2010 Mathematics Subject Classification: Primary 16W50, 17B70; Secondary 16R10.The main result is the classification, up to isomorphism, of all gradings by arbitrary abelian groups on the finitely generated algebras that are free in a nilpotent variety of algebras over an algebraically closed field of characteristic zero.The research was supported by an NSERC Discovery Grant #227060-0

Bahturin, Yuri

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Serdica Math. J. 38 (2012), 1–6GROUP GRADINGS ON FREE ALGEBRASOF NILPOTENT VARIETIES OF ALGEBRAS∗Yuri BahturinCommunicated by M. KochetovAbstract. The main result is the classification, up to isomorphism, of allgradings by arbitrary abelian groups on the finitely generated algebras thatare free in a nilpotent variety of algebras over an algebraically closed fieldof characteristic zero.1. Introduction. In this short note I put together two areas that havebeen of interest to me for an extended period of time: varieties of algebras andgroup gradings on algebras. Namely, we look at the gradings by abelian groupson relatively free algebras of finite rank in nilpotent varieties of algebras. Thesealgebras do not need to be associative or Lie but it is important that they arefinite-dimensional and the varieties in question must be defined by multihomo-geneous identities. If the base field of coefficients is infinite, then this is alwaystrue. Among algebras to which our results apply are free nilpotent associativeor Lie algebras, free metabelian nilpotent Lie algebras and many others. At the2010 Mathematics Subject Classification: Primary 16W50, 17B70; Secondary 16R10.Key words: graded algebra, nilpotent Lie algebra, grading.∗The research was supported by an NSERC Discovery Grant #227060-092 Yuri Bahturinsame time, our Theorem 1 may be of interest also beyond these two areas, forinstance in the study of nil-manifolds equipped with some groups of symmetries.An example of such applications of the gradings on algebras can be found in [1].While writing this note, we had an opportunity of using the forthcomingmonograph [2], which we believe will soon become a standard source in the theoryof gradings of algebras. We would like to thank the authors for this.2. Some basics. We recall that given an algebra A, two gradingsΓ : A =⊕g∈GAg and Γ′ : A =⊕h∈H A′h are called equivalent if there is anautomorphism ϕ ∈ AutA and a bijection β : Supp Γ → Supp Γ′ such that forall g ∈ G we have ϕ(Ag) = A′β(g). Here Supp Γ denotes the support : Supp Γ ={g ∈ G | Ag 6= 0}. We call ϕ a weak isomorphism if β can be extended to anisomorphism of groups G and H. If ϕ(Ag) = A′g for all g ∈ G (that is, β is theidentity map) then we call Γ and Γ′ isomorphic.Using the same notation as just above, we call Γ′ a refinement of Γ if forany h ∈ H there is g ∈ G such that A′h ⊂ Ag. (In this situation also Γ is calledthe coarsening of Γ′.) If at least for one h ∈ H, the latter containment is proper,we call the refinement proper. A grading Γ is called fine if it does not admitproper refinements.Let k be a field and Fn a finite-dimensional algebra which is free of finiterank n in a variety of algebras given by multihomogeneous identities (of degree> 1). Let X = {x1, . . . , xn} stand for the set of free generators of Fn. Givenα = (d1, . . . , dn) ∈ Zn, we consider the span Fαn of all the monomials in X whosedegree with respect to each variable xi equals di, i = 1, . . . , n. The subspaces Fαnform a Zn-grading of Fn:(1) Fn =⊕α∈ZnFαn .Since Fn is finite-dimensional, there is c such that Fαn = {0} as soon as |α| > c.Thus Fn is a nilpotent algebra.The automorphism group AutFn is an algebraic group and it containsan n-dimensional toral subgroup Dn whose elements are defined as follows. Pickt = (t1, . . . , tn) ∈ (k×)n. The map x1 7→ t1x1, . . . , xn 7→ tnxn uniquely extendsto an automorphism δ(t) of Fn. We set Dn = {δ(t) | t ∈ (k×)n}. Obviously,Dn ∼= (k×)n.3. Quasitori in AutFn. In the theory of gradings by abelian groups animportant role is played by the toral subgroups and their normalizers in AutFn.Let us define a subgroup Pn in AutFn as follows. Given a permutation σ onGroup gradings on free algebras of nilpotent varieties of algebras 3{1, 2, . . . , n} and t, as before, we consider the map x1 7→ t1xσ(1), . . . , xn 7→ tnxσ(n).The extension of this map to the whole of Fn will be denoted by δ(σ, t). Clearly,δ(t) = δ(id, t). The set of all δ(σ, t) where σ ∈ Sn and t ∈ (k×)n is a subgroup ofAutFn. We denote this subgroup by Pn.Proposition 1. If the ground field k is infinite, the normalizer of Dn inAutFn equals Pn.P r o o f. Suppose an automorphism µ belongs to the normalizer of Dn.On the elements of the generating set X the action of µ can be written asµ(xi) =n∑j=1ajixj +∑α,|α|>1wαi where for all i, j, α we have aji ∈ k, wαi ∈ Fαn .Pick δ(t) ∈ Dn. Then µ−1δ(t)µ ∈ Dn. In other words, there is δ(s) ∈ Dn suchthat δ(t)µ = µδ(s). Using the definition of µ we determine thatµδ(s)(xi) =n∑j=1siajixj +∑α,|α|>1siwαiwhileδ(t)µ(xi) =n∑j=1aji tjxj +∑α,|α|>1td11 · · · tdnn wαi .It follows that we must have aji tj = siaji for all i, j and also td11 · · · tdnn wαi = siwαifor any α with |α| > 1.From the equations of the first kind, provided that all ti are chosen pair-wise different, it follows that for each i = 1, . . . , n there is only one j such thataji 6= 0 and we must have si = tσ(i). Since µ is an automorphism, σ is a permu-tation on the set {1, . . . , n}.Now if wαi 6= 0 we must have tσ(i) = tα. Since t1, . . . , tn are chosendifferent from 0, it follows that tα6= 0. If this is the case for some α and i, wehave, for any (t1, . . . , tn) ∈ (k×)n that tσ(i) = td11 · · · tdnn where |α| = d1+· · ·+dn >1. If dj 6= 0 for some j 6= σ(i) then, replacing all tk, k 6= j, by 1 we obtaina contradiction 1 = tdjj , for all tj ∈ k×, k infinite. Otherwise, we have thattσ(i) = t|α|σ(i), for any tσ(i) ∈ k×, which is again a contradiction. Therefore, wemust have wαi = 0, for all σ and i.Our conclusion is that for each µ in the normalizer of D there exists apermutation σ ∈ Sn on the set {1, . . . , n} such thatµ(xi) = aσ(i)i xσ(i) for i = 1, . . . , n.4 Yuri BahturinThus we obtain µ = δ(σ, t) with t1 = aσ(1)1 , . . . , tn = aσ(n)n , hence µ ∈ Pn, asclaimed. The converse statement being obvious, our proof is complete. Corollary 1. If k is an infinite field, then Dn is a maximal torus inAutFn.P r o o f. We consider the subspace V =⊕|α|=1 Fαn . Every non-singularlinear transformation of V uniquely extends to an automorphism of Fn. All suchautomorphisms form a subgroup H of AutFn isomorphic GL(V ). Choosing X asa basis of V allows us to assume the existence of an isomorphism ι : H → GLn(k)such that ι(Dn) is the group of all non-singular diagonal matrices and ι(Pn)the subgroup of “monomial” matrices, that is, all matrices with exactly onenonsingular entry in each row and in each column. It is well known that thegroup of all diagonal matrices is a maximal torus in GLn(k). It follows that thecentralizer of ι(Dn) in GLn(k) is itself. But then also the centralizer of ι(Dn) inι(Pn) is itself. Since ι is an isomorphism, the centralizer of Dn in Pn is itself,proving that Dn is indeed a maximal torus in AutFn. The next corollary deals with quasitori. Given an algebraic group, a qua-sitorus is a subgroup consisting of semisimple elements, which is isomorphic to thedirect product of a torus and a finite abelian group. A typical example of impor-tance to us is the following. Let G be an abelian group and Γ : F =⊕g∈G(Fn)ga grading of Fn by G. Let us consider the “diagonal” subgroup D(Γ) ⊂ AutFn.This subgroup consists of the automorphisms of Fn whose restriction to any com-ponent (Fn)g of the grading Γ is a scalar linear transformation. Clearly, D(Γ) isa quasitorus in AutFn.Corollary 2. If k is an algebraically closed field, any quasitorus inAutFn is conjugate to a subgroup of Dn.P r o o f. Let Q be the quasitorus in question. According to [3] (see also[4]), any quasitorus is a subgroup in the normalizer of a maximal torus. Sincein an algebraic group any two maximal tori are conjugate, the same is true fortheir normalizers and thus using Proposition 1, we may assume that there existspi ∈ AutFn such piQpi−1 ⊂ Pn. Denote Q′ = piQpi−1.Next we use the isomorphism ι from the proof of Corollary 1. We ob-serve that ι(Q′) is a quasitorus in GLn(k). Since k is algebraically closed, thereexists ψ ∈ GLn(k) such that ψι(Q′)ψ−1 ⊂ ι(Dn). Set ρ = ι−1(ψ) ∈ H. Thenι(ρQρ−1) ⊂ ι(Dn) and thus ρQρ−1 ⊂ Dn. Setting τ = ρpi ∈ AutFn, we obtainthat τQτ−1 ⊂ Dn, as claimed. Group gradings on free algebras of nilpotent varieties of algebras 54. Gradings of Fn. From now on we assume that k is an algebraicallyclosed field and finally turn our attention to the gradings on Fn by arbitraryabelian groups. Consider one of such gradings, Γ : F =⊕g∈G(Fn)g, by anabelian group G. Since Fn is finite-dimensional, only finitely many elements of Gare used for the grading. This allows us, without loss of generality, to assume thatG is a finitely generated group. We consider Ĝ, the group of characters of G withvalues in k×. This group is a quasitorus. As is well known, Ĝ acts by semisimpleautomorphisms on Fn, according to the rule χ ∗ w = χ(g)w, for any χ ∈ Ĝ andw ∈ (Fn)g. As a result, we have a homomorphism ϕ : Ĝ→ AutFn, which maps Ĝinto the quasitorus D(Γ). By Corollary 2, there is an automorphism τ : Fn → Fnsuch that τD(Γ)τ−1 ⊂ Dn. We can also write that D(Γ) ⊂ τ−1Dnτ .Now let us consider the decomposition of Fn as the sum of subspacesτ−1(Fα). This is a grading of Fn by Zn isomorphic to our standard Zn-grading.At the same time, this is the eigenspace decomposition of Fn under the action ofthe maximal torus τ−1Dnτ . Since D(Γ) ⊂ τ−1Dnτ , we have that each eigenspaceof the decomposition of Fn with respect to the action of the quasitorus D(Γ) isthe sum of several subspaces τ−1(Fα).The eigenspace decomposition of Fn with respect to ϕ(Ĝ) is Γ if k is ofcharacteristic 0. If k has characteristic p then this is still Γ provided that G hasno elements of order p. Otherwise, this is the coarsening of Γ induced by thequotient map G → G/Sp where Sp is the Sylow p-subgroup of G and p is thecharacteristic of k.Now given α ∈ Zn, there is a character ω(α) : τ−1Dnτ → k× such thatτ−1(Fα) is the eigenspace of the action of τ−1Dnτ corresponding to ω(α). Therestriction of ω(α) to ϕ(Ĝ) is a character of this quasitorus, which can be identifiedwith an element g = gH, g ∈ G, of G/H where H is either trivial or Sp, asexplained above. Since the restriction map is a homomorphism, it follows thatν : α → g is a homomorphism of Zn to G/H. A simple calculation shows that(Fn)g =∑ν(α)=g Fα where (Fn)g =∑h∈H(Fn)gh.The above discussion brings us to the following main result of this note.Theorem 1. Let k be an algebraically closed field, Fn a (nonzero) finite-dimensional algebra which is free of rank n in a variety of algebras over k. LetΓ : F =⊕g∈G(Fn)g be a grading of Fn by an abelian group G. If k is ofcharacteristic zero or if G has no elements of order p where p is the characteristicof k then Γ is isomorphic to a G-grading induced from the standard Zn-grading(1) by a homomorphism Zn → G. Otherwise, the same is true for the coarseningof Γ obtained by taking the quotient of G with respect to its Sylow p-subgroup.To obtain all gradings, up to isomorphism, of Fn by an abelian group6 Yuri BahturinG (assuming char k = 0 or char k = p and G has no elements of order p) wesimply have to choose n elements g1, . . . , gn from G and assign to each w ∈ Fαthe degree d(w) = gd11 · · · gdnn where α = (d1, . . . , dn) ∈ Zn. In particular, d(x1) =g1, . . . , d(xn) = gn. The isomorphism class of such grading is determined by theelements of G that occur in the n-tuple (g1, . . . , gn), counting multiplicity.Our last result deals with the fine gradings on Fn by abelian groups.We note that, in characteristic zero, Theorem 1 implies that any such grading isisomorphic to a coarsening of the standard grading, hence we obtain the following:Corollary 3. Let k be an algebraically closed field of characteristic zero,Fn a (nonzero) finite-dimensional algebra which is free of rank n in a variety ofalgebras over k. Then, up to equivalence, the standard grading (1) is the onlyfine abelian group grading of Fn.REFERENCES[1] Y. Bahturin, M. Goze. Z2 × Z2-symmetric spaces. Pacific J. Math. 236(2008), 1–21.[2] A. Elduque, M. Kochetov. Gradings on Simple Lie Algebras, manuscript.[3] V. P. Platonov. The theory of algebraic linear groups and periodic groups.Trans Amer. Math. Soc. Ser. (2 ) 69 (1968), 61–159.[4] T. A. Springer, R. Steinberg. Conjugacy classes. In: Seminar on Alge-braic Groups and Related Finite Groups (The Institute for Advanced Study,Princeton, N.J., 1968/69), Lecture Notes in Mathematics, vol. 131, Springer,Berlin, 1970, 167–266.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’s, NL, A1C5S7, CanadaandDepartment of AlgebraFaculty of Mechanics and MathematicsMoscow State University119922 Moscow, Russiae-mail: bahturin@mun.ca Received June 20, 2012

Group Gradings on Free Algebras of Nilpotent Varieties of Algebras

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Group Gradings on Free Algebras of Nilpotent Varieties of Algebras

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