GRöBNER-SHIRSHOV BASES AND PBW THEOREMS Текст научной статьи по специальности «Математика»

УДК 512.5

Grobner-Shirshov Bases and PBW Theorems

Leonid A. Bokut*

Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P. R. China Yuqun Chen^

School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P. R. China

Received 18.09.2013, received in revised form 02.10.2013, accepted 04.11.2013 We review some applications of Grobner-Shirshov bases, including PBW theorems, linear bases of free universal algebras, normal forms for groups and semigroups, extensions of groups and algebras, embedding of algebras.

Keywords: Grobner-Shirshov basis, Composition-Diamond lemma, PBW theorem, normal form, group, semigroup, extension.

Introduction

Combinatorial algebra seminar at South China Normal University was organized by the authors in March 2006. Since then, there were some 30 Master Theses and 4 PhD Theses, about 40 published papers in JA, IJAC, PAMS, JPAA, Comm. Algebra, Algebra Coll., Siberian Math. J., Science in China and other Journals and Proceedings. There were organized 2 International Conferences (2007, 2009) with E. Zelmanov as Chairman of the Program Committee and several Workshops. We are going to review some of the papers.

Our main topic is Grobner-Shirshov bases method for different varieties (categories) of linear (Q-) algebras over a field k or a commutative algebra K over k: associative algebras (including group (semigroup) algebras), Lie algebras, dialgebras, conformal algebras, pre-Lie (Vinberg right (left) symmetric) algebras, Rota-Baxter algebras, metabelian Lie algebras, L-algebras, semiring algebras, category algebras, etc. There are some applications particularly to new proofs of some known theorems.

As it is well known, Grobner-Shirshov (GS for short) bases method for a class of algebras based on a Composition-Diamond lemma for the class. A general form of a Composition-Diamond Lemma over a field k is as follows.

Composition-Diamond lemma for a class of algebras Let M(X) be a free algebra of a category M of algebras over a field k, (N(X), <) a linear basis (normal words) of M(X) with an "admissible" well order and S C M(X). Let Id(S) be the ideal of M(X) generated by S and s the leading term of the polynomial s. Then the following statements are equivalent.

* [email protected] [email protected] © Siberian Federal University. All rights reserved

(i) S is a GS basis (i.e., each "composition" of polynomials from S is "trivial").

(ii) If f G Id(S), then the maximal word of f has a form f = (asb), s G S, a,b G N(X).

(iii) Irr(S) = {u G N(X)|u = (asb),s G S,a,b G N(X)} is a linear basis of M(X|S) = M (X )/Id(S).

There are two kinds of compositions, a composition (si, s2)w of two monic polynomials relative to w = "lcm"(si, s2) G N(X) and a composition Comw (s) of one polynomial relative to w = (vs) or w = (sv), v, w G N(X). Namely, for monic polynomials si, s2,

(si, s2)w = lcm(si, s2)k^Si - lcm(si, ^2,

and Comw (s) = vs or sv correspondingly.

For example, for words si = u, s2 = v G X*, lcm(u,v) G {ucv, c G X* (a trivial lcm); u = avb, a,b G X* (an inclusion lcm);ub = av, a,b G X*, deg(ub) < deg(u) + deg(v) (an intersection lcm)}. In these cases, (si,s2)w = sics2 — sics2, si — as2b, sib — as2 correspondingly.

For algebras over a field, a composition (si,s2)w relative to a trivial lcm w is trivial mod(si, s2; w).

A polynomial f is trivial mod(S, w) if f is a linear combination of "normal S-words" (asb), s G S, a,b G N(X), such that (asb) = (asb) < w.

The statement (i) ^ (ii) is the main statement of a Composition-Diamond lemma since others are much easier to prove. For two known cases, conformal algebras and dialgebras, (i) and (ii) are not equivalent.

The case of Lie algebras is an exception since s / N(X), the maximal word of a Lie polynomial s is the maximal word of s as associative polynomial after working out all Lie brackets.

For algebras over a commutative algebra K one needs to deal with a "double free" algebra Mk[Y] (X), a free M-algebra over a polynomial algebra. In this case

lcm(uY uX ,vY vX) = lcm(uY ,vY )lcm(uX ,vX)

and there are generally infinitely many compositions for given si, s2. It is since we need to use a "trivial" lcm(uX,vX) = (uX)cX(vX). For algebras over a field, the composition corresponding to a trivial (uX)cX(vX) is trivial. But for algebras over a commutative algebra it is not the case if lcm(uY,vY) = uYvY.

Recently some new Composition-Diamond lemmas are given: for tensor product free algebras k(X) <g> k(Y) [13], for Lie algebras over commutative algebras [14], for metabelian Lie algebras [42], for semirings [24], for Rota-Baxter algebras [16], for L-algebras [17], for Vinberg-Koszul-Gerstenhaber right-symmetric (pre-Lie) algebras [18], for categories [20], for dialgebras [22], for associative algebras with multiple operations [25], for associative n-conformal algebras [27], for associative conformal algebras [29], for Lie superalgebras [31], for differential algebras [43], for A-differential associative algebras with multiple operators [60], for commutative algebras with multiple operators and free commutative Rota-Baxter algebras [61], etc.

By using the above and the known Composition-Diamond lemmas, some applications are obtained: for embeddings of algebras [23, 52], for free inverse semigroups [28], for conformal algebras [30], for relative Grobner-Shirshov bases of algebras and groups [36], for extensions of groups and algebras [40,41], for some word problems [10,44], for some Lie algebras [47], for partially commutative Lie algebras [50,59], for braid groups [7-9,48,56], for PBW theorems [14,18,22,35,45,46,49], etc.

For development of Grobner-Shirshov bases, one may refer to surveys: [12,26,32-34].

1. Grobner-Shirshov bases for associative algebras and Lie algebras

Grobner-Shirshov bases for Lie algebras is established by Shirshov [63,64] for the free Lie algebras (with deg-lex order) in 1962 (see also Bokut [5]). In 1976, Bokut [6] specialized the approach of Shirshov to associative algebras (see also Bergman [1]). For commutative polynomials, this is due to Buchberger [37,38].

1.1. Composition-Diamond lemma for associative algebras

Let k(X) be the free associative algebra over a field k generated by X and (X*, <) a well-ordered free monoid generated by X, S C k(X) such that every s G S is monic (s is monic if the coefficient of the leading word of s is 1).

Let us prove (i) ^ (ii) and define a GS basis.

n

Let f aiaisibi G Id(S) where each ai G k, ai,bi G X*, si G S. Let wi = ai~sibi, wi = i=1

W2 = • • • = Wl > Wl + 1 > • • •.

For l = 1, it is ok.

For l > 1, w1 = a111b1 = a2s2b2, common multiple of s1 by definition,

wi = cwd, w = "lcm" (~s1,~s2), aisibi = w|Si^Si, i = 1, 2,

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

Then a1s1b1 — a2s2b2 = ^w^^^ — )d = c(s1,s2)wd. By definition of GS basis,

(s1, s2)w = 0 mod(S, w). So, a^b1 — a2s2b2 = 0 mod(S, w1). We can decrease l. By induction on l and w1, f = asb, a,b G X*, s G S.

1.2. Composition-Diamond lemma for Lie algebras over a field

Let S C Lie(X) C k(X) be a nonempty set of monic Lie polynomials, (X*, <) deg-lex order, s means the maximal word of s as non-commutative polynomial. Then compositions are defined as follows

(s1, s2)w = Msr^si]sT — [w|s2^S2 W, w G ALSW(X)

associative composition with the special Shirshov bracketing, where ALSW(X) is the set of associative Lyndon-Shirshov words on X.

Composition-Diamond lemma for Lie algebras over a field ( [11,63,64]). The following statements are equivalent.

(i) S is a Lie GS basis in Lie(X) (any composition is trivial modulo (S,w)).

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

(iii) Irr(S) = {[u] G NLSW (X) | u = asb, s G S, a,b G X *} is a linear basis for Lie(X |S).

2. PBW theorems

There are 8 PBW (Poincare-Birkhoff-Witt) theorems that are proved by using GS bases and Composition-Diamond lemmas.

2.1. Lie algebras—associative algebras (Shirshov)

Let L = Liek (X|S) be an arbitrary Lie algebra with generators X and defining relations S and U(L) = k(X|S(-)) the universal enveloping associative algebra of L. Then

(i) S is a Lie GS basis S is an associative GS basis.

(ii) In this case, a linear basis of U(L) is

utu2 • • • ut, ut < u2 < • • • < ut (lex-order), ui G Irr(S) n ALSW(X).

One uses Shirshov factorization theorem:

u G X*, 3! u = ut ••• ut, ui < • •• < ut, ui G ALSW(X).

2.2. Lie algebras—pre-Lie algebras (Segal)

Pre-Lie algebras are defined by an identity (x, y, z) = (x, z, y), where (x, y, z) = (xy)z — x(yz).

Let X = {xili G I} be a linear basis of the Lie algebra L and [xi, xj] = ^ ajxt =: {xi, xj} the multiplication table of the linear basis X. Then L has a presentation

L = Lie(xi, i G Il[xi, xj] = {xi, xj}, i, j G I) = Lie(X|S)

and the universal enveloping pre-Lie algebra of L

Upre-Lie (L)= pre-Lie (X |S(-)).

Then Sis a GS basis of Upre-Lie(L), L c Upre-Lie(L) and Irr(S) is a linear basis of Upre-Lie(L) by Composition-Diamond lemma for pre-Lie algebras (Bokut-Chen-Li [18]).

2.3. Leibniz algebras—dialgebras (Aymon, Grivel)

Dialgebra: a H (b h c) = a H b H c, (a H b) h c = a h b h c, a h (b H c) = (a h b) H c and H associative.

Leibniz identity: [[a, b],c] = [[a, c],b] + [a, [b, c]]. Di-commutator: [a, b] = a H b — b h a.

L = Lei(xi, i G I\[xi,xj] = {xi,xj}, i,j G I), UDialg (L) = D(X |S(-)).

A GS basis for UDialg (L) is given by Bokut-Chen-Liu [22] and then a linear basis for UDialg (L) by Composition-Diamond lemma for dialgebras which implies L c UDialg (L).

2.4. Akivis algebras—non-associative algebras (Shestekov)

Any nonassociative algebra is an Akivis algebra relative the commutator [x, y] = xy — yx and the associator (x, y, z) = (xy)z — x(yz).

Akivis identity: [[x, y], z] + [[y, z], x] + [[z, x],y] = (x, y, z) + (z, x, y) + (y, z, x) — (x, z, y) — (y, x, z) — (z, y, x).

A = A(xi, i G 11 [xi, xj] = {xi, xj}, (xi, xj, xt) = {xi, xj, xt}, i, j, t G I), U (A) = k{X |S(-)},

S= {[xi,xj] = {xi, xj}, (xi,xj,xt) = {xi, xj, xt}, i, j, t G I}. A GS basis of U(A) is given by Chen-Li [46] and then A c U(A).

2.5. Sabinin algebras—modules (Perez-Izquierdo)

Let (V, (;)) be a Sabinin algebra,

S(V) = T (V )/span(xaby — xbay + x(1)(x(2); a,b)y\x,y G T (V ),a,b G V) = mod(XII)k(x) as k(X)-modules

the universal enveloping module for V, where I = {xab — xba + J2 x(1)(x(2); a,b)Ix G X*,a > b, a, b G X}.

Then I is a GS basis (Chen-Chen-Zhong [45]) and then V C S(V).

2.6. Dendriform algebras—Rota-Baxter algebras (Chen-Mo, Kolesnikov)

Rota-Baxter identity:

P(x)P(y) = P(P(x)y) + P(xP(y)), Vx,y G A.

Dendriform identities: (x — y) — z = x — (y ^ z + y y z), (x y y) ^ z = x y (y ^ z), (x ^ y + x y y) y z = x y (y y z).

U(D) = RB(XIxiP(xj) = {xi xj}, P(xi)xj = {xi y xj},xi,xj G X). Then D C U(D), see Chen-Mo [49].

2.7. Shirshov's, Cartier's, Cohn's counter examples to PBW for Lie algebras over commutative algebra

Shirshov [62] 1953 and Cartier [39] 1958 give counter examples to PBW for Lie algebras over commutative algebra. Cohn [57] 1963 posts the conjecture:

Cp = LieK(x1,x2,x3 I y3x3 = y2x2 + y1x1), K = k[y1, y2, y3Iyi = 0,i = 1, 2, 3].

Lp can not be embedded into its universal enveloping associative algebra.

Bokut-Chen-Chen [14] establish GS bases theory for Lie algebras over a commutative algebra. We prove Cohn's conjecture is correct for p = 2, 3, 5.

2.8. "1/2 PBW theorem" (Bokut-Fong-Ke)

Let k be a field with the characteristic 0 and C a k-algebra with operations a(n)b, n > 0, and D(a). Then C is called a conformal algebra, if

(1) (the locality condition). For any a,b G C, there exist N (a, b) ^ 0 such that a(n)b = 0 for n > N (a, b).

(2) D(a(n)b) = D(a)(n)b + a(n)D(b) and D(a)(n)b = —na(n — 1)b.

Let C be a conformal algebra. Then C is an associative conformal algebra if

(As) (a(n)b)(m)c =^( — 1)s C ns a(n — s)(b(m + s)c);

s>0

C is a Lie conformai algebra if

(Lie) (anti-commutativity) (a[n]b) = {b[n]a},

where {b[n]a} = J2(-l)n+sD(s)(b[n + s]a), D(s) = \/s\Ds

s>0

(Jacoby) ((a[n]b)[m]c) = ^2(-l)sCn((a[n - s](b[m + s]c)) - (b[m + s](a[n - s]c))).

s>0

Grobner-Shirshov bases for conformai associative algebras (n-conformal associative algebras) are established in Bokut-Fong-Ke [29] and Bokut-Chen-Zhang [27]. There is a 1/2 PBW-theorem for Lie conformal algebras: Let

L = LieCon({ai, i G I},N|ai[n]aj ^^ akjak, i ^ j, n < N)

be a Lie conformal algebra with the linear basis {ai, i G I} over k[D] and the locality N. Let

U (L) = AsCon({ai, i G I },N |S = {s™j = ai(n)aj — {a,j (n)ai} — a,i[n]a,j, i ^ j, n < N})

be the universal enveloping associative conformal algebra for L.

Then any composition (sn mod(S,w), w = ai(n)a,j(m)ak.

Then any composition (slj,s™k)w = sfj(m)ak — ai(n)s'lJik, i > j > k, n,m < N is trivial

There is also a 1/2 PBW theorem between n-conformal Lie algebras and n-conformal associative algebras, see [27].

3. Linear bases of free universal algebras

-Bases of free Lie algebras

M. Hall and A.I. Shirshov use constructions and check axioms. One may use anti-commutative Grobner-Shirshov bases for a free Lie algebra. Hall basis (Bokut-Chen-Li [19]): Lie(X) = AC(X|Si), Si is a anti-commutative GS basis, Irr(Si) =Hall basis in X.

Lyndon-Shirshov basis (Bokut-Chen-Li [21]): Lie(X) = AC(X|S2), S2 is a anti-commutative GS basis, Irr(S2) =Lyndon-Shirshov basis in X.

-Loday basis of a free dialgebra

D(X) = L(XIS), L-identity: (a h b) 4 c = a h (b 4 c), S an L-GS basis (GS basis as L-algebra), Irr(S) =Loday basis in X (Bokut-Chen-Huang [17]).

-Bases of a free dentriform algebra

Den(X) = L(X|S), Irr(S)=a linear basis of Den(X) (Bokut-Chen-Huang [17]).

-Bases of a free Rota-Baxter algebra (Rota, Cartier) Via GS method for Q-algebras (Bokut-Chen-Qiu [25]).

-Free inverse semigroup (Polyakova-Schain)

An associative GS basis is given by (Bokut-Chen-Zhao [28]), Irr(S) is a normal form of free inverse semigroup.

-Free idempotent semigroup (Chen-Yang [53]).

4. Normal forms for groups and semigroups

-Braid groups

in Artin-Burau generators (Bokut-Chanikov-Shum [9]); in Artin-Garside generators (Bokut [7]); in Birman-Ko-Lee generators (Bokut [8]); in Adyan-Thurston generators (Chen-Zhong [56]).

-Chinese monoid (Chen-Qiu [51])

-Plactic monoid (Bokut-Chen-Chen-Li [15]).

-HNN extension Britton Lemma and Lyndon-Schupp normal form lemma for an HNN-extension of a group was proved using an associative Composition-Diamond lemma relative to a "S-partially" monomial order of words (Chen-Zhong [54]).

-One-relator groups In (Chen-Zhong [55]), some one-relator groups were studying by means of groups with the standard normal forms (the standard GS bases) in the sense (Bokut [3,4]). It is known that any one-relator group can be effectively embedded into 2-generator one-relator group G = gp(x,yIxi1 yj1 .. .xikyjk), k > 1 is the depth. It is proved that a group G of depth < 3 is computably embeddable into a Magnus-Moldavanskii tower of HNN-extensions with the standard normal form of elements. There are quite a lot of examples that support an old conjecture that the result is valid for any depth.

5. Schreier extensions of groups and algebras

In (Chen [40]), it is dealing with a Schreier extension

1 ^ A ^ C ^ B ^ 1

of a group A by B. M. Hall [58] mentioned that for any B it is difficult to find an analogous conditions. Actually this problem was solved in [40] using the GS bases technique. As applications there were given above conditions for cyclic and free abelian cases, as well as for the case of HNN-extensions.

The same kind of result was established for Schreier extensions of associative algebras (Chen [41]).

6. Embedding of algebras

In Bokut-Chen-Mo [23], we were dealing with the problem of embedding of countably generated associative and Lie algebras, groups, semigroups, ^-algebras into (simple) 2-generated ones. We proved some known results (of Higman-Neuman-Neuman, Evance, Malcev, Shirshov) and some new ones using GS bases technique. For example

Theorem 1. Every countable Lie algebra is embeddable into simple 2-generated Lie algebra.

Theorem 2. Every countable differential algebra is embeddable into a simple 2-generated differential algebra.

G. Bergman (Private communication, 2013 [2]) gave an idea how to avoid the restriction on cardinality of the ground field. Now Qiuhui Mo proved that the Bergman's idea works.

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Базисы Грёбнера-Ширшова и PBW-теоремы

Леонид А.Бокуть Ючжун Чень

Статья является обзором некоторых приложений Грёбнера-Ширшова базисов, включая PBW

теоремы, линейные базисы свободных универсальных алгебр, нормальных форм для групп и полугрупп, 'расширения групп и алгебр, вложения алгебр.

Ключевые слова: базис Грёбнева-Ширшова, лемма о композиции-диаманте, PBW-теорема, нормальная форма, группа, полугруппа, расширение.