Gröbner-Shirshov Bases for Universal Algebras*

L A Bokut, Chen Yuqun

(1.School of Mathematical Sciences, South China Normal University, Guangzhou 510631,China; 2.Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk 630090, Russia)

Gröbner bases and Gröbner-Shirshov bases were invented independently by A.I. Shirshov for ideals of free (commutative, anti-commutative) non-associative algebras[1-2], free Lie algebras[2-3]and implicitly free associative algebras[2-3](see also [4-6]), by Hironaka[7]for ideals of the power series algebras (both formal and convergent), and by Buchberger[8]for ideals of the polynomial algebras.

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

Composition-DiamondlemmaLetM(X) be a free algebra generated by a setXof a categoryMof algebras overk, (N, ≤) a linear basis (normal words) ofM(X) with a monomial ordering ≤,S⊂M(X) andId(S) the ideal ofM(X) generated byS. Then the following statements are equivalent:

(a)Sis a Gröbner-Shirshov basis in(X).

In some exceptional cases (dialgebras, conformal algebras) (a)⟹(b)⟺(c), but not (b)⟹(a).

How to establish a CD-lemma for the free algebra(X)? Following the idea of Shirshov, one needs

1) to define appropriate linear basis (normal words)Nof(X);

2) to find a monomial ordering onN;

3) to find normalS-words;

4) to define compositions of elements inS(they may be compositions of intersection, inclusion and left (right) multiplication, or may be else);

5) to prove two key lemmas:

Lemma1LetSbe a GS basis. Then any element ofId(S) is a linear combination of normalS-words.

Let the normal wordsNof the free algebra(X) be a well-ordered set and 0≠f(X). Denote bythe leading word off.fis called monic if the coefficient ofis 1.

For example, letXbe a well-ordered set andX*the free monoid generated byX. We define the deg-lex ordering onX*: to compare two words first by length and then lexicographically. Then, such an ordering is monomial onX*.

LetS⊂(X),sS,uN. Then, roughly speaking,u|s=u|xis, wherexiis the individuality occurrence of the letterxiXinu, is called anS-word (ors-word). More precisely, the idealId(S) of(X) is the set of linear combination ofS-words.

andComw(s)=vsorsvcorrespondingly.

Given a monic subsetS⊂(X) andwN, anintersection (inclusion) compositionhis called trivial modulo (S,w) ifhcan be presented as a linear combination of normalS-words with leading words less thanw; a left (right) multiplication composition (or another kinds composition)his called trivial modulo (S) ifhcan be presented as a linear combination of normalS-words with leading words less than or equal to

The setSis a GS basis in(X) if all the possible compositions of elements inSare trivial moduloSand correspondingw.

If a subsetSof(X) is not a GS basis then one can add all nontrivial compositions of polynomials ofStoS. Continuing this process repeatedly, we finally obtain a GS basisScthat containsS. Such a process is called Shirshov algorithm.

Mainapplications

• normal forms;

• word problems; conjugacy problem;

• rewriting system;

• automata theory;

• embedding of algebras into simple algebras and two-generated algebras;

• extensions of groups and algebras;

• PBW type theorems;

• homology;

• Dehn function; complexity; growth function; Hilbert series; etc.

Since 2006, there were some 30 master theses and 4 PhD theses, more then 30 published papers in JA, IJAC, Comm Algebra, Algebra Coll 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 and published three survey papers[9-11]. Our main topic is GS bases method for different varieties (categories) of linear (Ω-) algebras over a fieldkor a commutative algebraKoverk: associative algebras (including group (semigroup) algebras), tensor product of free associative 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.

1 Gröbner-Shirshov bases theory for some new classes of (universal) algebras

In this section, we review some new CD-lemmas for tensor product of free associative algebras, Lie algebras over a commutative algebra, dialgebras, pre-Lie (Vinberg right (left) symmetric) algebras, Rota-Baxter algebras, metabelian Lie algebras,L-algebras, semiring algebras, associative algebras with multiple operations, differential algebras, category algebras, non-associative algebras over a commutative algebra,S-act algebras, etc. There are some applications for mentioned algebras.

In [12], GS bases method was initiated for a category ofk-algebra with the free objectkY⊗kX. HereN(X,Y) =Y*X*is a set of normal words ofkY⊗kX. For anyu=uYuX,v=vYvXN(X,Y),lcm(u,v)=lcm(uY,vY)lcm(uX,vX) and it is no need the composition of multiplication.

A key lemma in our proof is

Lemma3LetSbe a Lie GS basis inLiek[Y](X) and (asb) a normalS-word. Then

In [18], CD-lemma was established for metabelian Lie algebras.

A Lie algebraLis called a metabelian Lie algebra if (L2)2=0.

Let (A,∘ , ·,θ, 1) be a semiring, i.e., (A,∘ ,θ) is a commutative monoid, (A,·,1) is a monoid, andθ·a=a·θ=θ, · is distributive relative to ∘ from left and right: (a∘b)c=ac∘bc,c(a∘b)=ca∘cb. The semiring (A,∘ ,·,θ, 1) is commutative if (A, ·, 1) is a commutative monoid.

The class of semirings is a variety. So a free semiringRigXgenerated by a setXis defined as usual as for any variety of universal systems.

In [21], CD-lemma was proved for free semiring algebraskRigX|S. It gives GS bases method for semiringsRigX|S. The same was found for commutative semiringskRig[X|S]. As applications there were rediscovered normal forms of elements for two semirings from papers by Blass[22],R1=Rig[x|x=1∘x2] and Miore-Leinster[23],R2=Rig[x|x= 1∘x∘x2]. ForR1a GS basis isS1={1∘x2=x,x∘x4=x∘x3,x5=1∘x4, 1∘x3∘xn=xn,n=3,4}. ForR2a GS basis isS2={x4=1∘ 1·x2,x·x3=1·x2,1·x2·xn=xn,n=1,2,3}.

AnΩ-algebraAis ak-linear space with the linear operator setΩonA.

In [24], we establish GS bases method for associativeΩ-algebras, whereΩconsists ofn-ary operations,n≥1.

There are two kinds of compositions: intersection and inclusion.

As applications, we give linear bases for free Rota-Baxterk-algebra of weightand-differential algebra, where a Rota-Baxterk-algebra is an associative algebraRwith ak-linear operationP:R→Rsatisfying the Rota-Baxter relation:

P(x)P(y)=P(P(x)y+xP(y)+xy), ∀x,yR.

D(xy)=D(x)y+xD(y)+D(x)D(y), ∀x,yR.

In [25], GS bases method for Rota-Baxter algebras over a field of characteristic 0 is found.

There are four kinds of compositions: intersection, inclusion and left (right) multiplication.

As application, we prove that every countably generated Rota-Baxter algebra with weight 0 can be embedded into a two-generated Rota-Baxter algebra.

Another important application is PBW theorem for dendriform algebra which is a conjecture of L Guo.

A dendriform algebra is ak-linear spaceDwith two binary operationsand ≻ such that for anyx,y,zD,

(xy)z=x(yz+y≻z),
(x≻y)z=x≻(yz),
(xy+x≻y)≻z=x≻(y≻z).

Suppose that (D,,≻) is a dendriform algebra overkwith a linear basisX={xi|iI}. Letxixj={xixj},xi≻xj={xi≻xj}, where {xixj} and {xi≻xj} are linear combinations ofxX. ThenDhas an expression by generators and defining relations

D=D(X|xixj={xixj},xi≻xj={xi≻xj},xi,xjX).

Denote by

U(D)=RB(X|xiP(xj)={xixj},

whereRB(X) is the free Rota-Baxter algebra generated byX. ThenU(D) is the universal enveloping Rota-Baxter algebra ofD.

The following result is obtained[26]: every dendriform algebra over a field of characteristic 0 can be embedded into its universal enveloping Rota-Baxter algebra.

A non-associativeAis called a right-symmetric (or pre-Lie) algebra ifAsatisfies the following identity (x,y,z)=(x,z,y) for the associator (x,y,z)=(xy)z-x(yz).

In [27], GS bases method is found for Pre-Lie algebras. As an application, we give a GS basis for the universal enveloping right-symmetric algebra of a Lie algebra. From this it follows PBW theorem for Lie algebra and right-symmetric algebra (Segal’s theorem).

δ(ab)=δ(a)·b+a·δ(b).

GS bases method for differential algebras is established[28].

As applications, there are given linear bases for free Lie-differential algebras and free commutative-differential algebras, respectively.

AnL-algebra is ak-linear spaceLequipped with two binaryk-linear operations,≻:L⊗L→Lverifying the so-called entanglement relation:

(x≻y)z=x≻(yz), ∀x,y,zL.

In [29], GS bases method forL-algebras is found.

There are two kinds of compositions: inclusion and right multiplication.

As applications, we give linear bases of a free dialgebra (Loday’s theorem) and the free product of twoL-algebras, and the following embedding theorems forL-algebras are obtained: 1) Every countably generatedL-algebra over a fieldkcan be embedded into a two-generatedL-algebra. 2) EveryL-algebra over a fieldkcan be embedded into a simpleL-algebra. 3) Every countably generatedL-algebra over a countable fieldkcan be embedded into a simple two-generatedL-algebra. 4) Three arbitraryL-algebrasA,B,Cover a fieldkcan be embedded into a simpleL-algebra generated byBandCif |k|≤dim(B*C) and |A|≤|B*C|, whereB*Cis the free product ofBandC.

Ak-linear spaceDequipped with two bilinear multiplications ├ and ┤ is called a dialgebra, if both ├ and ┤ are associative and

a┤(b├c)=a┤b┤c,
(a┤b)├c=a├b├c,
a├(b┤c)=(a├b)┤c

In [30], GS bases method for dialgebras is given. As results, we give linear bases for the universal enveloping algebra of a Leibniz algebra (Aymon-Grivel’s theorem), the bar extension of a dialgebra, the free product of two dialgebras, and Clifford dialgebra. We obtain some normal forms for algebras mentioned the above.

GS bases method for small categories is established[31]. As applications, we give linear bases for the simplicial category (a classical result) and the cyclic category (A. Cohen’s theorem) respectively.

GS bases method for non-associative algebras over commutative algebras is given[32]. As an application, it is shown that each countably generated non-associative algebra over an arbitrary commutative algebraKcan be embedded into a two-generated non-associative algebra overK.

In [33], it is investigated the relationship between the GS bases in free associative algebras, free left modules and “double-free” left modules (that is, free modules over a free algebra). We first give the Chibrikov’s CD-lemma for modules and then we show that the Kang-Lee’s CD-lemma follows from this lemma. As applications, we give linear bases for the following modules: the highest weight module over a Lie algebrasl2, the Verma module over a Kac-Moody algebra, the Verma module over the Lie algebra of coefficients of a free conformal algebra, and a universal enveloping module for a Sabinin algebra.

LetAbe an associative algebra over a fieldkandSa monoid of linear operators onA. ThenAis called anS-act algebra ifAis anS-act with the actions(a) satisfying

In [35], the “double free”S-act algebra (i.e., a freeS-act algebra, whereSis a free semigroup) is constructed. Then GS bases theory forS-act algebras is established, whereSis an arbitrary semigroup. As an application, a GS basis of free Chinese monoid-act algebra is given and hence a linear basis of free Chinese monoid-act algebra is obtained.

2 Applications

In this section, we review some applications of GS bases method (CD-lemmas) mentioned in the above section and CD-lemma for associative algebras.

In [36], it is generalized the Shirshov’s composition lemma for associative algebras by replacing the monomial ordering for “S-partially monomial” ordering of words. As applications, we give a new proof of Britton lemma for HNN extensions of groups. Also a GS basis of the alternating group is obtained.

In [37], it is dealing with Schreier extensions of group

1→A→C→B→1

ofAbyB. IfBis a cyclic or a free abelian group, there are classical necessary and sufficient conditions forCto be a Schreier extension ofAbyBin terms of the factor set ofB. In [38], it is mentioned that for anyBit is difficult to find an analogous conditions. Actually this problem was solved in [37] using the GS bases technique. As applications there were given new proofs of 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[39]. An algebrais called a (singular) extension of the algebraMbyBifM2=0,Mis an ideal ofand/M≅Bas algebras. In [39], singular extensions are studied from the GS bases viewpoint. Namely, even though extensions correspond to cocycles, the reconstruction of an extension from a cocycle can be applied to every 2-cochain ofBwith coefficients inM. This means that every 2-cochainφgives rise to a certain associative algebraA(φ) presented by generators and relations. The main observation of the paper[39]is that the relations ofA(φ) form a GS basis if and only if the cochainφis a cocycle (so the algebraA(φ) is a singular extension ofBbyM). This implies the main result of this paper: an algorithmic procedure to check the extension condition.

LetUq(AN) be the Drinfeld-Jimbo quantum group of typeAN. In [40], by using GS bases, we give a simple (but not short) proof of the Rosso-Yamane theorem on PBW basis ofUq(AN) (see also [41]).

In [42], a simple analysis of the word problem for Novikov’sp1p2(P) and Boone’s groupsG(T,q) based on Gröbner-Shirshov bases technique is given (see also[43-44]).

In [45], a GS basis for the Chinese monoid is obtained and an algorithm for normal form of the Chinese monoid is given. It gives new proof of some results in Cassaigne, etc[46].

The same kind of result was established in [48] for Hall basis of a free Lie algebra for an ordering with condition [u]>[v] ifu>vin the deg-lex ordering.

In [49], a free inverse semigroup

FI(X)=sgpY=X∪X-1|aa-1a=a,aa-1bb-1=

was studied. A GS basis and normal forms were found for the semigroup. This gives simple proofs of substantial refinements of results in Polyakova-Schain[50].

In [51], by using GS bases, a straightforward proof for Artin-Markov normal form theorem for braid groups is given (see also[52]).

In [53], by using CD-lemma forL-algebras[29], we give a GS basis of the free dendriform diaglebra as a quotient algebra of anL-algebras. Then we obtain the Hilbert series and Gelfand-Kirillov dimension of the free dendriform dialgebra generated by a finite set.

In [54], by using GS bases, we show that in the following classes, each (resp. countably generated) algebra can be embedded into a simple (resp. two-generated) algebra: associative differential algebras, associativeΩ-algebras, associative-differential algebras. We show that in the following classes, each countably generated algebra over a countable fieldkcan be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associativeΩ-algebras, associative-differential algebras. Also we prove that any countably generated module over a free associative algebrakXcan be embedded into a cyclickX-module, where |X|>1. We give other proofs of the well known theorems: each countably generated group (resp. associative algebra, semigroup, Lie algebra) can be embedded into a two-generated group (resp. associative algebra, semigroup, Lie algebra) (Higman-Neumann-Neumann theorem, Malcev theorem, Shirshov theorem).

In [55], we prove that two-generator one-relator groups with depth less than or equal to 3 can be effectively embedded into a tower of HNN-extensions in which each group has the effective standard normal form in the sense of Bokut-Kukin[56]. We give an example to show how to deal with some general cases for one-relator groups.

In [58], by using the CD-lemma for non-associative algebras invented by A.I. Shirshov in 1962, we give GS bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I.P. Shestakov’s result that any Akivis algebra is linear and D. Segal’s result that the set of all good words inX**forms a linear basis of the free Pre-Lie algebraPLie(X) generated by the setX.

In [59], a Lie GS basis is found for any free partially commutative Lie algebra.

In [60], we give two explicit (quadratic) presentations of the plactic monoid in row and column generators correspondingly. Then we give direct independent proofs that these presentations are GS bases of the plactic algebra in deg-lex orderings of generators. From CD-lemma for associative algebras it follows that the set of Young tableaux is the Knuth normal form for plactic monoid.

Kukin construction. LetP=sgpx,y|ui=vi,iIbe a semigroup. Consider the Lie algebra

whereSconsists of the following relations:

Here, ⎣zu」 means the left normed bracketing.

Drinfeld-Kohno construction. Letn>2 be an integer. The Drinfeld-Kohno Lie algebraLnoveris defined by generatorstij=tjifor distinct indices 1≤i,j≤n-1, and relations

wherei,j,k,lare distinct.

In [61], GS bases for Drinfeld-Kohno Lie algebraLnand Kukin Lie algebraAPare obtained. As an application, we show thatLnis an iterated semidirect product of free Lie algebras. Another application is Kukin’s result: if semigroupPhas the undecidable word problem then the Lie algebraAPhas the same property. It gives another proof of the theorem of the first author that the word problem for Lie algebras is algorithmically undecidable (1972). For semigroups and groups there are famous results by Markov-Post (1947) and Novikov (1955) respectively.

3 Some prospects

We are trying to establish Gröbner-Shirshov bases theory for Novikov algebras,strict monoidal categories, modules over an associative conformal algebra,associativen-conformal algebras, Nijenhuis algebras, etc. We will give some new applications of the mentioned Composition-Diamond lemmas,for example, automatic structures for some semigroups,PBW theorems, homology for some semigroup algebras, Dehn functions for some groups, embedding algebras into two generated simple algebras, word problems for some algebras, and so on.

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