ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.284959 J. Algebra Comb. Discrete Appl. 4(2) • 165–180 Received: 14 June 2015 Accepted: 21 February 2016 Journal of Algebra Combinatorics Discrete Structures and Applications On the graded identities of the Grassmann algebra∗ Review Article Lucio Centrone Abstract: We survey the results concerning the graded identities of the infinite dimensional Grassmann algebra. 2010 MSC: 16R10, 16P90, 16S10, 16W50 Keywords: Graded polynomial identities, Grassmann algebra 1. Introduction All algebras we refer to are assumed to be associative with unit and all fields are assumed to be infinite unless explicitely written. Moreover, every group is an abelian group unless explicitely written. The Grassmann algebra is the algebra of the wedge product, also called an alternating algebra or exterior algebra. It fits in several places in mathematics and, in general, in sciences as well. We recall that if F is a field and X = {x1,x2, . . .} is a countable infinite set of variables, let F〈X〉 be the free associative algebra freely generated by X over F. We shall refer to the elements of F〈X〉 as polynomials in the set of variables X. If A is an F-algebra, we say that f(x1, . . . ,xn) ∈ F〈X〉 is a polynomial identity of A if f(a1, . . . ,an) = 0 for any a1, . . . ,an ∈ A. If A has a non-trivial polynomial identity we say that A is a polynomial identity algebra or PI-algebra and we denote by T(A) the set of all polynomial identities satisfied by A. It is well known that T(A) is an ideal of F〈X〉 invariant under all endomorphisms of F〈X〉, i.e., it is a T-ideal called the T-ideal of A. Concerning the mathematical aspects of the Grassmann algebra and, in particular, the algebraic ones, the Grassmann algebra E generated by an infinite dimensional vector space and its identities, play an important role in the structure theory of Kemer on varieties of associative algebras with polyno- mial identities [21, 22]. More precisely, Kemer proved that any associative PI-algebra over a field F of characteristic zero satisfies the same identities (is PI-equivalent) of the Grassmann envelope of a finite di- mensional associative superalgebra, i.e., they have the same T-ideal. Moreover for any associative algebra A, T(A) is finitely generated as a T-ideal. ∗ Partially supported by FAPESP grant 2013/06752-4, partially supported by FAPESP grant 2015/08961-5. Lucio Centrone; IMECC, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda 651, Campinas (SP), Brazil (email: centrone@ime.unicamp.br). 165 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 One of the main goals of the theory of PI-algebras is finding a complete set of generators of the T-ideal of a given algebra. For example, it is well known that if A is a commutative unitary algebra, then T(A) is generated by the Lie commutator [x,y] := xy − yx. The problem turns out to be very hard even for finite dimensional algebras. In fact, if we consider Mn(F) the matrix algebra over a field F of characteristic 0, we know that its T-ideal is well known only for the case n = 2 (see [31] and [15]), provided that the case n = 1 is trivial. If F has positive characteristic p 6= 2 we have a description of a finite basis of identities for M2(F) (see [24]). We note that some further partial results for M2(F) in the case of characteristic 2 were obtained in [16] and [23] but it is still unknown if the T-ideal of M2(F) is finitely generated or not in this case. Let us consider a generalization of the definition of polynomial identity for a G-graded algebra A = ⊕ g∈G A g, where G is any group. If we specialize the variable from X with a G-degree, to say ‖ · ‖, we obtain a set F〈X|G〉 of “graded polynomials”. Of course we may generalize the notion of polynomial identity with the notion of G-graded polynomial identity in a natural way. We say that f(x1, . . . ,xn) ∈ F〈X|G〉 is a G-graded polynomial identity of A if f(a1, . . . ,an) = 0 for all a1, . . . ,an ∈ ⋃ g∈G A g such that ai ∈ A‖xi‖. As well as in the ungraded (or ordinary) case we denote by TG(A) the set of all G-graded polynomial identities satisfied by A. It is well known that TG(A) is an ideal of F〈X|G〉 invariant under all the G-graded endomorphisms of F〈X|G〉, i.e., it is a TG-ideal called the TG-ideal of A. In [38] Vasilovsky gives a complete description of the TZn-ideal of Mn(F) for a particular Zn-grading and for all n ≥ 2 in characteristic 0 whereas Azevedo in [3] obtained the same results without any restriction on the ground field. We recall that the works by Vasilovsky and by Azevedo are a generalization of the work by Di Vincenzo for 2 by 2 matrices (see [11]). As well as in the ordinary case, if A is an associative algebra graded by a finite group G, then TG(A) is finitely generated as a TG-ideal (see [1]). Coming back to the Grassmann algebra E, we know that in the ordinary case T(E) is generated by the triple commutator [x,y,z] := [[x,y],z] as shown in the papers by Latyshev [27] and [26] by Krakovski and Regev. For this purpose, we want to point out that the latter two papers deal with characteristic 0 only even if the argument used in [27] is still valid in positive characteristic as the argument used in [17] Theorem 5.1.2. For this purpose, see also the paper by Giambruno and Koshlukov [18]. In light of what we said until now and with the intent of a future use, in this paper we want to collect the results concerning the graded identities (and other related topics) of the Grassmann algebra trying to be as exaustive as possible. We recall that in [36] the author collected the results related to Z2-gradings of E. 2. Graded PI-algebras We introduce the terminology for the study of graded polynomial identities. We start off with the following definition. In the sequel every algebra is associative with unit and every field is infinite unless explicitely written. Definition 2.1. Let G be a group and A be an algebra over a field F. We say that the algebra A is G- graded if there exist subspaces Ag, g ∈ G such that A = ⊕ g∈G A g as a vector space and for all g, h ∈ G, one has AgAh ⊆ Agh. It is easy to note that if a is any element of A it can be uniquely written as a finite sum a = ∑ g∈G ag, where ag ∈ Ag. We shall call the subspaces Ag the G-homogeneous components of A. Accordingly, an element a ∈ A is called G-homogeneous if exists g ∈ G such that a ∈ Ag. If B ⊆ A is a subspace of A, B is G-graded if and only if B = ⊕ g∈G(B ∩A g). Analogously one can define G-graded algebras, subalgebras, ideals, etc. We say that a G-grading on A is homogeneous if there exists a linear basis B of A such that every element of B is a homogeneous element of A. Let {Xg | g ∈ G} be a family of disjoint countable sets of indeterminates. Set X = ⋃ g∈G X g and denote by F〈X|G〉 the free associative algebra freely generated by X over F . An indeterminate x ∈ X is said to be of homogeneous G-degree g, written ‖x‖ = g, if x ∈ Xg. We always write xg if x ∈ Xg. The homogeneous G-degree of a monomial m = xi1xi2 · · ·xik is defined to be ‖m‖ = ‖xi1‖·‖xi2‖·· · ··‖xik‖. For 166 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 every g ∈ G, denote by F〈X|G〉g the subspace of F〈X|G〉 spanned by all monomials having homogeneous G-degree g. Notice that F〈X|G〉gF〈X|G〉g ′ ⊆ F〈X|G〉gg ′ for all g,g′ ∈ G. Thus F〈X|G〉 = ⊕ g∈G F〈X|G〉g is a G-graded algebra. The elements of the G-graded algebra F〈X|G〉 are referred to as G-graded poly- nomials or, simply, graded polynomials. In order to simplify the notation we shall sometimes use yi’s to denote variables of homogeneous degree 1G, zi’s to denote variables of homogenous degree different than 1G and xi’s for any variables without distinguish their homogenous degree. Definition 2.2. If A is a G-graded algebra, a G-graded polynomial f(x1, . . . ,xn) is said to be a graded polynomial identity of A if f(a1,a2, . . . ,an) = 0 for all a1,a2, . . . ,an ∈ ⋃ g∈G A g such that ak ∈ A‖xk‖, k = 1, . . . ,n. We shall write f ≡ 0 in order to say that f is a graded polynomial identity for A. If A satisfies a non-trivial graded identity we say that A is a PI G-graded algebra. When the algebra A is graded by the trivial group (in fact it has no grading) we refer to polynomial identities of A and T-ideal of A. Given an algebra A graded by a group G, we define TG(A) := {f ∈ F〈X|G〉|f ≡ 0 on A}, the set of G-graded polynomial identities of A. Definition 2.3. An ideal I of F〈X|G〉 is said to be a TG-ideal if it is invariant under all F- endomorphisms ϕ : F〈X|G〉→ F〈X|G〉 such that ϕ(F〈X|G〉g) ⊆ F〈X|G〉g for all g ∈ G. Hence TG(A) is a TG-ideal of F〈X|G〉. On the other hand, it is easy to check that all TG-ideals of F〈X|G〉 are of this type. If S ⊆ F〈X|G〉, we shall denote by 〈S〉TG the TG-ideal generated by the set S, i.e., the smallest TG-ideal containing S. Moreover, given a set of polynomials S ⊆ F〈X|G〉, we say that I is the TG-ideal generated by S, if I is the smallest TG-ideal containing S. In this case we say that S is a basis for I or that the elements of I follow from or are consequences of the elements of S. If S is a finite set generating the T-ideal I we say I is finitely based. Notice that being a basis for a T-ideal does not imply being a minimal basis. The theory of PI G-graded algebras in characteristic zero passes through the representation theory of the symmetric group. We consider the following Sn-modules. Definition 2.4. Let PGn = span〈x g1 σ(1) x g2 σ(2) · · ·xgn σ(n) |gi ∈ G,σ ∈ Sn〉, then the elements in PGn are called multilinear polynomials of degree n of F〈X|G〉. It turns out that PGn is a left Sn-module under the natural left action of the symmetric group Sn; we denote the Sn-character of the factor module PGn /(P G n ∩TG(A)) by χGn (A), and by cGn (A) its dimension over F . We say that ( χGn (A) ) n∈N is the G-graded cocharacter sequence of A ( cGn (A) ) n∈N is the G-graded codimension sequence of A 167 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 Now, for lg1, . . . , lgr ∈ N let us consider the blended components of the multilinear polynomials in the indeterminates labeled as follows: xg11 , . . . ,x g1 lg1 , then xg2lg1 +1, . . . ,x g2 lg1 +lg2 and so on. We denote this linear space by PGlg1,...,lgr . Of course, this is a left Slg1 ×···×Slgr -module. We shall denote by χ G lg1,...,lgr (A) the character of the module PGlg1,...,lgr (A)/(P G lg1,...,lgr (A) ∩TG(A)) and by cGlg1,...,lgr (A) its dimension. When the algebra A is graded by the trivial group (in fact it has no grading) we refer to cocharacter sequence of A and codimension sequence of A. For a more detailed account on PI-algebras, see Chapters 1 and 3 of [20] or [17]. Since the ground field F is infinite, a standard Vandermonde-argument yields that a polynomial f is a G-graded polynomial identity for A if and only if its multihomogeneous components are identities as well. Moreover, since char(F) = 0, the well known multilinearization process shows that the TG-ideal of a G-graded algebra A is determined by its multilinear polynomials, i.e. by the various PGlg1,...,lgr (A). We remark that, given the cocharacter χGlg1,...,lgr (A), the graded cocharacter χ G n (A) is known as well. More precisely, the following is due to Di Vincenzo (see [12], Theorem 2). Proposition 2.5. Let A be a G-graded algebra with graded cocharacter sequences χGlg1,...,lgr (A). Then χGn (A) = ∑ (lg1, . . . , lgr ) lg1 + . . . + lgr = n χGlg1,...,lgr (A)↑Sn. Moreover cGn (A) = ∑ (lg1, . . . , lgr ) lg1 + . . . + lgr = n ( n lg1, . . . , lgr ) cGlg1,...,lgr (A). Actually, if A is a G-graded PI-algebra, it is more convenient studying PGn (A) than the whole TG(A) ∩ PGn (A). In fact, the latter grows factorially while a graded generalization of a well celebrated work by Regev (see [32]) says that PGn (A) grows at most exponentially. 3. The Grassmann algebra In this section we define the Grassmann algebra and we list some results about its ordinary polynomial identities and the polynomial identities of some related algebras. Definition 3.1. Let X = {x1,x2, . . .} and let us consider F〈X〉. If I is the two-sided ideal of F〈X〉 generated by the set of polynomials {xixj + xjxi|i,j ≥ 1}, we set E := F〈X〉/I. Then we say that E is the infinite dimensional Grassmann algebra. Indeed if the set X is finite we define analogously the finite dimensional Grassmann algebra. We denote by L the vector space spanned by the ei := xi + I’s and we call it underlying vector space of E and we write E = E(L). Moreover, if w = ei1 · · ·eit is a monomial in the ei’s we say t is the length of w and we write l(w) = t. The set of different ei’s appearing in w is called support of w. We observe that E has the following presentation: E = 〈1,e1,e2, . . . |eiej = −ejei, for all i,j ≥ 1〉. Remark 3.2. Of course over a field of characteristic 2, the Grassmann algebra turns out to be commu- tative. Hence in the sequel every field is supposed to have characteristic p 6= 2. 168 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 The set B = {1,ei1 · · ·eik|1 ≤ i1 < · · · < ik} is a basis of E over F. It is convenient to write E in the form E = E0 ⊕E1 where E0 := span{1,ei1 · · ·ei2k|1 ≤ i1 < · · · < i2k,k ≥ 0}, E1 := span{1,ei1 · · ·ei2k+1|1 ≤ i1 < · · · < i2k+1,k ≥ 0}. It is easily checked that E0E0 + E1E1 ⊆ E0 and E0E1 + E1E0 ⊆ E1. This says that the decomposition E = E0 ⊕E1 is a Z2-grading of E, called natural or canonical Z2-grading of E. Notice that E0 coincides with the center of E whereas it is not true if L has finite dimension. For example, if L has dimension d, d odd, then e1 · · ·ed annihilates any element of E, then it is central but it does not fit in E0. The next is a well known fact. Proposition 3.3. E satisfies the identity [[x,y],z] ≡ 0. Let us suppose E over a field of characteristic zero, then the triple commutator is the only generator of T(E). In fact we have the following. Theorem 3.4. (Latyshev [27], Krakowski and Regev [26]) The T-ideal of E is generated by the polynomial [x1,x2,x3]. We list some results concerning the polynomial identities of some algebras related to E. Unless otherwise stated the base field is supposed to be of characteristic 0. In what follows we shall denote by UTn(R) the F-algebra of n×n upper triangular matrices with entries of the F-algebra R. Theorem 3.5. (Berele and Regev [6]) The T-ideal of UTn(E) is generated by the polynomial [x1,x2,x3] · · · [x3n−2,x3n−1,x3n]. More precisely, in [6] Theorem 2.8 the authors give a more general version of the previous result. We also recall that in [28] Latyshev proved that T(UTn(E)) is finitely based. Theorem 3.6. (Popov [30]) The T-ideal of E ⊗E is generated by the polynomials [x1,x2, [x3,x4],x5], [[x1,x2],x 2 2]. We also want to point out that in [30] the author described the structure of the relatively free algebra of E ⊗E. If we consider the finite dimensional Grassmann algebra, we have the next result by Di Vincenzo. Theorem 3.7. (Di Vincenzo [10]) Let E be the Grassmann algebra generated by the k dimensional underlying vector space Lk. Then the T-ideal of E is generated by the polynomials [x1,x2,x3], [x1,x2] · · · [x2t−1,x2t], where t = [k/2] + 1 and [a] is the integer part of a. Even if the characteristic zero is the most investigated case, we also have several works in positive characteristic. Here we cite the analog of Theorem 3.4 for any infinite field. 169 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 Theorem 3.8. The T-ideal of E is generated by the polynomial [x1,x2,x3]. Notice that the T-ideal of E does not depend on the characteristic of the field. This is not the case of the Grassmann algebra over a finite field or, as we shall see later, of the (Z2)-graded case. We may cite the work by Regev [34] which gave a lot of information about the T-ideal of E, introducing the so called class identities whose definition goes far from the intent of this survey. By the way it will be interesting to note the following. Proposition 3.9. Let E be the infinite dimensional Grassmann algebra over a field of characteristic p and let us consider E∗ = E −{1}, then E∗ satisfies the identity xp. We close this section by citing a famous result by Olsson and Regev about the cocharacters and the codimensions of the infinite dimensional Grassmann algebra in characteristic 0. We recall that a partition of the non-negative integer n is a sequence of integers λ = (λ1, . . . ,λr) such that λ1 ≥ ···≥ λr > 0 and λ1 + · · ·+ λr = n. In this case we shall write λ ` n. We assume two partitions λ = (λ1, . . . ,λr) and µ = (µ1, . . . ,µs) to be equal if r = s and λ1 = µ1, . . . ,λr = µr. When λ = (λ1, . . . ,λk1+···+kp) and λ1 = · · · = λk1 = µ1, . . . ,λk1+···+kp−1+1 = · · · = λk1+···+kp = µp, we accept the notation λ = (µk11 , . . . ,µ kp p ). Definition 3.10. Given a partition λ = (λ1, . . . ,λr), we associate to λ its Young diagram [λ] having r rows such that its i-th row contains λi squares. Moreover we denote by λ′ the partition associated to the transpose diagram of [λ]. Theorem 3.11. (Olsson and Regev [29] for cocharacters and Krakowsky and Regev [26] for codimensions) The cocharacter sequence of the Grassmann algebra is the following: χn(E) = n∑ k=1 (k,1n−k), where n ≥ 1. Moreover its codimension sequence is such that for each n ≥ 1 we have cn(E) = 2n−1. 4. Z2-graded identities In this section we collect the results concerning the Z2-graded identities of E based on the works by the author [7] and Di Vincenzo and da Silva [14]. For the sake of completeness we want to cite the papers [37] and [2] by Anisimov in which the author computes the sequence of involutive codimensions of Grassmann algebra for some special involutions (see [37]), then generalized in [2]. In the latter paper the author gives also an explicit form of the sequence of involutive codimensions of the Grassmann algebra 170 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 for arbitrary involution (exept one case) and for some other groups. The work by Anisimov has been completed by da Silva in [35] for the remaining case. It is also interesting to say that the non-homogeneous G-gradings on E are unknown as well as their corresponding ideals of graded identities. In the sequel for G-grading we mean homogeneous G-grading. In order to simplify the notation we shall use the symbols y’s for variables of Z2-degree 0 and the symbols z’s for variables of Z2-degree 1 as declared in Section 2. Let E = E(L) be the infinite dimensional Grassmann algebra with underlying vector space L and let G be an abelian group. If G is finite and BL = {e1,e2, . . .} is a basis of L, let ϕ : BL → G be any map. Then ϕ induces a homogeneous G-grading on E and viceversa. In this section we consider homogeneous Z2-gradings only. Let us consider the map ϕ : L → Z2 such that ei 7→ 1. The map ϕ gives out the natural grading over E. In this case, let E0 be the homogeneous component of Z2-degree 0 and let E1 be the component of degree 1. It is easy to see that E0 is the center of E and ab + ba = 0 for all a,b ∈ E1. This means that E satisfies the following graded polynomial identities: [y1,y2], [y1,z1], z1z2 + z2z1. Moreover, the latter generates the whole TZ2-graded ideal of E endowed with its natural Z2-grading in the case of characteristic 0. In fact we have the following. Theorem 4.1. (Giambruno, Mischenko, Zaicev [19]) The TZ2-graded ideal of E endowed with its natural Z2-grading is generated by the polynomials [y1,y2], [y1,z1], z1z2 + z2z1, if the characteritic of the ground field is 0. Now, let us consider the Z2-gradings over E induced by the maps degk∗, deg∞, and degk, defined respectively by degk∗(ei) = { 1 for i = 1, . . . ,k 0 otherwise, deg∞(ei) = { 1 for i odd 0 otherwise, degk(ei) = { 0 for i = 1, . . . ,k 1 otherwise. We shall denote by Ek∗, E∞, Ek the Grassmann algebra endowed with the Z2-grading induced by the maps degk∗, deg∞, and degk. We denote by Ed any of the superalgebras Ek∗, E∞, Ek without distinguish them. Let f = z ri1 i1 · · ·zrisis [zj1,zj2] · · · [zjt−1,zjt] and consider the set S := {different homogeneous variables appearing in f}. If h = |S|, then S = {zi1, . . . ,zih}. We consider now T = {j1, . . . ,jt}⊆ S 171 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 and let us denote the previous polynomial by fT (zi1, · · · ,zih). For m ≥ 2 let gm(zi1, . . . ,zih) = ∑ T |T| even (−2)− |T| 2 fT , moreover we set g1(z1) = z1. Let F be an infinite field of characteristic p > 2, then we have the next results (see [7]). Theorem 4.2. Let k ∈ N. If p > k, then all Z2-graded polynomial identities of Ek∗ are consequences of the graded identities: [x1,x2,x3], z1 · · ·zk+1. On the other side, if p ≤ k, all Z2-graded polynomial identities of Ek∗ are consequences of the graded identities: [x1,x2,x3], z1 · · ·zk+1, zp. Theorem 4.3. All the Z2-graded polynomial identities of E∞ are consequences of the graded identities: [x1,x2,x3], z p. Theorem 4.4. Let k ∈ N and set X = Y ∪ Z. Then if p > k all the Z2-graded identities of Ek are consequences of the graded identities: • [x1,x2,x3], • [y1,y2] · · · [yk−1,yk][yk+1,x] (if k is even) • [y1,y2] · · · [yk,yk+1] (if k is odd) • gk−l+2(z1, . . . ,zk−l+2)[y1,y2] · · · [yl−1,yl] (if l ≤ k) • [gk−l+2(z1, . . . ,zk−l+2),y1][y2,y3] · · · [yl−1,yl] (if l ≤ k, l is odd) • gk−l+2(z1, . . . ,zk−l+2)[z,y1][y2,y3] · · · [yl−1,yl] (if l ≤ k, l is odd) If p ≤ k we have to add to the list above the identity • zp From Theorem 4.4 it turns out that a minimal basis of the Z2-graded identities of E either in positive characteristic or in characteristic zero is generated by the polynomials [y1,y2], [y1,z1], z1z2 + z2z1. The case of characteristic zero was the first case which has been considerated and it was completely solved by Di Vincenzo and da Silva in [14]. Their generators in the three cases are the ones above without the polynomial zp. We observe that the identity zp comes from the fact that the E1 component of the Grassmann algebra lies in E∗, then we use Proposition 3.9. In [14] the authors described the sequence of Z2-graded cocharacters and codimensions in the case of characteristic 0. We collect below their results. We shall adopt the following notation. Let λs = (l−s,1s) ` l, µt = (1+t,1m−t−1) ` m be the hook partition of l and m with leg s and arm t respectively. 172 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 Theorem 4.5. Let k ∈ N, then for each n ∈ N the n-th Z2-graded codimension of Ek∗ is cZ2n (Ek∗) = 2 n−1 k∑ t=0 ( n t ) . Let k ∈ N, then the Z2-graded cocharacter sequence of Ek∗ is given by χl,0(Ek∗) = l−1∑ s=0 λs ⊗∅ if l ≥ 1; χ0,m(Ek∗) = m−1∑ t=0 ∅⊗µt if m ≥ 1; χl,m(Ek∗) = l−1∑ s=0 m−1∑ t=0 2(λs ⊗µt) if l ≥ 1, 1 ≤ m ≤ k; χl,m(Ek∗) = 0 if l ≥ 0, m ≥ k + 1. Theorem 4.6. For each n ∈ N the n-th Z2-graded codimension of E∞ is cZ2n (E∞) = 4 n−1 2 . The Z2-graded sequence of E∞ is given by χl,0(E∞) = l−1∑ s=0 λs ⊗∅ if l ≥ 1; χ0,m(Ek∗) = m−1∑ t=0 ∅⊗µt if m ≥ 1; χl,m(Ek∗) = l−1∑ s=0 m−1∑ t=0 2(λs ⊗µt) if m,l ≥ 1; Theorem 4.7. Let k ∈ N, then for each n ∈ N we have cn−m,m(Ek) = k∑ t=0 ( n−1 t ) if m ≥ 1; cn,0(Ek) = e(k)∑ t=0 ( n−1 t ) , where e(k) = { k if k is even k −1 if k is odd. 173 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 Let k ∈ N, then the Z2-graded cocharacter sequence of Ek is given by χl,0(Ek) = l−1∑ s=0 λs ⊗∅ if l ≤ k; χl,0(Ek) = e(k)∑ s=0 λs ⊗∅ if l ≥ k + 1; χ0,m(Ek) = m−1∑ t=0 ∅⊗µt if t ≤ k; χ0,m(Ek) = e(k)∑ t=0 ∅⊗µt if t ≥ k + 1; χl,m(Ek) = l−1∑ s=0 m−1∑ t=0 ms,t(λs ⊗µt) if l,m ≥ 1, where ms,t =   2 if s + t ≤ k −1 1 if s + t = k 0 otherwise. 5. G-graded identities of E Now we consider the more general case of a homogeneous G-grading of E, where G is a finite abelian group. We show that in order to study the G-graded identities of E we may reduce to G′-gradings, where G′ is a group having a smaller number of elements than G. We give the proofs of the main results. The complete contents related to this section may be found in [8]. We also recall that from now on every field is supposed to be of characteristic 0. Finally, if H / G we shall denote by gH or simply by g the coset of g modulo H, where g ∈ G. Let us consider the following homomorphism between free graded algebras π : F〈X|G〉→ F〈Y |G/H〉, such that for every g ∈ G and for every i ∈ N, π(xgi ) = y gH i , where H is a subgroup of G. Definition 5.1. Let G be a finite abelian group and suppose E is G-graded. We say that the subgroup H of G has the property P when for any h ∈ H, Eh has infinitely many elements of even length with pairwise disjoint support. The importance of this property is given by the following proposition. Proposition 5.2. Let H < G having the property P and let f ∈ F〈X|G〉 be a multilinear polynomial. Then f ∈ TG(E) if and only if π(f) ∈ TG/H(E). 174 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 Proof. We have to prove just the only if part. Let f = f(x g1 1 ,x g1 2 , . . . ,x g1 lg1 , . . . ,x gr∑r−1 i=1 lgi +1 , . . . ,x gr∑ r i=1 lgi ) ∈ TG(E) and let F = π(f). Let ϕ be any G/H-graded substitution, hence ϕ(ygHj ) = ∑ h∈H a gh j , and by the multilinearity of f, we can consider only substitutions ϕ such that ygHj 7→ a gh j , for some h ∈ H and for any j. Now, we observe that every homogeneous component Eh has infinitely many elements of even length with pairwise disjoint supports because H satisfies the property P, then for every j and for every h ∈ H exists bh −1 j of even length such that ‖b h−1 j ‖ = h −1. For every h ∈ H, wgj = a gh j b h−1 j is a homogeneous element of degree g in the G-grading of E. Let us consider a new substitution ψ such that xgj 7→ w g j . This is a G-graded substitution. Now, since f ∈ TG(E), 0 = f(w g1 1 , . . . ,w gr lgr ) = ∏ h∈H,j b h−1 j ·F(a gh j ) because the bh −1 j ’s are in Z(E) and this implies F(a gh j ) = 0. We observe that if Lg is infinite dimensional and |H| = n is odd, then H = 〈g〉 has the property P. Moreover even if G is a finite abelian group and H = 〈g|dimF Lg = ∞ and o(g) is odd〉, then H has the property P. We shall adopt the following notation: if H is a subgroup of G, we shall denote by g the translate gH ∈ G/H. We have the following. Theorem 5.3. Let G be a finite abelian group of odd order and let H = 〈g|dimF Lg = ∞〉. Then the following properties hold: 1. for any multilinear polynomial f(x1, . . . ,xn) ∈ F〈X〉 one has that f ∈ TG(E) if and only if π(f) ∈ TG/H(E). 2. In the quotient grading of E, Lg is infinite dimensional if and only if g = 1G/H. Now let us consider the following subsets of G : I = {g ∈ G|dimF Lg = ∞}, I1 = {g ∈I|o(g) is odd}, I2 = I−I1 and I3 = {g2|g ∈I2}−I1. We have the following. Theorem 5.4. Let G be a finite abelian group and let H = 〈g|g ∈I1∪I3〉. Then the following properties hold: 1. for any multilinear polynomial f = f(x1, . . . ,xn) ∈ F〈X〉 one has f ∈ TG(E) if and only if π(f) ∈ TG/H(E). 2. In the quotient grading of E, if Lg is infinite dimensional, then g2 = 1G/H. 175 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 Proof. (1). Let h ∈ H, then there exist a1, . . . ,ar ∈I1, b1, . . . ,bs ∈I3 and positive integers such that h = am11 · · ·a mr r b mr+1 1 · · ·b mr+s s . Let ar+1, . . . ,ar+s ∈I2 such that bi = a2r+i, then dimF L ai = ∞ for any i = 1, . . . ,r + s. Let us denote by Ei the Grassmann algebra generated by the subspace La mi i . For any i = 1, . . . ,r + s, Ei contains infinite elements wi1,w i 2, . . . ,w i m, . . . of even length with pairwise disjoint supports. Moreover, for all m ≥ 1 we have that ∥∥wim∥∥ = amii if i = 1, . . . ,r and ∥∥wim∥∥ = bmii−r for i = r + 1, . . . ,r + s. We consider in Eh the elements um = w1m · · ·wr+sm , m ≥ 1; clearly the elements {um|m ≥ 1} have pairwise disjoint supports and they have even length. Now H has the property P and the assertion comes by Proposition 5.2. (2). Let g = gH ∈ G/H be such that Lg = ⊕ h∈H L gh is infinite dimensional. Since G is finite there exists g′ ∈ gH such that Lg ′ is infinite dimensional. If o(g′) is odd, then g′ ∈ H and so gH = g′H = 1G/H. If o(g′) is even, then g′2 ∈ H and so (gH)2 = (g′H)2 = 1G/H. In light of Theorems 5.3 and 5.4, we list the results about the G-graded identities of E in the case dimF L 1G = ∞. Theorem 5.5. Let G = {g1, . . . ,gr} be a finite abelian group with g1 = 1G. Suppose that Lg1 has infinite dimension. Let lg1, lg2, . . . , lgr ∈ N such that lg1 + lg2 + . . . + lgr = m. Then Plg1,...,lgr ⊆ TG(E) or for any f ∈ Plg1,lg2,...,lgr one has f(x g1 1 , . . . ,x g1 lg1 , . . . ,x gr∑r−1 i=1 lgi +1 , . . . ,x gr∑ r i=1 lgi ) ∈ TG(E) if and only if f(x1, . . . ,xm) ∈ T(E). Theorem 5.6. Let G = {g1, . . . ,gr} be a finite abelian group with g1 = 1G. Let L be a G-homogeneous vector space over L such that dimF Lg1 = ∞ and dimF Lgi = ki < ∞, if i 6= 1. If E = E(L) is the Grassmann algebra generated by L, then TG(E) is generated as a TG-ideal by the following polynomials: 1. [u1,u2,u3] for any choice of the G-degree of the variables u1,u2,u3. 2. monomials of P0,t2,...,tr such that ∑r i=2 ti = 1 + ∑r i=2 ki 3. monomials of P0,t2,...,tr such that ∑r i=2 ti < 1 + ∑r i=2 ki and P0,t2,...,tr ⊆ TG(E). As a consequence of the previous results, we have, up to combinatorics, the following description of the G-graded cocharacters in the case L1G is infinite dimensional. Corollary 5.7. Let G = {g1, . . . ,gr} be a finite abelian group with g1 = 1G. If Lg1 has infinite dimension and lg1, lg2, . . . , lgr ∈ N such that lg1 + lg2 + . . . + lgr = m, then clg1,...,lgr (E) = 0 or clg1,...,lgr (E) = 2 m−1 and in this last case, Plg1,...,lgr (E) and Pm(E) are isomorphic Slg1 ×···×Slgr -modules. 176 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 Let us consider now the set S(ϕ) = { (lg1, lg2, . . . , lgr ) ∈ N r|Plg1,lg2,...,lgr ⊆ TG(E) } . We note that if L1G is the only homogeneous subspace of L such that dimF L1G = ∞, then S(ϕ) 6= ∅. S(ϕ) allows us to give the complete description of the sequence of the graded cocharacters and codimensions of E. In fact, we have the following proposition. Theorem 5.8. Let G = {g1, . . . ,gr} be a finite abelian group and L be a G-homogeneous vector space with linear basis {e1,e2, . . .}. Let ϕ : BL → G be a map such that |ϕ−1(1G)| = ∞ and consider E, the G-graded Grassmann algebra obtained by ϕ. Then χGlg1,...,lgr (E) = 2|G|−1 ∑lg1−1 a1=0 ∑lg2−1 a2=0 · · · ∑lgr−1 ar=0 λa1 ⊗λa2 ⊗···⊗λar if (lg1, . . . , lgr ) /∈ S(ϕ), where λai is the hook partition of leg ai and arm lgi −ai + 1. Moreover cGn (E) = 2 n−1 ∑ (lg1, . . . , lgr ) /∈ S(ϕ) lg1 + . . . + lgr = n ( n lg1, . . . , lgr ) . 6. Other results In what follows we recall some results about the (graded) polynomial identities of structures related to the Grassmann algebra. We shall consider a G-graded algebra A and the canonical Z2-grading of the Grassmann algebra E = E0 ⊕E1 over a field of characteristic 0, and compare the G-graded identities of A with the G×Z2- graded identities of the G×Z2-graded algebra A⊗E with homogeneous components given by (A⊗E)(g,i) := Ag ⊗Ei. Notice that the free algebra F〈X|G×Z2〉 is both a G-graded algebra and a Z2-graded algebra. Refer- ring to the Z2-grading of F〈X|G×Z2〉one defines the map ζ as follows. Let m be a multilinear monomial in F〈X|G×Z2〉 and let i1 < · · · < ik be the indexes of the variables with odd Z2-degree occurring in m. Then, for some σ in the symmetric group Sk({i1, . . . , ik}), we may write m = m0zσ(i1)m1zσ(i2) · · ·mk−1zσ(ik)mk, where m0, . . . ,mk are multilinear monomials in even variables only and zij are odd variables. Then, as in Kemer [22], Di Vincenzo and Nardozza [13] define ζ(m) := (−1)σm. Note that ζ(ζ(m)) = m. We define a similar map from the free G-graded algebra to the free G×Z2-graded algebra. Definition 6.1. Let J ⊆ N. Let ϕJ : F〈X〉→ F〈X|G×Z2〉 be the unique G-homomorphism defined by the map ϕJ(x g) = { x(g,0) if i /∈ J x(g,1) if i ∈ J. Also, for a multilinear monomial m ∈ PGn , define ζJ(m) := ζ(ϕJ(m)). The map ϕJ depends on J, of course. We may extend the map ζJ by linearity to the space of all G-graded multilinear polynomials PGn . If f ∈ PGn , then ζJ(f) is a multilinear element of F〈X|G×Z2〉. We have the following result (see Theorem 11 of [13]). Theorem 6.2. Let S be a system of multilinear generators for TG(A). Then the system {ζJ(f) ∈ F〈X〉 |f ∈ S,J ⊆ N} is a set of multilinear generators for TG×Z2(A⊗E). 177 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 If we consider the case of positive characteristic, the previous result is verified for the algebra UT2(E) of upper triangular 2×2 matrices. Theorem 6.3. (C. and da Silva [9]) Let F be a field of characteristic p > 2 and E be graded with its natural Z2-grading. Let us set S := {[y1,y2], [y1,z1], z1z2 ◦z2z1}, then TZ2(UT2(E)) = {ζJ(f)|f ∈ S}. We observe that the previous result does not depend on the characteristic of the field. Moreover TZ2(UT2(E)) = TZ2(E)TZ2(E) as in the ordinary case. We have the next related result about the G×Z2-cocharacters of A⊗E. Theorem 6.4. (Di Vincenzo and Nardozza [13]) Let n ∈ N and k1, l1, . . . ,kr, lr ∈ N such that ∑r i=1 ki + li = n and consider H = Sk1 ×Sl1 ×···Skr ×Slr . If (χGn (A))↓H = ∑ mλ1,µ1,...,λr,µrλ1 ⊗µ1 ⊗···⊗λr ⊗µr, then χG×Z2n (A⊗E) = ∑ ∑ i (ki+li)=n ∑ λi ` ki µi ` li mλ1,µ1,...,λr,µrλ1 ⊗µ ′ 1 ⊗···⊗λr ⊗µ ′ r. We give a short account of the structure theory of T-ideals developed by Kemer [22]. Definition 6.5. The T-ideal S of F〈X〉 is called T-semiprime or verbally semiprime if any T-ideal U such that Uk ⊆ S for some k, lies in S, i.e. U ⊆ S. The T-ideal P is T-prime or verbally prime if the inclusion U1U2 ⊆ P for some T-ideals U1 and U2 implies U1 ⊆ P or U2 ⊆ P. Let E = E0 ⊕E1 be endowed with its canonical Z2-grading, then the vector subspace of Ma+b(E), Ma,b(E) := {( r s t u ) |r ∈ Ma(E(0)),s ∈ Ma×b(E(1)),u ∈ Mb(E(0)) } is an algebra. The building blocks in the theory of Kemer are the polynomial identities of the matrix algebras over the field and over the Grassmann algebra and the algebras Ma,b(E). In fact, we have the following theorem. Theorem 6.6. 1. For every T-ideal U of F〈X〉 there exist a T-semiprime T-ideal S and a positive integer k such that Sk ⊆ U ⊆ S. 2. Every T-semiprime T-ideal S is an intersection of a finite number of T-prime T-ideals Q1, . . . ,Qm, S = Q1 ∩·· ·∩Qm. 3. A T-ideal P is T-prime if and only if P coincides with one of the following T-ideals: T(Mn(F)), T(Mn(E)), T(Ma,b(E)), (0), F〈X〉. We recall that if two algebras A and B satisfy the same polynomial identities we say that A is PI-equivalent to B and denote by A ∼ B. An important corollary to the structure theory of Kemer is the Tensor Product Theorem (TPT) which follows from the result by Kemer [22]. 178 L. Centrone / J. Algebra Comb. Discrete Appl. 4(2) (2017) 165–180 Theorem 6.7. The tensor product of two verbally prime algebras is PI equivalent to a verbally prime algebra. More precisely, let a,b,c,d ∈ N such that a ≥ b and c ≥ d and F be a field of characteristic 0, then: 1. Ma,b(E)⊗E ∼ Ma+b(E); 2. Ma,b(E)⊗Mc,d(E) ∼ Mac+bd,ad+bc(E); 3. M1,1(E) ∼ E ⊗E. The remaining PI equivalences follow from the isomorphism of the corresponding algebras. An alternative proof of the TPT can be be found in the paper by Regev [33]. In [33] we also have the proof that the TPT is still valid for multilinear identities in the case E is the infinite dimensional Grassmann algebra over an infinite field of characteristic p 6= 2. In the papers [4], [5] and [25] the authors deal with graded identities for certain gradings on some of the verbally prime algebras. In the paper [25] the authors constructed an appropriate model for the relatively free algebra in the variety of algebras determined by E⊗E when the field F has characteristic p > 2. This model is the generic algebra of A = F ⊕M1,1(E∗). It turned out that E ⊗E and A satisfy the same graded and hence ordinary polynomial identities. In [4] the authors used the properties of A in order to show that that T(M1,1(E)) ( T(E⊗E). 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