ), α ⊥ β〉,

where r(α) means a letter with the index α.

Then for the braid group with n + 1 generators, Bn+1∼= B′

n+1. Indeed, define θ :Bn+1 → B′

n+1, σi 7→ r(si) and θ′ : B′n+1 → Bn+1, r(α) 7→ α|si 7→σi

. Then two mappingsare homomorphisms and θθ′ = lB′

n+1, θ′θ = lBn+1 . Hence,

Bn+1 = gp〈r(α), α ∈ Sn+1 \ 1 | r(α)r(β) = r(αβ), α ⊥ β〉.

Let X = r(α), α ∈ Sn+1 \ 1. The generator X of Bn+1 is called Adyan-Thurstongenerator.

Then the positive braid semigroup in generator X is B+n+1 = sgp〈X | r(α)r(β) =

r(αβ), α ⊥ β〉.

Let s1 < s2 < · · · < sn. Define r(α) < r(β) if and only if |α| > |β| or |α| = |β|, α <lex β.It is clear that such an order on X is well-ordered. We will use the deg-lex order on X∗.

Theorem 3.11 ([79]) A Grobner-Shirshov basis of B+n+1 in Adyan-Thurston generator

X relative to the deg-lex order on X∗ is:

r(α)r(β) = r(αβ), α ⊥ β; r(α)r(βγ) = r(αβ)r(γ), α ⊥ β ⊥ γ.

Theorem 3.12 ([79]) A Grobner-Shirshov basis of Bn+1 in Adyan-Thurston generatorX relative to the deg-lex order on X∗ is:

(1) r(α)r(β) = r(αβ), α ⊥ β,

(2) r(α)r(βγ) = r(αβ)r(γ), α ⊥ β ⊥ γ,

(3) r(α)∆ε = ∆εr(α′), α′ = α|si 7→sn+1−i,

(4) r(αβ)r(γµ) = ∆r(α′)r(µ), α ⊥ β ⊥ γ ⊥ µ, r(βγ) = ∆,

(5) ∆ε∆−ε = 1.

Corollary 3.13 (Adyan-Thurston) The normal forms for Bn+1 are∆kr(α1) · · · r(αs) (k ∈

Z), where r(α1) · · · r(αs) is minimal in deg-lex order.

3.2 Grobner-Shirshov basis for Chinese monoid

The Chinese monoid CH(X,<) over a well-ordered set (X,<) has the following presenta-tion CH(X) = sgp〈X|S〉, where X = xi|i ∈ I and S consists of the following relations:

xixjxk = xixkxj = xjxixk for every i > j > k,

xixjxj = xjxixj , xixixj = xixjxi for every i > j.

Theorem 3.14 ([75]) With the deg-lex order on X∗ the following relations (1)-(5) formsa Grobner-Shirshov basis of the Chinese monoid CH(X).

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(1) xixjxk − xjxixk,

(2) xixkxj − xjxixk,

(3) xixjxj − xjxixj,

(4) xixixj − xixjxi,

(5) xixjxixk − xixkxixj,

where xi, xj, xk ∈ X, i > j > k.

Let Λ be the set which consists of words on X of the form un = w1w2 · · ·wn, n ≥ 0,where

w1 = xt111

w2 = (x2x1)t21xt22

2

w3 = (x3x1)t31(x3x2)

t32xt333

· · ·

wn = (xnx1)tn1(xnx2)

tn2 · · · (xnxn−1)tn(n−1)xtnn

n

with xi ∈ X , x1 < x2 < · · · < xn and all exponents non-negative.

Corollary 3.15 ([63]) The set Λ is normal forms of elements of the Chinese monoidCH(X).

3.3 Grobner-Shirshov basis for free inverse semigroup

Let S be a semigroup. s ∈ S is called an inverse of t ∈ S if sts = s, tst = t. An inversesemigroup is a semigroup in which every element t has a unique inverse, denoted by t−1.

Let X−1 = x−1|x ∈ X with X ∩ X−1 = ∅. Denote X ∪ X−1 by Y . We define theformal inverses for elements of Y ∗ by the rules

1−1 = 1, (x−1)−1 = x (x ∈ X),

(y1y2 · · · yn)−1 = y−1

n · · · y−12 y−1

1 (y1, y2, · · · , yn ∈ Y ).

It is well known that,

FI(X) = sgp〈Y | aa−1a = a, aa−1bb−1 = bb−1aa−1, a, b ∈ Y ∗〉

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

We give definitions in Y ∗ of formal idempotent, (prime) canonical idempotent andordered (prime) canonical idempotent. Let < be a well order on Y .

(i) The empty word 1 is an idempotent.

(ii) If h is an idempotent and x ∈ Y , then x−1hx is both an idempotent and a primeidempotent.

31

(iii) If e1, e2, · · · , em (m > 1) are prime idempotents, then e = e1e2 · · · em is an idempo-tent.

(iv) An idempotent w ∈ Y ∗ is called canonical if w has no subword of the form x−1exfx−1,where x ∈ Y , both e and f are idempotents.

(v) A canonical idempotent w ∈ Y ∗ is called ordered if for any subword e = e1e2 · · · emof w, fir(e1) < fir(e2) < · · · < fir(em), where each ei is an idempotent and m > 2and fir(u) is the first letter of u ∈ Y ∗.

Theorem 3.16 ([38]) Let S be the set of the following two kinds of polynomials in k〈Y 〉:

• ef − fe, where e, f are ordered prime canonical idempotents such that ef > fe;

• x−1e′xf ′x−1 − f ′x−1e′, where x ∈ Y, x−1e′x and xf ′x−1 are ordered prime canonicalidempotents.

Then, with deg-lex order on Y ∗, S is a Grober-Shirshov basis for free inverse semigroupsgp〈Y |S〉.

Theorem 3.17 ([38]) Normal forms of elememts of the free inverse semigroup sgp〈Y |S〉are

u0e1u1 · · · emum ∈ Y ∗,

where m ≥ 0, u1, · · · , um−1 6= 1, u0u1 · · ·um has no subword of form yy−1 for y ∈Y , e1, · · · , em are ordered canonical idempotents, and the first (last, respectively) lettersof the factors of ei (1 ≤ i ≤ m) are not equal to the first (last, respectively) letter ofui (ui−1, respectively).

The above normal form is analogous to “semi-normal” forms that were given in Poliakova-Schein [152], 2005.

3.4 Grobner-Shirshov bases for plactic monoids

Let X = x1, . . . , xn be a set of n elements with the order x1 < · · · < xn. Then M.P.Schutzenberger calls Pn = sgp〈X|T 〉 a plactic monoid (see also M. Lothaire [131], Chapter5), where T consists of the Knuth relations

xixkxj = xkxixj (xi ≤ xj < xk),

xjxixk = xjxkxi (xi < xj ≤ xk).

A nondecreasing word R ∈ X∗ is called a row and a strictly decreasing word C ∈ X∗

is called a column, for example, x1x1x3x5x5x5x6 is a row and x6x4x2x1 is a column.

Let R, S ∈ A∗ be two rows. Then R dominates S if |R| ≤ |S| and each letter of R islarger than the corresponding letter of S, where |R| is the length of the word R.

A (semistandard) Young tableau on A (see [125]) is a word w = R1R2 · · ·Rt ∈ U∗

such that Ri dominates Ri+1, i = 1, . . . , t − 1. For example, x4x5x5x6 · x2x2x3x3x5x7 ·x1x1x1x2x4x4x4 is a Young tableau.

In this section, we give GS bases for plactic monoids in two sets of generators, rowwords and column words.

32

3.4.1 Plactic monoids in row generators

Let Pn = sgp〈X|T 〉 be a plactic monoid. Suppose that X = 1, 2, . . . , n with 1 < 2 <· · · < n. Let N be the set of non-negative integers. Let R ∈ X∗ be a row. For convenience,denote R = (r1, r2, . . . , rn), where ri is the number of letter i (i = 1, 2, . . . , n), for example,R = 111225 = (3, 2, 0, 0, 1, 0, . . . , 0).

Let U be the set of all rows in X∗. We order the set U∗ as follows.

Let R = (r1, r2, . . . , rn) ∈ U . Then |R| = r1 + · · ·+ rn is the length of R in X∗.

We first order U : for any R, S ∈ U , R < S if and only if |R| < |S| or |R| = |S|and (r1, r2, . . . , rn) > (s1, s2, . . . , sn) lexicographically. Clearly, this is a well order on U .Then, we order U∗ by the deg-lex order.

Lemma 3.18 ([25]) Let (φ1, . . . , φn) ∈ U . Denote Φp =∑p

i=1 φi (1 ≤ p ≤ n), whereφ represents any lowercase symbol and Φ the corresponding uppercase symbol. Let W =(w1, w2, . . . , wn), Z = (z1, z2, . . . , zn) ∈ U . DenoteW ′ = (w′

1, w′2, . . . , w

′n), Z

′ = (z′1, z′2, . . . , z

′n),

where w′1 = 0, w′

p = min(Zp−1−W ′p−1, wp), W ′

p = min(Zp−1,W′p−1+wp), z′q = wq + zq −

w′q, Z ′

q = Wq +Zq −W ′q, n ≥ p ≥ 2, n ≥ q ≥ 1. Then either W ′, Z ′ ∈ U or W ′ ∈ U and

Z ′ = (0, 0, . . . , 0), W · Z = W ′ · Z ′ in Pn = sgp〈X|T 〉 and W ′ · Z ′ is a Young tableau onX. Moreover,

Pn = sgp〈X|T 〉 ∼= sgp〈U |Γ〉,

where Γ = W · Z = W ′ · Z ′ | W,Z ∈ V .

Theorem 3.19 ([25]) With the deg-lex order on U∗, the set Γ is a Grobner-Shirshovbasis for the plactic algebra k〈U | Γ〉.

Corollary 3.20 (Knuth [120]) The set of semi-Schensted Young tableaux on X are nor-mal forms of elements of the plactic monoid Pn.

3.4.2 Plactic monoids in column generators

Let Pn = sgp〈X|T 〉 be a plactic monoid. Suppose that X = 1, 2, . . . , n with 1 <2 < · · · < n. Let C ∈ X∗ be a column. Let ci be the number of letter i in C. Thenci ∈ 0, 1, i = 1, 2, . . . , n. We denote C = (c1; c2; . . . ; cn). For example, C = 6421 =(1; 1; 0; 1; 0; 1; 0; . . . ; 0).

Let V = C | C is a column in X∗.

Let R = (r1; r2; . . . ; rn) ∈ V and wt(R) = (|R|, r1, . . . , rn). We order V : for anyR, S ∈ V , R < S if and only if wt(R) < wt(S) lexicographically. Then, we order V ∗ bythe deg-lex order.

Let (φ1; . . . ;φn) ∈ V . Denote Φp =∑p

i=1 φi, 1 ≤ p ≤ n, where φ represents anylowercase symbol defined above and Φ the corresponding uppercase symbol.

Lemma 3.21 ([25]) Let W = (w1;w2; . . . ;wn), Z = (z1; z2; . . . ; zn) ∈ V . Define W ′ =(w′

1;w′2; . . . ;w

′n), Z

′ = (z′1; z′2; . . . ; z

′n), where z

′1 = min(w1, z1), z

′p = min(Wp−Z

′p−1, zp), Z

′p =

min(Wp, Z′p−1 + zp), w′

q = wq + zq − z′q, W ′q = Wq + Zq − Z ′

q, n ≥ p ≥ 2, n ≥ q ≥ 1.Then W ′, Z ′ ∈ V , W · Z = W ′ · Z ′ in Pn = sgp〈X|T 〉 and W ′ · Z ′ is a Young tableau onX. Moreover,

Pn = sgp〈X|T 〉 ∼= sgp〈V |Λ〉,

where Λ = W · Z = W ′ · Z ′ | W,Z ∈ V .

33

Theorem 3.22 ([25]) With the deg-lex order on V ∗, Λ is a finite Grobner-Shirshov basisfor the plactic algebra k〈V | Λ〉.

Thus, by using CD-lemma for associative algebras and Theorem 3.22, we also have theCorollary 3.20.

4 Grobner-Shirshov bases for Lie algebras

In this chapter, we first give another approach to the LS basis and the Hall basis, respec-tively, of a free Lie algebra by using Shirshov CD-lemma for anti-commutative algebras.Then by using LS basis, we construct the classical GS bases theory for Lie algebras over afield. Finally, we mention GS bases for Lie algebras over a commutative algebra in whichsome interesting applications are given.

4.1 Lyndon-Shirshov basis and Lyndon-Shirshov words within

anti-commutative algebras

In this section, we give another approach to the LS basis (and the Hall basis) of a freeLie algebra.

A linear space A together with a bilinear product x · y is called an anti-commutativealgebra if it satisfies the following identity x2 = 0, so, x · y = −y · x for any x, y ∈ A.

Let X be a well-ordered set and X∗∗ the set of all non-associative words.

We define three orders ≻lex, >deg−lex

and >n−deg−lex

on X∗∗. For (u), (v) ∈ X∗∗,

(u) = ((u1)(u2)) ≻lex (v) = ((v1)(v2)) ((u2) or (v2) is empty if |(u)| = 1 or |(v)| = 1) iffone of the following cases holds:

(a) u1u2 > v1v2 in the lex order,

(b) u1u2 = v1v2 and (u1) ≻lex (v1),

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

(u) = ((u1)(u2)) >deg−lex (v) = ((v1)(v2)) iff one of the following cases holds:

(a) u1u2 > v1v2 in the deg-lex order,

(b) u1u2 = v1v2 and (u1) >deg−lex (v1),

(c) u1u2 = v1v2, (u1) = (v1) and (u2) >deg−lex (v2).

(u) >n−deg−lex

(v) iff one of the following cases holds:

(a) |(u)| > |(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(v2)).

34

We define regular words (u) ∈ X∗∗ by induction on |(u)|:

(i) xi ∈ 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], if (u) is regular.

The set N(X) of all regular words onX forms a linear basis of the free anti-commutativealgebra AC(X) on X .

The following result gives an alternating approach to the definition of LS words as“roots” of associative supports u of normal words [u].

Theorem 4.1 ([31]) Let [u] be a regular word of an anti-commutative algebra AC(X).Then u = vm, where v is a Lyndon-Shirshov word in X, m ≥ 1. Moreover, the set ofassociative supports of N(X) includes the set of all Lyndon-Shirshov words in X.

Let f, g be monic polynomials in AC(X) with a fixed order >deg−lex on X∗∗. If thereexist a, b ∈ X∗ such that [w] = [f ] = [a[g]b], then the composition of inclusion of f and gis defined by (f, g)[w] = f − [a[g]b].

A monic subset S of AC(X) is called a GS basis in AC(X) if any composition (f, g)[w]

of inclusion in S is trivial modulo (S, [w]).

Theorem 4.2 (Shirshov CD-lemma for anti-commutative algebras, cf. [182]) Let S ⊂AC(X) be a nonempty set of monic polynomials under the order >deg−lex on X∗∗. Thenthe following statements are equivalent.

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

(ii) f ∈ Id(S) ⇒ [f ] = [a[s]b] for some s ∈ S and a, b ∈ X∗, where [asb] is a normalS-word.

(iii) Irr(S) = [u] ∈ N(X)|[u] 6= [a[s]b] a, b ∈ X∗, s ∈ S and [asb] is a normal S-wordis a linear basis of the algebra AC(X|S) = AC(X)/Id(S).

Let S1 be a subset of the free anti-commutative algebra AC(X) defined as

S1 = ([u][v])[w]− ([u][w])[v]− [u]([v][w]) |

[u], [v], [w] ∈ N(X) and [u] ≻lex [v] ≻lex [w].

It is easy to prove that the free Lie algebra Lie(X) = AC(X)/Id(S1).

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

Theorem 4.3 ([31]) Under the order deg−lex, the subset S1 of AC(X) is a anti-commutativeGrobner-Shirshov basis in AC(X). Then Irr(S1) is all non-associative LS words in X.So, LS monomials form a linear basis of the free Lie algebra Lie(X).

Remark. ([29]) Denote S2 the set of S1 by replacing ≻lex with >n−deg−lex

. Then underthe order >

n−deg−lex, the subset S2 of AC(X) is also a anti-commutative GS basis. The

set Irr(S2) is exactly the set of all Hall words in X . It forms a linear basis of a free Liealgebra Lie(X).

35

4.2 Composition-Diamond lemma for Lie algebras over a field

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

Let X = xi|i ∈ I be a well-ordered set with xi > xt if i > t for any i, t ∈ I. Foru = xi1xi2 · · ·xim ∈ X∗, let the length of u be m, denoted by |u| = m; xβ = min(u) =minxi1 , xi2 , · · · , xim; fir(u) = xi1 ; X ′(u) = xj

i = xi xβ · · ·xβ︸ ︷︷ ︸j

|i > β, j ≥ 0. We order

X ′(u) by the following way:

xj1i1> xj2

i2⇔ i1 > i2 or (i1 = i2, j2 > j1).

Suppose u = xr1 xβ · · ·xβ︸ ︷︷ ︸m1

· · ·xrt xβ · · ·xβ︸ ︷︷ ︸mt

. Then we define u′ = xm1r1· · ·xmt

rtin (X ′(u))∗.

We use two linear orders on X∗: for any u, v ∈ X∗,

(i) (lex order) 1 ≻ t if t 6= 1 and, by induction, if u = xiu1 and v = xjv1 then u ≻ v ifand only if xi > xj or xi = xj and u1 ≻ v1;

(ii) (deg-lex order) u > v if |u| > |v|, or |u| = |v| and u ≻ v.

We regard Lie(X) as the Lie subalgebra of the free associative algebra k〈X〉, whichis generated X under the Lie bracket [u, v] = uv − vu. Given f ∈ k〈X〉, denote f theleading word of f with respect to the deg-lex order, f = αf f − r

f, αf ∈ k. A polynomial

f is monic if αf = 1.

Definition 4.4 ([133, 177]) w ∈ X∗\1 is an associative Lyndon-Shirshov word (ALSWfor short) if

(∀u, v ∈ X∗, u, v 6= 1) w = uv ⇒ w > vu.

We denote the set of all ALSW’s on X by ALSW (X).

We cite some useful properties of ALSW’s following Shirshov [183, 177, 178, 181].Property (VIII) was given by Shirshov [178] and Chen-Fox-Lyndon [64]. Property (VI)was implicitly used in Shirshov [181].

(I) If w ∈ ALSW (X) then an arbitrary proper prefix of w cannot be a suffix of w.

(II) If w = uv ∈ ALSW (X), where u, v 6= 1 then u ≻ w ≻ v.

(III) If u, v ∈ ALSW (X) and u ≻ v then uv ∈ ALSW (X).

(IV) (Shirshov factorization theorem) An arbitrary associative word w can be uniquelyrepresented as w = c1c2 . . . cn, where c1, . . . , cn ∈ ALSW (X) and c1 c2 . . . cn.

(V) If u1u2 and u2u3 are ALSW’s then u1u2u3 is also an ALSW provided u2 6= 1.

(VI) If an associative word w is represented as in (IV) and v is a LS subword of w,then v is a subword of one of the words c1, c2,. . .,cn.

(VII) (Shirshov key property of ALSW’s) u is an ALSW in X∗ if and only if u′ is anALSW in (X ′(u))∗.

It allows one to prove the remaining properties by induction on the length.

(VIII) If an ALSW w = uv and v is its longest proper ALSW end of w, then u is anALSW as well.

The following Shirshov definition of LS Lie monomials is equivalent to the definitionfrom Chen-Fox-Lyndon [64].

36

Definition 4.5 ([177]) A non-associative word (u) in X is a non-associative LS word(NLSW for short), denoted by [u], if

(i) u is an ALSW;

(ii) if [u] = [(u1)(u2)] then both (u1) and (u2) are NLSW’s (from (I) it then follows thatu1 ≻ u2);

(iii) if [u] = [[[u11][u12]][u2]] then u12 u2.

We denote the set of all NLSW’s on X by NLSW (X).

In fact, NLSW’s may be defined as Hall-Shirshov words relative to lex order of non-associative words (for definition of Hall-Shirshov words see [179], also [164]).

We cite some useful properties (IX)-(XX) of NLSW’s following Shirshov [177]. Prop-erty (IX) is from Shirshov [177] and Chen-Fox-Lyndon [64], and Property (XI) fromShirshov [178] and Chen-Fox-Lyndon [64].

(IX) NLSW (X) forms a linear basis of Lie(X) ⊂ k〈X〉.

(X) For an ALSW w, there is a unique “down-to-up” bracketing [w] such that [w] isNLSW: w ⇒ w′ ⇒ w′′ ⇒ · · · ⇒ w(k) = [w], where each time one joins a minimal letter toa previous one by left normed Lie bracketing, cf. (VII).

(XI) In (X), [w] is equivalent to Shirshov [178] and Chen-Fox-Lyndon [64] “up-to-down” bracketing: [w] = [[u][v]] if |w| > 1, where v is the longest proper associative LSend of w.

(XII) For an ALSW w, considering NLSW [w] as a polynomial in k〈X〉, we have[w] = w.

(XIII) (Shirshov special bracketing) Suppose that w = aub, where w, u ∈ ALSW (X).Then

(i) [w] = [a[uc]d], where b = cd and possibly c = 1.

(ii) Represent c in the form c = c1c2 . . . cn, where c1, . . . , cn ∈ ALSW (X) and c1 ≤ c2 ≤. . . ≤ cn. Replacing [uc] by [. . . [[u][c1]] . . . [cn]] we obtain the word

[w]u = [a[. . . [[[u][c1]][c2]] . . . [cn]]d]

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

(iii) [w]u = w. Moreover, [w]u = a[u]b+∑i

αiai[u]bi, where each αi ∈ k and aiubi < aub.

(XIV) (Shirshov Lie elimination of a leading word) Let f, s be two monic Lie polyno-mials and f = asb for some a, b ∈ X∗. Then f1 = f − [asb]s is a Lie polynomial with aless leading word, i.e. f1 < f .

(XV) (Shirshov “double” special bracketing) Suppose that w = aubvc, where w, u, v ∈ALSW (X). Then there exists bracketing [w]u,v such that [w]u,v = [a[u]b[v]c]u,v and

[w]u,v = w.

(XVI) (Shirshov’s algorithm to recognize Lie polynomials, cf. Dynkin-Specht-Wever’sand Friedrich’s ones) If s ∈ Lie(X) ⊂ k〈X〉, then s is an ALSW and s1 = s− αs[s] is aLie polynomial with a less maximal word (in deg-lex order), s1 < s.

37

Definition 4.6 Let S ⊂ Lie(X) with each s ∈ S monic, a, b ∈ X∗ and s ∈ S. If asb isan ALSW, then we call [asb]s = [asb]s|[s] 7→s a special normal S-word (or special normals-word), where [asb]s is defined as (XIII) (ii). A Lie S-word (asb) is called a normalS-word if (asb) = asb. Any special normal s-word is a normal s-word by (XIII) (iii).

Suppose that f, g ∈ S. Then, there are two kinds of Lie compositions:

(i) If w = f = agb for some a, b ∈ X∗, then the polynomial 〈f, g〉w = f− [agb]g is calledthe composition of inclusion of f and g with respect to w.

(ii) If w is a word such that w = f b = ag for some a, b ∈ X∗ with deg(f) + deg(g) >deg(w), then the polynomial 〈f, g〉w = [fb]f − [ag]g is called the composition ofintersection of f and g with respect to w; w is an ALSW due to (V).

Let h be a Lie polynomial and w ∈ X∗. We shall say that h is trivial modulo (S, w),denoted by h ≡Lie 0 mod(S, w), if h =

∑i αi(aisibi), where each (aisibi) is a normal

S-word and aisibi < w.

The set S is called a GS basis in Lie(X) if any composition in S is trivial modulo Sand corresponding w.

(XVII) Let s ∈ Lie(X) be monic and (asb) a normal S-word. Then (asb) = asb +∑i αi(aisbi), where each aisbi < asb.

A proof of (XVII) follows from CD-lemma for associative algebras since s is anassociative GS basis by (I).

(XVIII) Let f, g be monic Lie polynomials. Then

〈f, g〉w − (f, g)w ≡ass 0 mod(f, g, w).

Proof. If 〈f, g〉w and (f, g)w are compositions of intersection, where w = f b = ag, then,by (XII) and (XIII), we may assume that

〈f, g〉w = [fb]f − [ag]g = fb+∑

I1

αiaifbi − ag −∑

I2

βjajgbj ,

where aif bi, aj gbj < fb = ag = w. It follows that 〈f, g〉w− (f, g)w ≡ass 0 mod(f, g, w).Similarly, for the case of the compositions of inclusion, we have the same conclusion.

Theorem 4.7 ([49, 50]) Let S ⊂ Lie(X) ⊂ k〈X〉 be a nonempty set of monic Liepolynomials. Then S is a Grobner-Shirshov basis in Lie(X) if and only if S is a Grobner-Shirshov basis in k〈X〉.

Proof. Note that, by the definitions, for any f, g ∈ S, they have composition in Lie(X)if and only if so do in k〈X〉.

Suppose that S is a GS basis in Lie(X). Then, for any composition 〈f, g〉w, we have〈f, g〉w =

∑I1

αi(aisibi), where (aisibi) are normal S-words and aisibi < w. By (XVII),

〈f, g〉w =∑I2

βjcjsjdj , where each cj sjdj < w. Thus, by (XVIII), we get (f, g)w ≡ass

0 mod(S, w). So, S is a GS basis in k〈X〉.

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Conversely, assume that S is a GS basis in k〈X〉. Then by CD-lemma for associativealgebras, 〈f, g〉w = asb < w for some a, b ∈ X∗, s ∈ S. Then h = 〈f, g〉w − α[asb]s ∈Idass(S), h < 〈f, g〉w and h is a Lie polynomial. By induction on 〈f, g〉w, we have〈f, g〉w ≡Lie 0 mod(S, w).

Theorem 4.8 (CD-lemma for Lie algebras over a field) Let S ⊂ Lie(X) ⊂ k〈X〉be nonempty set of monic Lie polynomials. Let Id(S) be the ideal of Lie(X) generated byS. Then the following statements are equivalent.

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

(ii) f ∈ Id(S)⇒ f = asb for some s ∈ S and a, b ∈ X∗.

(iii) Irr(S) = [u] ∈ NLSW (X) | u 6= asb, s ∈ S, a, b ∈ X∗ is a linear basis forLie(X|S).

Proof. (i)⇒ (ii). Denote Idass(S) and IdLie(S) the ideals of k〈X〉 and Lie(X) generatedby S respectively. By noting that IdLie(S) ⊆ Idass(S) and by using Theorem 4.7 and CD-lemma for associative algebras, the result follows.

(ii) ⇒ (iii). Suppose∑

αi[ui] = 0 in Lie(X|S) where each [ui] ∈ Irr(S) and u1 >u2 > · · · , that is,

∑αi[ui] ∈ IdLie(S). Then each αi must be 0. Otherwise, say, α1 6= 0.

Then∑

αi[ui] = u1 and by (ii), [u1] 6∈ Irr(S), a contradiction. On the other hand, bythe following (XIX), as a vector space, Irr(S) generates Lie(X|S).

The following property is similar to Lemma 2.2.

(XIX) For any S ⊂ Lie(X), f ∈ Lie(X), f has a presentation

f =∑

αi[ui] +∑

βj [ajsjbj ]sj ,

where each αi, βj ∈ k, [ui] ∈ Irr(S), [ui] ≤ f, [ajsjbj ]sj special normal S-word and

[ajsjbj ]sj ≤ f .

(iii)⇒ (i). This part follows from (XIX).

(XX) Let (asb) be a normal s-word and w = asb. Then (asb) ≡ [asb]s mod(s, w). Itfollows that h ≡Lie 0 mod(S, w) is equivalent to h =

∑i αi[aisibi]si , where [aisibi]si are

special normal S-words and aisibi < w.

Proof. By noting that for any monic Lie polynomial s, s is a GS basis in Lie(X).Now, by (XIV) and CD-lemma for Lie algebras, we have (asb) ≡ [asb]s mod(s, w).

Remark: Both associative and Lie CD-lemmas are valid if we replace the base field k byan arbitrary commutative ring K with identity since we assume that all GS bases consistof monic polynomials. In fact, by the same reason, all CD-lemmas in this survey are validif we replace the base field k by an arbitrary commutative ring K with identity. If this isthe case, then the (iii) in a CD-lemma should be: K(X|S) is a free K-module with basisIrr(S). But in general case, Shirshov algorithm does not work.

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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 4.9 Let n > 2 be an integer number. The Drinfeld-Kohno Lie algebra Ln

over Z is defined by generators tij = tji for distinct indices 1 ≤ i, j ≤ n−1, and relations[tijtkl] = 0, [tij(tik + tjk)] = 0, where i, j, k, l are distinct.

Then Ln has a presentation Ln = LieZ(T |S), where T = tij| 1 ≤ i < j ≤ n − 1 andS consists of the following relations

[tijtkl] = 0, if k < i < j, k < l, l 6= i, j, (18)

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

[tjktik]− [tiktij ] = 0, if i < j < k. (20)

Now we order T : tij < tkl if either i < k or (i = k and j < l). Let < be the deg-lexorder on T ∗.

Theorem 4.10 ([71]) Let S = (18), (19), (20) as before, < the deg-lex order on T ∗.Then S is a Grobner-Shirshov basis for Ln.

Corollary 4.11 A linear basis of the Drinfeld-Kohno Lie algebra Ln is

Irr(S) = [tik1tik2 · · · tikm ] | tik1tik2 · · · tikm is an ALSW in T ∗, m ∈ N.

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

A.A. Markov [138], E. Post [158], A. Turing [185], P.S. Novikov [149] and W.W. Boone [53]constructed finitely presented semigroups and groups with the undecidable word problem.This result also follows from the Higman theorem [101] that any recursive presented groupis embeddable into finitely presented group. A weak analogy of Higman theorem for Liealgebras was proved in [15] that was enough for existence of a finitely presented Lie algebrawith the undecidable word problem. In this section, we will give a Kukin’s construction[123] of a Lie algebra AP for any semigroup P such that if P has the undecidable wordproblem then AP has the same property.

Let P = sgp〈x, y|ui = vi, i ∈ I〉 be a semigroup. Consider the Lie algebra

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

where S consists of the following relations:

(1) [xx] = 0, [xy] = 0, [yx] = 0, [yy] = 0,

(2) [xz] = −[zx], [yz] = −[zy],

(3) ⌊zui⌋ = ⌊zvi⌋, i ∈ I.

Here, ⌊zu⌋ means the left normed bracketing.

Let the order x > y > z > x > y and > the deg-lex order on x, y, x, y, z∗. Let ρ bethe congruence on x, y∗ generated by (ui, vi), i ∈ I. Let

40

(3)′ ⌊zu⌋ = ⌊zv⌋, (u, v) ∈ ρ with u > v.

Lemma 4.12 ([71]) With the above notation, the set S1 = (1), (2), (3)′ is a GS basisin Lie(x, y, x, y, z).

Proof: For any u ∈ x, y∗, by induction on |u|, ⌊zu⌋ = zu. All possible compositions inS1 are intersection of (2) and (3)′, and inclusion of (3)′ and (3)′.

For (2) ∧ (3)′, w = xzu, (u, v) ∈ ρ, u > v, f = [xz] + [zx], g = ⌊zu⌋ − ⌊zv⌋. We have

〈[xz] + [zx], ⌊zu⌋ − ⌊zv⌋〉w = [fu]f − [xg]g

≡ ⌊([xz] + [zx])u⌋ − [x(⌊zu⌋ − ⌊zv⌋)]

≡ ⌊zxu⌋ + ⌊xzv⌋ ≡ ⌊zxu⌋ − ⌊zxv⌋ ≡ 0 mod(S1, w).

For (3)′ ∧ (3)′, w = zu1 = zu2e, e ∈ x, y∗, (ui, vi) ∈ ρ, ui > vi, i = 1, 2. We have

〈⌊zu1⌋ − ⌊zv1⌋, ⌊zu2⌋ − ⌊zv2⌋〉w ≡ (⌊zu1⌋ − ⌊zv1⌋)− ⌊(⌊zu2⌋ − ⌊zv2⌋)e⌋

≡ ⌊⌊zv2⌋e⌋ − ⌊zv1⌋ ≡ ⌊zv2e⌋ − ⌊zv1⌋ ≡ 0 mod(S1, w).

Thus, the set S1 = (1), (2), (3)′ is a GS basis in Lie(x, y, x, y, z).

Corollary 4.13 (Kukin [123]) Let u, v ∈ x, y∗. Then

u = v in the semigroup P ⇔ ⌊zu⌋ = ⌊zv⌋ in the Lie algebra AP .

Proof: Suppose that u = v in the semigroup P . Without loss of generality,we mayassume that u = au1b, v = av1b for some a, b ∈ x, y∗ and (u1, v1) ∈ ρ. For any r ∈x, y, by the relations (1), we have [xr] = 0 and so ⌊zxc⌋ = ⌊[zx]c⌋ = [⌊zc⌋x], ⌊zyc⌋ =[⌊zc⌋y] for any c ∈ x, y∗. From this it follows that in AP , ⌊zu⌋ = ⌊zau1b⌋ = ⌊⌊zau1⌋b⌋ =

⌊⌊zu1←−a ⌋b⌋ = ⌊zu1

←−a b⌋ = ⌊zv1←−a b⌋ = ⌊zav1b⌋ = ⌊zv⌋, where for any xi1xi2 · · ·xin ∈

x, y∗, ←−−−−−−−−xi1xi2 · · ·xin =: xinxin−1 · · ·xi1 and xi1xi2 · · ·xin =: xi1 xi2 · · · xin . Moreover, (3)′

holds in AP .

Suppose that ⌊zu⌋ = ⌊zv⌋ in the Lie algebra AP . Then both ⌊zu⌋ and ⌊zv⌋ have thesame normal form in AP . Since S1 is a GS basis in AP , both ⌊zu⌋ and ⌊zv⌋ can be reducedto the same normal form of the form ⌊zc⌋ for some c ∈ x, y∗ only by the relations (3)′.This implies that in P , u = c = v.

By the above corollary, if the semigroup P has the undecidable word problem then sodoes the Lie algebra AP .

4.3 Composition-Diamond lemma for Lie algebras over commu-

tative algebras

Let X = xi|i ∈ I be a well-ordered set, consider Lie(X) ⊂ k〈X〉 the free Lie algebraunder the Lie bracket [u, v] = uv − vu.

Let Y = yj|j ∈ J be a well-ordered set. Then [Y ], the free commutative monoidgenerated by Y , is a linear basis of k[Y ].

41

Regard Liek[Y ](X) ∼= k[Y ] ⊗ Lie(X) as the Lie subalgebra of k[Y ]〈X〉 ∼= k[Y ] ⊗ k〈X〉the free associative algebra over polynomial algebra k[Y ], which is generated by X underthe Lie bracket [u, v] = uv − vu.

Then NLSW (X) forms a k[Y ]-basis of Liek[Y ](X).

Denote by [Y ]X∗ = βt|β ∈ [Y ], t ∈ X∗. For u = βt ∈ [Y ]X∗, denote uX = t, uY = β.

Order [Y ], X∗ by deg-lex order >Y, >

Xrespectively. Now define an order > on [Y ]X∗:

for u, v ∈ [Y ]X∗,

u > v if (uX >XvX) or (uX = vX and uY >

YvY ).

For any f ∈ Liek[Y ](X), f has an expression f =∑

αiβi[ui], where αi ∈ k, βi ∈ [Y ]and [ui] ∈ NSLW (X).

Then consider f =∑

αiβi[ui] =∑

gj(Y )[uj], gj(Y ) ∈ k[Y ] as a polynomial in k-algebra k[Y ]〈X〉, the leading word f of f in k[Y ]〈X〉 is of the form βu, where β ∈[Y ], u ∈ ALSW (X). f is called monic (k-monic) if the coefficient of f is equal to 1. f iscalled k[Y ]-monic defined similarly.

Remind that for any ALSW w, there is a unique bracketing way such that [w] is aNLSW.

Let S ⊂ Liek[Y ](X) be a monic subset, s ∈ S and a, b ∈ X∗.

Let (u) be a non-associative words on X with a fixed occurence of some xi and s ∈ S.Then (u)xi 7→s is called an S-word. |u| is also the s-length of (u)xi 7→s.

Each S-word has a form of (asb), a, b ∈ X∗, s ∈ S.

If asXb ∈ ALSW (X), then we have the special bracketing [asXb]sX of asXb relative tosX . We define [asb]s = [asXb]sX |[sX ] 7→s to be a special normal s-word (or special normalS-word).

An S-word (u) = (asb) is normal s-word, denoted by ⌊u⌋s, if (asb)X

= asXb. Thefollowing condition is a sufficient one.

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

(ii) If ⌊u⌋s is normal S-word with s-length k and [v] ∈ NLSW (X) such that |v| = l,

then [v]⌊u⌋s when v > ⌊u⌋X

s and ⌊u⌋s[v] when v < ⌊u⌋X

s are normal S-words ofs-length k + l.

Let f, g be monic polynomials of Liek[Y ](X) and L = lcm(fY , gY ).

There are four kinds of compositions.

C1) Inclusion composition. If fX = agXb for some a, b ∈ X∗, then C1〈f, g〉w = LfY f −

LgY[agb]g (w = LfX = LagXb).

C2) Intersection composition. If fX = aa0, gX = a0b, a, b, a0 6= 1, then C2〈f, g〉w =

LfY [fb]f −

LgY[ag]g (w = LfXb = LagX).

C3) External composition. If gcd(fY , gY ) 6= 1, then for any a, b, c ∈ X∗ such that w =LafXbgXc ∈ TA = βt|β ∈ [Y ], t ∈ ALSW (X), C3〈f, g〉w = L

fY [afbgXc]f−

LgY[afXbgc]g.

C4) Multiplication composition. If fY 6= 1, then for any special normal f -word[afb]f (a, b ∈ X∗), C4〈f〉w = [afXb][afb]f (w = afXbaf b).

42

Given a k-monic subset S ⊂ Liek[Y ](X) and w ∈ [Y ]X∗ (not necessary in TA), anelement h ∈ Liek[Y ](X) is called trivial modulo (S, w) if h can be presented as a k[Y ]-linearcombination of normal S-words with leading words less than w. S is a Grobner-Shirshovbasis in Liek[Y ](X) if all the possible compositions of elements in S are trivial modulo Sand corresponding w.

Theorem 4.14 ([24] CD-lemma for Lie algebras over commutative algebras) Let S ⊂Liek[Y ](X) be nonempty set of monic polynomials and Id(S) be the ideal of Liek[Y ](X)generated by S. Then the following statements are equivalent.

(i) S is a Grobner-Shirshov basis in Liek[Y ](X).

(ii) f ∈ Id(S)⇒ f = asb ∈ TA for some s ∈ S and a, b ∈ [Y ]X∗.

(iii) Irr(S) = [u] | [u] ∈ TN , u 6= asb, for any s ∈ S, a, b ∈ [Y ]X∗ is a linear basisfor Liek[Y ](X|S) = (Liek[Y ](X))/Id(S).

Here, TA = βt | β ∈ [Y ], t ∈ ALSW (X), TN = β[t] | β ∈ [Y ], [t] ∈NLSW (X).

Let (L, [ ]) = LieK(X|S) be an arbitrary Lie algebra over the commutative algebraK = k[Y |R]. Then U(L) = K〈X|S(−)〉 = k[Y ]〈X|S(−), RX〉, where S(−) is just S butsubstitute all [uv] by uv − vu, is the universal enveloping associative algebra of L.

A Lie algebra L over a commutative algebra K is called special if it is embeddable intoits universal enveloping associative algebra. Otherwise it is called non-special.

There are known classical examples by A.I. Shirshov (1953) and P. Cartier (1958)of non-special Lie algebras over commutative algebras over GF (2), which were provedusing ad hoc methods. P.M. Cohn (1963) suggested another non-special Lie algebra overcommutative algebras over a field of characteristic p > 0.

Example 4.15 (Shirshov (1953)) Let k = GF (2) and K = k[yi, i = 0, 1, 2, 3|y0yi =yi (i = 0, 1, 2, 3), yiyj = 0 (i, j 6= 0)]. Let L = LieK(xi, 1 ≤ i ≤ 13|S1, S2), where

S1 = [x2x1] = x11, [x3x1] = x13, [x3x2] = x12,

[x5x3] = [x6x2] = [x8x1] = x10, [xixj ] = 0 (i > j);

S2 = y0xi = xi (i = 1, 2, . . . , 13),

y1x1 = x4, y1x2 = x5, y1x3 = x6, y1x12 = x10,

y2x1 = x5, y2x2 = x7, y2x3 = x8, y2x13 = x10,

y3x1 = x6, y3x2 = x8, y3x3 = x9, y3x11 = x10,

y1xk = 0 (k = 4, 5, . . . , 11, 13),

y2xt = 0 (t = 4, 5, . . . , 12),

y3xl = 0 (l = 4, 5, . . . , 10, 12, 13).

Then L = LieK(X|S1, S2) = Liek[Y ](X|S1, S2, RX) and S = S1 ∪ S2 ∪ RX ∪ y1x2 =y2x1, y1x3 = y3x1, y2x3 = y3x2 is a GS basis in Liek[Y ](X), which implies x10 is in thelinear basis of L by Theorem 4.14, i.e. x10 6= 0 in L.

43

On the other hand, the universal enveloping algebra of L has a presentation:

UK(L) = K〈X|S(−)1 , S2〉 ∼= k[Y ]〈X|S

(−)1 , S2, RX〉.

But the GS completion (see Mikhalev-Zolotyhk [146]) of S(−)1 ∪ S2 ∪RX in k[Y ]〈X〉 is

SC = S(−)1 ∪ S2 ∪ RX ∪ y1x2 = y2x1, y1x3 = y3x1, y2x3 = y3x2, x10 = 0.

Thus, L is not special.

Example 4.16 (Cartier (1958)) Let k = GF (2), K = k[y1, y2, y3|y2i = 0, i = 1, 2, 3] and

L = LieK(xij , 1 ≤ i ≤ j ≤ 3|S), where S = [xiixjj] = xji (i > j), [xijxkl] = 0 y3x33 =y2x22 + y1x11. Then L is not special over K.

Proof. S ′ = S ∪ y2i xkl = 0 (∀i, k, l) ∪ S1 is a GS basis in Liek[Y ](X), where

S1 = y3x23 = y1x12, y3x13 = y2x12, y2x23 = y1x13, y3y2x22 = y3y1x11,

y3y1x12 = 0, y3y2x12 = 0, y3y2y1x11 = 0, y2y1x13 = 0.

Then, y2y1x12 ∈ Irr(S ′), so y2y1x12 6= 0 in L.

But in UK(L) = K〈X|S(−)〉 ∼= k[Y ]〈X|S(−), y2i xkl = 0 (∀i, k, l)〉, we have

0 = y23x233 = (y2x22 + y1x11)

2 = y22x222 + y21x

211 + y2y1[x22, x11] = y2y1x12.

Thus, L 6→ UK(L).

Conjecture (Cohn [85]) Let K = k[y1, y2, y3|ypi = 0, i = 1, 2, 3] be an algebra of truncated

polynomials over a field k of characteristic p > 0. Let

Lp = LieK(x1, x2, x3 | y3x3 = y2x2 + y1x1).

Then Lp is not special. Lp is called Cohn’s Lie algebras.

In UK(Lp) we have

0 = (y3x3)p = (y2x2)

p + Λp(y2x2, y1x1) + (y1x1)p = Λp(y2x2, y1x1),

where Λp is a Jacobson-Zassenhaus Lie polynomial. P.M. Cohn conjectured that Λp(y2x2, y1x1) 6=0 in Lp. To prove it, we need to know a GS basis of Lp up to degree p in X . We did itfor p = 2, 3, 5. For example, Λ2 = [y2x2, y1x1] = y2y1[x2x1] and a GS basis of L2 up todegree 2 in X is

y3x3 = y2x2 + y1x1, y2i xj = 0 (1 ≤ i, j ≤ 3), y3y2x2 = y3y1x1, y3y2y1x1 = 0,

y2[x3x2] = y1[x3x1], y3y1[x2x1] = 0, y2y1[x3x1] = 0.

So, y2y1[x2x1] ∈ Irr(SC).

The same but much longer computations show that Λ3 6= 0 in L3 and Λ5 6= 0 in L5.Thus, we have the following theorem.

Theorem 4.17 ([24]) Cohn’s Lie algebras L2, L3 and L5 are non-special.

44

Theorem 4.18 ([24]) For an arbitrary commutative k-algebra K = k[Y |R], if S is aGrobner-Shirshov basis in Liek[Y ](X) such that for any s ∈ S, s is k[Y ]-monic, thenL = LieK(X|S) is special.

Corollary 4.19 ([24]) Any Lie K-algebra LK = LieK(X|f) with one monic definingrelation f = 0 is special.

Theorem 4.20 ([24]) Suppose that S is a finite homogeneous subset of Liek(X). Thenthe word problem of LieK(X|S) is solvable for any finitely generated commutative k-algebraK.

Theorem 4.21 ([24]) Every finitely or countably generated Lie K-algebra can be embed-ded into a two-generated Lie K-algebra, where K is an arbitrary commutative k-algebra.

5 Grobner-Shirshov bases for Ω-algebras and beyond

Up to now, some new CD-lemmas for Ω-algebras: associative conformal algebras [39] andn-conformal algebras [37]; tensor product of free algebras [23]; matabelian Lie algebras[67]; associative Ω-algebras [35]; color Lie superalgebras and Lie p-superalgebras [141,142]; Lie superalgebras [143], standard bases in linear structures [126, 127]; conformalLie algebras [80]; associative differential algebras [68]; associative Rota-Baxter algebras[26]; L-algebras [27]; dialgebras [32]; pre-Lie algebras [28]; semirings [34]; operads [88];commutative intergro-differential algebras [90]; differential type operators [97]; difference-differential modules and difference-differential dimension polynomials [198]; λ-differentialassociative Ω-algebras [160]; commutative associative Rota-Baxter algebras [161]; and soon.

Let us formulate CD-lemma for pre-Lie algebras, see [28].

A non-associative A is called a pre-Lie (or right-symmetric) algebra if A satisfies thefollowing identity (x, y, z) = (x, z, y) for the associator (x, y, z) = (xy)z − x(yz). It is aLie admissible algebra in a sense that A(−) = (A, [xy] = xy − yx) is a Lie algebra.

Let X = xi|i ∈ I be a well-ordered set. We order X∗∗ by induction on the lengths ofthe words (u) and (v):

(i) If |((u)(v))| = 2, then (u) = xi > (v) = xj if and only if i > j.

(ii) If |((u)(v))| > 2, then (u) > (v) if and only if one of the following cases holds:

(a) |(u)| > |(v)|.

(b) If |(u)| = |(v)|, (u) = ((u1)(u2)) and (v) = ((v1)(v2)), then (u1) > (v1) or((u1) = (v1) and (u2) > (v2)).

We now cite the definition of good words (see [172]) by induction on length:

1) x is a good word for any x ∈ X .

2) non-associative word ((v)(w)) is called a good word if

45

(a) both (v) and (w) are good words,

(b) if (v) = ((v1)(v2)), then (v2) ≤ (w).

We denote (u) by [u], if (u) is a good word.

Let W be the set of all good words in the alphabet X and RS〈X〉 the free right-symmetric algebra over a field k generated by X . Then W forms a linear basis of thefree pre-Lie algebra RS〈X〉, see [172]. Daniyar Kozybaev, Leonid Makar-Limanov, UalbaiUmirbaev [122] has proved that the deg-lex order on W is monomial.

Let S ⊂ RS〈X〉 be a set of monic polynomials and s ∈ S. An S-word (u)s is called anormal S-word if (u)s = (asb) is a good word.

Let f, g ∈ S, [w] ∈ W and a, b ∈ X∗. Then there are two kinds of compositions.

(i) If f = [agb], then (f, g)f = f − [agb] is called composition of inclusion.

(ii) If (f [w]) is not good, then f · [w] is called composition of right multiplication.

Theorem 5.1 ([28] CD-lemma for right-symmetry algebras) Let S ⊂ RS〈X〉 be anonempty set of monic polynomials and the order < be defined as before. Then the fol-lowing statements are equivalent.

(i) S is a Grobner-Shirshov basis in RS〈X〉.

(ii) f ∈ Id(S) ⇒ f = [asb] for some s ∈ S and a, b ∈ X∗, where [asb] is a normalS-word.

(iii) Irr(S) = [u] ∈ W |[u] 6= [asb] a, b ∈ X∗, s ∈ S and [asb] is a normal S-word is alinear basis of the algebra RS〈X|S〉 = RS〈X〉/Id(S).

As an application, we have a GS basis for universal enveloping right-symmetric algebraof a Lie algebra.

Theorem 5.2 ([28]) Let (L, [ ]) be a Lie algebra with a well-ordered linear basis X =ei| i ∈ I. Let [eiej] =

∑m

αmij em, where αm

ij ∈ k. We denote∑m

αmij em by eiej. Let

U(L) = RS〈eiI | eiej − ejei = eiej, i, j ∈ I〉

be the universal enveloping right-symmetric algebra of L. Let

S = fij = eiej − ejei − eiej, i, j ∈ I and i > j.

Then the set S is a Grobner-Shirshov basis in RS〈X〉.

By Theorems 5.1 and 5.2, we immediately have the following PBW theorem for Liealgebra and right-symmetric algebra.

Corollary 5.3 ([172]) A Lie algebra L can be embedded into its universal envelopingright-symmetric algebra U(L) as a subalgebra of U(L)(−).

46

Recently, by using CD-lemmas mentioned before and some others combinatorial meth-ods, some applications are given: for Novikov’s and Boone’s group [11, 12, 69]; forCoxeter groups [51, 129]; for Leavitt path algebras [1]; for center-by-metabelian Lie al-gebras [188]; for free metanilpotent Lie algebras, Lie algebras and associative algebras[98, 144, 189, 190]; for Poisson algebras [136]; for quantum Lie algebras and related prob-lems [115, 118]; for PBW-bases [114, 117, 135]; for extensions of groups and associativealgebras [65, 66]; for (color) Lie (p)- superalgebras [5, 41, 81, 82, 93, 94, 95, 145, 200, 201];for Hecke algebras and Specht modules [111]; for Ariki-Koike algebras [112]; for linearalgebraic approach [113]; for HNN groups [77]; for some one-relator groups [78]; for em-beddings of algebras [33, 73]; for free partially commutative Lie algebras [74, 157]; forquantum groups of type Dn, E6, G2 [150, 163, 195, 196]; etc.

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