Graded identities of matrix algebras and the universal graded algebra

We consider fine G-gradings on M_n(C) (i.e. gradings of the matrix algebra over the complex numbers where each component is 1 dimensional). Groups which provide such a grading are known to be solvable. We consider the T-ideal of G-graded identities and show that it is generated by a special type of binomial identities which we call elementary. In particular we show that the ideal of graded identities is finitely generated as a T-ideal. Next, given such grading we construct a universal algebra U_{G,c} in two different ways: one by means of polynomial identities and the other one by means of a generic two-cocycle (this parallels the classical constructions in the non-graded case). We show that a suitable central localization of U_{G,c} is Azumaya over its center and moreover, its homomorphic images are precisely the G-graded forms of M_n(C). Finally, we consider the ring of central quotients Q(U_{G,c}) (this is an F-central simple algebra where F=Frac(Z) and Z is the center of of U_{G,c}). Using an earlier results of the authors (see E. Aljadeff, D. Haile and M. Natapov, Projective bases of division algebras and groups of central type, Israel J. Math.146 (2005) 317-335 and M. Natapov arXiv:0710.5468v1 [math.RA]) we show that this is a division algebra for a very explicit (and short) family of nilpotent groups. As a consequence, for groups G such that Q(U_{G,c}) is not a division algebra, one can find a non identity polynomial p(x_{i,g}) such that p(x_{i,g})^r is a graded identity for some integer r. We illustrate this phenomenon with a fine G-grading of M_6(C) where G is a semidirect product of S_3 and C_6.

In the last decade, group gradings and graded identities of finite dimensional central simple algebras have been an active area of research. We refer the reader to Bahturin, et al [6] and [7]. There are two basic kinds of group grading, elementary and fine. It was proved by Bahturin and Zaicev [7] that any group grading of M n (C) is given by a certain composition of an elementary grading and a fine grading. In this paper we are concerned with fine gradings on M n (C) and their corresponding graded identities.
Let R be a simple algebra, finite dimensional over its center k and G a finite group. We say that R is fine graded by G if R ∼ = ⊕ g∈G R g is a grading and dim k (R g ) ≤ 1. Thus any component is either 0 or isomorphic to k as a k-vector space. It is easy to show that Supp(R), the subset of elements of G for which R g is not 0, is a subgroup of G. Moreover R is strongly graded by Supp(R), namely R g R h = R gh for every g, h ∈ Supp(R). Since every group Γ containing Supp(R) provides a fine grading of R (just by putting R g = 0 for g outside Supp(R)) we will restrict our attention to the case where G = Supp(R). In this case it has been shown (see Bahturin and Zaicev [7]) that the existence of such a grading is equivalent to R being isomorphic to a twisted group algebra k c G where [c] is an element in H 2 (G, k × ) and G acts trivially on k × .
A group G is said to be of central type if it admits a cocycle c, [c] ∈ H 2 (G, C × ), for which the twisted group algebra C c G is central simple over C. By definition such a cocycle c is called nondegenerate. As we have just seen fine gradings arise from such groups and cocycles. Groups of central type appear in the theory of projective representations of finite groups and in the classification of finite dimensional Hopf algebras. Using the classification of finite simple groups, Howlett and Isaacs [12] proved that any group of central type is solvable.
Given a (fine) G-grading on M n (C) we consider the set of graded identities: let k be a subfield of C and let Ω = {x ig : i ∈ N, g ∈ G} be a set of indeterminates. Let Σ(k) = k Ω be the noncommutative free algebra generated by Ω over k. A polynomial p(x ig ) ∈ Σ(k) is a graded identity of M n (C) if the polynomial vanishes upon every substitution of the indeterminates x ig by elements of degree g in M n (C). It is clear that the set of graded identities is an ideal of Σ(k). Furthermore, it is a graded T -ideal. Recall that an ideal (in Σ(k)) is a graded T -ideal if it is closed under all G-graded endomorphisms of Σ(k). In section two we show that all graded identities are already defined over a certain finite cyclotomic field extension Q(µ) of Q. Moreover Q(µ) is minimal and unique with that property. We refer to Q(µ) as the field of definition of the graded identities. We consider the free algebra Σ(Q(µ)) and denote by T (Q(µ)) the T -ideal of graded identities of M n (C) in Σ(Q(µ)). We introduce a set of special graded identities which we call "elementary". We show that a certain finite subset of the set of elementary identities generate T (Q(µ)). In particular T (Q(µ)) is finitely generated as a T -graded ideal.
In section three we examine the algebra U G = Σ(Q(µ))/T (Q(µ)), which we call the universal G-graded algebra. First we introduce another algebra analogous to the ring of generic matrices in the classical theory. We show this algebra is isomorphic to U G and thus are able to prove that the center Z = Z(U G ) is a domain and that if F denotes the field of fractions of Z, then the algebra Q(U G ) = F ⊗ Z U G is an F -central simple algebra of dimension equal to the order of G. In particular U G is a prime ring. Moreover we show there is a certain multiplicatively closed subset M in Z such that the central localization M −1 U G is an Azumaya algebra over its center S = M −1 Z. The simple images of this Azumaya algebra are the graded forms of M n (C), that is the G-graded central simple L-algebras B such that B ⊗ L C is isomorphic as a graded algebra to M n (C), where L varies over all subfields of C. We also give quite explicit determinations of S and F . Note that these algebras depend on the given grading on M n (C) and so we should write U G,c , M −1 U G,c and Q(U G,c ), where c is the given nondegenerate two-cocycle. We will omit the c except in those cases where we need to emphasize the particular grading.
Unlike the case of classical polynomial identities, the central simple algebra Q(U G ) is not necessarily a division algebra. Based on earlier work of the authors [2] we in fact show that Q(U G,c ) is a division algebra if and only if the group G belongs to a certain explicit list Λ of groups. This is independent of the cocycle c, a fact we will return to below. The list Λ consists of a very special family of nilpotent groups. Roughly speaking these are the nilpotent groups for which each Sylow-p subgroup is the direct product of an abelian group of the form A × A (called of symmetric type) and possibly a unique nonabelian group of the form C p n ⋉ C p n . For p = 2 an extra family of non abelian groups can occur, namely C 2 × C 2 n−1 ⋉ C 2 n . The precise definition of Λ is given in the last section.
The main result of the final section is that for every group G on the list Λ the automorphism group of G acts transitively on the cohomology classes represented by nondegenerate two-cocycles. It follows that the algebras Q(U G,c ) are all isomorphic for a fixed G (but not graded isomorphic).
For a group G of central type let ind(G) denote the maximum over all nondegenerate cocycles c of the indices of the simple algebras Q(U G,c ). We have just seen that if G is not on the list, then ind(G) is strictly less than the order of G. In a forthcoming paper, Aljadeff and Natapov [4], it is shown that in fact the groups on the list are the only groups responsible for the index. The precise result is stated in the last section (Theorem 21). It follows from this theorem that if G is a p-group of central type then ind(G) ≤ max(ord(H) 1/2 ) where the maximum is taken over all p-groups H on the list that are sub-quotients of G.
If Q(U G ) is not a division algebra, there are graded identities over the field of definition that are the product of nonidentities. We present an explicit example of a grading on M 6 (C) (for the group G = S 3 ⋉ C 6 ) for which there is a graded identity over the field of definition that is the cube of a nonidentity.
Even in the case where Q(U G ) is a division algebra it is possible that this universal algebra does not remain a division algebra under extension of the coefficient field. In other words the algebra Q(U G ) ⊗ Q(µ) C may not be a division algebra. This means that there can be gradings on matrices for which the algebra Q(U G ) is a division algebra and yet there are graded identities over C that are the products of nonidentities. In the last section we compute explicitly the index of this extended algebra Q(U G ) ⊗ Q(µ) C for groups G on the list Λ.

Graded Identities.
Let M n (C) have a fine grading by the finite group G. In this section we investigate the Tideal of G-graded identities on M n (C). As we have seen in the introduction the existence of such a grading implies that G is of central type and that there is a nondegenerate cocycle c such that M n (C) is isomorphic to the twisted group algebra C c G. We want to make this more precise. Let A = M n (C). The G-grading induces a decomposition A ∼ = ⊕A g where each homogeneous component is 1-dimensional over C. Every homogeneous component A g is spanned by an invertible element u g (which we also fix from now on) and hence any element in A g is given by λ g u g where λ g ∈ C. The multiplication is given by u g u h = c(g, h)u gh where c(g, h) ∈ C × is a two-cocycle. We let [c] ∈ H 2 (G, C × ) be the corresponding cohomology class. In this way we identify A (as a G-graded algebra) with the twisted group algebra C c G. Replacing c by a cohomologous cocycle c ′ produces a twisted group algebra C c ′ G that is isomorphic as a G-graded algebra to A.
We want to obtain information about the T -ideal of identities of M n (C). We begin with a special family of identities. As in the introduction we let Σ(C) = C Ω denote the free algebra generated by Ω over C, where Ω = {x ig : i ∈ N, g ∈ G}.
Let Z 1 = x r 1 g 1 x r 2 g 2 · · · x r k g k and Z 2 = x s 1 h 1 x s 2 h 2 · · · x s l h l be two monomials in Σ(C). We say Z 1 and Z 2 are congruent if the following three properties are satisfied: a) g 1 · · · g k = h 1 · · · h k b) k = l (equal length) c) there exists an element π ∈ Sym(k) such that for 1 ≤ i ≤ k, s i = r π(i) and h i = g π(i) .
We say Z 1 and Z 2 are weakly congruent if they satisfy condition (a). Clearly congruence and weakly congruence are equivalence relations.
Let c(g 1 , g 2 , . . . , g k ) be the element in C × determined by u g 1 u g 2 · · · u g k = c(g 1 , . . . , g k )u g 1 ···g k in C c G. For any two congruent monomials Z 1 = x r 1 g 1 x r 2 g 2 · · · x r k g k and Z 2 = x r π(1) g π(1) x r π(2) g π(2) · · · x r π(k) g π(k) consider the binomial It is easy to see that B is a graded identity. We will call it an elementary identity. We refer to c(B) = c(g 1 ,...,g k ) c(g π(1) ,...,g π(k) ) in C × as the coefficient of the elementary identity B. This element c(B) can be understood homologically. Let F = F (Ω) be the free group generated by Ω = {x ig : i ∈ N, g ∈ G}. Let ν : F −→ G (x ig −→ g) be the natural map and R = ker(ν). If π ∈ Sym(k) and Z 1 = x r 1 g 1 x r 2 g 2 · · · x r k g k , Z 2 = x r π(1) g π(1) x r π(2) g π(2) · · · x r π(k) g π(k) are congruent, then the element z = Z 1 (Z 2 ) −1 lies in [F, F ]∩ R. By the Hopf formula the Schur multiplier M(G) of G is given by M(G) = [F, F ] ∩ R/[F, R] and so we may consider the element z ∈ M(G). Moreover the Universal Coefficients Theorem provides an isomorphism φ from H 2 (G, C × ) to Hom(M(G), C × ), Proposition 1. Let the notation be as given above.   Proof. Statement (a) is clear. For the proof of (b) we consider the central extension induced by the map ν and let be the central extension that corresponds to the cocycle c (i.e. {u g } g∈G is a set of representatives in Γ and u g u h = c(g, h)u gh ). It is easy to see that the map γ : F/[F, R] → Γ given by γ(x ig ) = u g induces a map of extensions and it is well known its restriction γ : [F, F ] ∩ R/[F, R] = M(G) → C × is independent of the presentation and corresponds to [c] by means of the Universal Coefficient Theorem. That is φ([c]) = γ. It follows that φ([c])(z) = γ(z) = u g 1 · · · u gr (u g π(1) · · · u g π(r) ) −1 = c(g 1 ,...,g k ) c(g π(1) ,...,g π(k) ) = c(B) as desired. Statements (c) and (d) are direct consequences of (b). This completes the proof of the proposition.
From the Proposition it follows that the set J is defined over Q(µ). Our next task is to show that the the T -ideal of graded identities T (C) is spanned by J and that Q(µ) is the field of definition for the graded identities. We will need the following terminology: Let k be a subfield of C and let p(x r i g i )= λ r 1 g 1 ,r 2 g 2 ,...,r k g k x r 1 g 1 x r 2 g 2 · · · x r k g k be a polynomial in Σ(k) (⊆ Σ(C)). We say p(x r i g i ) is reduced if the monomials x r 1 g 1 x r 2 g 2 · · · x r k g k are all different. If p(x r i g i ) is reduced we say it is weakly homogeneous if its monomials are all weakly congruent and we say p(x r i g i ) is homogeneous is if its monomials are all congruent. It is clear that any reduced polynomial can be written uniquely as the sum of its weakly homogeneous components and that each weakly homogeneous component can be written uniquely as the sum of its homogeneous components. Proposition 2. If p(x r i g i ) is a graded identity then its homogeneous components are graded identities as well. Moreover, any graded identity is a linear combination of elementary identities.
Proof. Let us show first that its weakly homogeneous components are graded identities. Indeed, the replacement of x r i g i by u g i ∈ C c G maps weakly homogeneous components of p(x r i g i ) to different homogeneous components (in the graded decomposition of C c G). Since these are linearly independent, they all must be 0. Next, assume that we have at least two homogeneous components in a weakly homogeneous component which we denote by Π. Then there exists an indeterminate y = x r j g j which appears with different multiplicities in (at least) two different monomials. Let s ≥ 1 be the maximal multiplicity. We decompose Π into (at most) s + 1 components U s , . . . , U 0 where U i consists of the monomials of Π that contain y with multiplicity i. Of course, by induction, the result will follow if we show that U s is a graded identity. Suppose not. Then there is a substitution which does not annihilate U s . Note that since Π is weakly homogeneous the image of all components are multiples of u g in C c G and hence there is an evaluation that maps U i to λ i u g with λ s not zero. We may multiply the evaluation for y by a central indeterminate z. Then we get a non zero polynomial in z, whose coefficients are λ i and for any evaluation of z we get zero. This is of course impossible in a field of characteristic zero.
To complete the proof of the proposition it suffices to show that every homogeneous identity is a linear combination of elementary identities. But this is clear since any two monomials which are congruent determine an elementary identity (and monomials are not identities).
Proposition 3. If L is a subfield of C such that Q(µ) ⊆ L ⊆ C then the set J spans T (L) over L. Conversely, if L ⊆ C is a field of definition for T (C), that is T (C) = T (L) ⊗ L C, then L contains Q(µ).
Proof. Fix a monomial Z = x r 1 g 1 x r 2 g 2 · · · x r k g k . Let Z = Z 1 , Z 2 . . . , Z d be the distinct monomials that are congruent to Z and let W be the d-dimensional C-vector space spanned by these monomials. Each Z i , for 1 ≤ i < d determines an elementary identity Z i − c i Z d and these identities form a basis for the subspace Y of graded identities in W . In particular the dimension of Y over C is d − 1. Furthermore if γ = {c 1 , . . . , c d−1 } then clearly Y is defined over Q(γ). But γ ⊆ µ and so it follows that T (C) is defined over Q(µ) and hence over any field L that contains Q(µ).
For the converse let W L denote the L-span of the monomials Z 1 , . . . , Z d and let Y L = Y ∩ W L . We claim that if L is a field of definition for Y (that is, Y = Y L ⊗ L C), then L ⊇ Q(γ). Because Z is arbitrary, it will follow from the claim that L contains µ. To prove the claim, let f 1 , . . . , f d−1 be a basis of Y L over L. Express f 1 , . . . , f d−1 using the monomials Z i . The coefficient matrix is a (d − 1) × d matrix over L. This matrix may be row reduced to the normal form ( The column vector C ′ is uniquely determined. In other words there are unique scalars a i in L, for 1 ≤ i < d, such that the vectors Z i − a i Z d form a basis for Y L over L. On the other hand the identities Z i − c i Z d form a basis of Y L(γ) over L(γ). It follows that a i = c i ∈ L for all i.
We now show that the T -ideal of graded identities is finitely generated. Let n = ord(G). Consider the set V = {x ig : 1 ≤ i ≤ n , g ∈ G} of indeterminates and let E be the set of elementary identities of length ≤ n (that is where the monomials are of length ≤ n) and such that its indeterminates are elements of V . Clearly E is a finite set.
Theorem 4. The ideal of graded identities is generated as a T -ideal by E. In particular the ideal is finitely generated as a T -ideal.
Proof. Denote by I ′ the T -ideal generated by E. We will show that I ′ contains all graded identities. Clearly it is enough to show that I ′ contains all elementary identities. Let We assume therefore that k > n and proceed by induction on k. The first observation is that if there exist i, t, with 1 ≤ i, t ≤ k − 1 i = π(t) and i + 1 = π(t + 1) then we can reduce the length of the word: We let g = g i g i+1 and replace x r i g i x r i+1 g i+1 by x rg , where 1 ≤ r ≤ n and x rg does not appear in B. The resulting identity (one has to check that the coefficient is right) has shorter length and so we may assume this identity is in I ′ . But we can obtain the longer one from the shorter one by the substitution of x r i g i x r i+1 g i+1 for x rg , so we are done in this case.
Next, note that since k > n, the pigeonhole principle applied to the expressions g 1 · · · g s , s = 1, . . . , k shows that there are integers 1 ≤ i < j ≤ n such that g i g i+1 · · · g j = 1. Observe also, that if γ is a cyclic permutation of the numbers i, i + 1, . . . , j then we have g γ(i) · · · g γ(j) = 1. That means that if we replace the string x r i g i x r i+1 g i+1 · · · x r j g j by a cyclic permutation of the variables in the first monomial and leave the second monomial alone, we do not change our identity modulo I ′ . Also for all g ∈ G, we have gg i g i+1 · · · g j = g i g i+1 · · · g j g. This means that modulo I ′ we can move the string x r i g i x r i+1 g i+1 · · · x r j g j anywhere in the monomial. So, combining these two things we can cyclically move the substring and move it anywhere in the first monomial. But by doing these moves we can arrange it so that some two variables that are next to each other in the second monomial are also next to each other (in the same order) in the first monomial. This reduces us to the first case.

The Universal Algebra.
In this section we determine the basic structural properties of the universal G-graded algebra U G and relate that algebra to the other algebras described in the introduction. The results are analogous to results in the theory of (non-graded) polynomial identities. We will see for example that U G is a prime ring and that its algebra of central quotients, the universal G-graded algebra, is central simple.
We start with the following lemma.
Lemma 5. Let c be a nondegenerate cocycle on G and let C c G be the twisted group algebra. Let L be a subfield of C.
(a) Let β : G × G → L × be a two-cocycle and let L β G be the twisted group algebra. There is a homomorphism η : L β G → C c G over L of G-graded algebras if and only if the cocycles c and β are cohomologous in C. In particular Im(η) is a G-graded form of C c G.
It follows that β and c are cohomologous over C. A similar calculation shows the other direction.
(b) We have seen in Proposition 1 that φ([c]) = µ where φ is the isomorphism between H 2 (G, C × ) and Hom(M(G), C × ). Hence if β is cohomologous to c over C then L must contain µ by part (d) of Proposition 1.
For the converse we may assume that L = Q(µ). By the naturality of the Universal Coefficient Theorem we have a commutative diagram The first step in the analysis of U G is the construction of a counterpart to the ring of generic matrices. Let G be a group of central type of order n 2 and let c : G × G → C × be a nondegenerate two-cocycle. For every g ∈ G let t ig for i = 1, 2, 3, . . . be indeterminates.
For each g ∈ G let t g = t 1g . We will assume u 1 = 1 (i.e. the cocycle c is normalized). Let k denote the field generated over Q by the indeterminates and the values of the cocycle. Note that by Lemma 5, k contains Q(µ). Consider the twisted group algebra k c G. Because the cocycle is nondegenerate Q(c(g, h)) c G is a central simple Q(c(g, h))-algebra and hence k c G is a central simple k-algebra. Now let U G denote the Q(µ)-subalgebra of k c G generated by the elements t ig u g for all i and all g ∈ G. We want to describe U G . Let Z denote the center of U G . Proposition 6. (a) The center Z of U G is the Q(µ)-subalgebra of U G generated by the following set of elements: Proof. The ring kU G contains the elements u g for all g ∈ G and so kU G = k c G, which has center k. Hence the center of U G is U G ∩ k. But this intersection is precisely the Q(µ) span of the given set of monomials. This proves both parts.
. This is a two-cocycle, cohomologous to c over k. Because c is nondegenerate, so is s and hence the algebra k s G is k-central simple. In fact it is convenient to view the algebra k s G as equal to k c G. It is spanned over k by the elements t g u g .
Let Y be the subgroup of k × generated by the values of s and let S = Q(µ)[Y ], a subring of k. The S-subalgebra of k s G generated by the elements t g u g is the twisted group algebra S s G. Let M = {t g 1 u g 1 t g 2 u g 2 · · · t gm u gm | g 1 g 2 · · · g m = 1}. Let F ⊆ k denote the field of fractions of Z.

Proposition 7. (a) The set M is a multiplicatively closed subset of Z and M
. Observe that t e and t g t g −1 c(g, g −1 ) are in M and hence t ig t g −1 c(g, g −1 )/t g t g −1 c(g, g −1 ) = t ig /t g is in M −1 Z. Next, the elements Similar calculations show that the opposite inclusion also holds.
(b) This follows easily from part (a).
Let ψ denote the Q(µ)-algebra homomorphism from the free algebra Σ(Q(µ)) to k c G given by ψ(x ig ) = t ig u g . The image of this map is clearly U G .
In particular the universal algebra U G is a prime ring with center isomorphic to Z and and its ring of central Proof. This is clear since a polynomial p(x ig ) ∈ Σ(Q(µ)) is an identity of C c G if and only if p(λ ig u g ) = 0 for any λ ig ∈ C and this is equivalent to p(t ig u g ) = 0 where {t ig } are central indeterminates.
Because of this isomorphism we will from now on drop the bars on Z, F , etc.
We want to say more about F , the field of fractions of the center of the universal algebra U G . The group Y is finitely generated and hence of the form Y t Y f where Y t is the torsion subgroup of Y and Y f is a finitely generated free abelian group. Because Y is a subgroup of k × it follows that Y t is cyclic. We will see in the next proposition that in fact Y t = µ. Let y 1 , y 2 , . . . , y m ∈ k denote a basis for Y f . (We will see later -in Proposition 14 -that the rank of Y f is equal to n, the order of G, and so m = n.) , the ring of Laurent polynomials in y 1 , . . . , y m over Q(µ).
. In particular F is isomorphic to the field of rational functions in countably many variables over Q(µ).
We use the notation and set-up of part (b) of Proposition 1. Let d be the element of the free group F given by , the variables t g in the expression for λ must all cancel and so d ∈ [F, F ] and λ = c(g 1 , h 1 )) ǫ 1 c(g 2 , h 2 )) ǫ 2 · · · c(g k , h k )) ǫ k . In particular we may consider the element d in M(G). Then again as in part (b) of Proposition 1 we have φ([c])(d) = γ(d) = c(g 1 , h 1 )) ǫ 1 c(g 2 , h 2 )) ǫ 2 · · · c(g k , h k )) ǫ k = λ and so λ lies in µ.

Part (b) is clear. Part (c) follows from part (b) and from the fact that
Our next goal is to study the algebra S s G. We will see that it is Azumaya and that its simple images are the graded forms of C c G = M n (C). Proof. (a) We first show that the two-cocycles is nondegenerate. We use the following characterization of nondegeneracy (see Isaacs [13,Problem 11.8]): a two-cocycle δ : G × G → C × is nondegenerate if and only if for every element g ∈ G there is an element h ∈ G such that g and h commute but in the algebra C δ G the elements u g and u h do not commute. Let g ∈ G. Because c is nondegenerate there is an element h ∈ G that centralizes g and such that c(g, h)c(h, g) −1 = 1. Moreover c(g, h)c(h, g) −1 ∈ µ and s(g, h)s(h, g) −1 = c(g, h)c(h, g) −1 . Because S contains the field Q(µ), the restriction of the canonical homomorphism from S toS is injective on µ and sos(g, h)s(h, g) −1 = 1. Hences is nondegenerate. It follows thatSsG is a central simpleS-algebra. It is clear that the canonical homomorphism from S s G toSsG is surjective with kernel mS s G. Let L be any subfield of C. Recall that every G-graded form of C c G over L is a twisted group algebra L β G where β is a two-cocycle cohomologous to c over C.
We will show now that S s G is universal with respect to such forms. The following observation about the Hopf formula M(G) = [F, F ] ∩ R/[F, R] will be useful (and is probably well known). We use the notation and discussion preceding Proposition 1.
Proof. Let y ∈ [F, F ] ∩ R. We can write y = x ǫ 1 g 1 x ǫ 2 g 2 · · · x ǫm gm where ǫ i = ±1 for 1 ≤ i ≤ m and where for each group element g that appears the elements x g and x −1 g appear the same number of times (and so m is even) and where the product g ǫ 1 1 g ǫ 2 2 · · · g ǫm m equals one. Let m = 2r. We claim that we may assume ǫ i = 1 if 1 ≤ i ≤ r and ǫ i = −1 if r + 1 ≤ i ≤ m. The lemma follows easily from this claim. To prove the claim, suppose ǫ i = −1 for some i, 1 ≤ i ≤ r. We consider y = x ǫ 1 g 1 x ǫ 2 g 2 · · · x ǫm gm = g r+1 · · · x ǫm gm . Now for any element g ∈ G the element x g x g −1 commutes modulo [F, R] with every element of F . Hence we can move the element x g i x g −1 i in y and obtain an element that is equivalent modulo [F, R] to y and for which the element x −1 g i is replaced by x g i −1 . Continuing in this way proves the claim.
Proposition 12. Let L be a subfield of C which contains Q(µ). Let ϕ : S −→ L be a Q(µ)-algebra homomorphism. Let β(g, h) = ϕ(s(g, h)). Then ϕ induces a G-graded homomorphismφ : S s G −→ L β G and L β G is a G-graded form of C c G. In particular β is cohomologous to c over C.
Conversely if L β G is a G-graded form of C c G, then there is homomorphism ϕ of S into L such that γ = ϕ (s(g, h) is a cocycle on G cohomologous to β over L and ϕ induces a homomorphism from S s G onto ϕ(S) γ G, a G-graded form of L β G.
For the converse assume L β G is a G-graded form of C c G. Then L β G satisfies the graded identities of M n (C) and so there is a graded homomorphism from U G sending each x ig to u g in L β G. But under this map the images of the elements of M are nonzero elements in L and so we get an induced homomorphism from S s G that takes S into L. The result follows.
We can use this result to obtain a parametrization of the G-graded forms over L. We begin with a homomorphism ϕ from S to L and the two-cocycle β given by β(g, h) = ϕ (s(g, h)). In the previous proposition we say that β is cohomologous to c over C and so L β G is a form of M n (C). We will now show how to produce all other forms over L. Let V the free abelian group (of rank ord(G)) on symbols r g , g ∈ G. Let U be the subgroup generated by the elements r g r h /r gh . Note that V /U is a finite group, isomorphic to G/G ′ = G ab via the map that sends r g to gG ′ . For each element ψ in Hom(U, L × ) let β ψ be given by β ψ (g, h) = β(g, h)ψ(r g r h /r gh ). Note that there is a canonical homomor-phism from Hom(V, L × ) to Hom(U, L × ). We will let Im((Hom(V, L × )) denote its image in Hom(U, L × ).
Proposition 13. The following hold: (a) For every ψ ∈ Hom(U, L × ) the function β ψ is a two-cocycle cohomologous to β over C. In particular L β ψ G is a G-graded form of M n (C).

(d) The function ψ → β ψ induces a one-to-one correspondence between the group
Hom(U, L × )/ Im(Hom(V, L × ) and the set of G-graded isomorphism classes of G-graded forms of C c G over L.
Proof. (a) It is clear that β ψ is a cocycle. To show that β and β ψ are cohomologous over C it suffices to show that they determine the same element in Hom(M(G), The argument is quite similar to the proof of the corresponding fact in Proposition 12, so we will omit it.
(b) If L γ G is a G-graded form of M n (C) then we have seen that γ is cohomologous to β over C so there is a cochain λ : G → C × such that for all g, h ∈ G, we have γ(g, h) = (λ(g)λ(h)/λ(gh))β(g, h). It follows that for all g, h ∈ G, λ(g)λ(h)/λ(gh) is in L. The cochain λ clearly determines a homomorphism from V to C × (sending r g to λ(g)) and this homomorphism restricts to a homomorphism from ψ from U to L × (given by ψ(r g r h /r gh ) = λ(g)λ(h)/λ(gh)) which satisfies γ = β ψ .
(d) This follows from the first three parts.
We can connect this parametrization with cohomology as follows. We have a short exact sequence of groups Applying the functor Hom(−, L × ) we obtain a long exact sequence in cohomology where the surjectivity of the last map follows from the fact that Ext(V, L × ) = 0 because V is a free abelian group. We also have the universal coefficients sequence: Combining these equations shows that Hom(U, L × )/ Im(Hom(V, L × ) is isomorphic to Ext 1 (G ab , L × ), the kernel of the map from H 2 (G, L × ) to Hom(M(G), L × ).
The last result in this section is the determination of the rank of Y f , the free part of the subgroup of k × generated by the values of s. The computation involves the groups U and V . We first remark that because V /U is finite, the rank of U is equal to the rank of V , which is clearly n, the order of G.

Proposition 14.
The group Y f is isomorphic to U. In particular the rank of Y f is n, the order of G.
Proof. Recall that Y denotes the subgroup of k × generated by the values of the cocycle s. Let H denote the subgroup of k × generated by the values of the cocycle c. Then by Proposition 9 Y ∩ H = Y ∩ C × = µ. LetŨ be the subgroup of k × generated by set {t g t h /t gh | g, h ∈ G}. Clearly the groupŨ is isomorphic to U. Now we have Corollary 15. The field of fractions of S has transcendence degree n over Q(µ).
Proof. This follows from the description of S given in Proposition 9.

The universal G-graded central simple algebra.
We have seen that each fine grading by a group G on M n (C) gives rise to a nondegenerate two-cocycle c on the group G (which must be of central type) such that M n (C) is graded isomorphic to C c G. We have also seen that the universal G-graded algebra U G is a prime ring with ring of central quotients Q(U G ) graded isomorphic to the F -central simple algebra F s G where s is the generic cocycle obtained from c and F = Q(µ)(y 1 , . . . , y m )( t ig tg | i ≥ 1, g ∈ G) (see Proposition 9). In this section we obtain information about the index of Q(U G ) and about the dependence of Q(U G ) on the given cocycle c.
Proposition 16. The index of Q(U G ) is equal to the maximum of the indices of the Ggraded forms L β G where L varies over all subfields of C that contain Q(µ) and β varies over all nondegenerate cocycles over L cohomologous to c over C.
Proof. Let M denote the maximum of the indices of the forms. We first claim that ind(Q(U G )) ≤ M: In fact we can specialize the universal algebra U G by sending x g to r g u g where the elements {r g | g ∈ G} are algebraically independent complex numbers. The resulting algebra U is isomorphic to U G and tensoring with the field of fractions of its center gives a central simple algebra that is isomorphic to Q(U G ). Clearly the index of this form equals the index of Q(U G ) and so we get the inequality. We proceed to show that the index of Q(U G ) = F s G is greater than or equal to M. Recall (Proposition 9) that the center S of S s G is a ring of Laurent polynomials over the field Q(µ). If M is any maximal ideal of S then S M is a regular local ring. By Proposition 12 it suffices to show that if M is any maximal ideal of S then the index of the algebra S s G ⊗ S F = F s G is greater than or equal to the index of the residue algebra S s G/MS s G. This is a well known fact. We outline a proof: Let r 1 , r 2 , . . . , r m be a system of parameters of the regular local ring R = S M and let F s G = M t (D) where D is an F -central division algebra (so ind(F s G) = ord(G) 1/2 /t). The localization R (r 1 ) is a discrete valuation ring with residue field the field of fractions of the regular local ring R 1 = R/(r 1 ). The ring R s (r 1 ) G is an Azumaya algebra over the discrete valuation ring R (r 1 ) and hence is isomorphic to M t (A) where A is an Azumaya algebra such that A ⊗ F = D (see Reiner [16,Theorem 21.6]). It follows that R s 1 G is an Azumaya algebra and if we let Q(R 1 ) denote the field of fractions of R 1 , then R s 1 G ⊗ Q(R 1 ) is isomorphic to M t (D 1 ) where D 1 a central simple Q(R 1 )algebra, the residue algebra of D. The image of M in R 1 is generated by the images of r 2 , r 3 , . . . , r m and these m − 1 elements form a system of parameters for R 1 . We can therefore repeat the process and eventually obtain that S s M G/MS s M G is isomorphic to an algebra of the form M t (E) for some central simple S/M-algebra E. Hence the index of The question for which groups G of central type there is a cocycle c such that Q(U G,c ) is a division algebra was answered in Aljadeff, et al [2] and Natapov [15]: We consider the following list of p-groups, called Λ p : 1. G is abelian of symmetric type, that is G ∼ = (C p n i × C p n i ), G 1 = C p n ⋊ C p n = π, σ | σ p n = π p n = 1 and σπσ −1 = π p s +1 where 1 ≤ s < n and 1 = s if p = 2, and G 2 is an abelian group of symmetric type of exponent ≤ p s , = π, σ, τ π 2 n+1 = σ 2 n = τ 2 = 1, στ = τ σ, σπσ −1 = π 3 , τ πτ −1 = π −1 and G 2 is an abelian group of symmetric type of exponent ≤ 2.
We let Λ be the collection of nilpotent groups such that for any prime p, the Sylow-p subgroup is on the Λ p . (c) There is a nondegenerate cocycle c on G such that the resulting G-graded algebra M n (C) has a G-graded form that is a division algebra.

(d) The group G is on the list Λ.
Proof. The equivalence of (a) and (b) follows from the fact that F s G is isomorphic to Q(U G,c ). The equivalence of (b) and (c) follows from the previous proposition. Finally the equivalence of (c) and (d) follows from Aljadeff et al [2,Corollary 3] and Natapov [15,Theorem 3].
Our main result in this section is that if G is a group on the list Λ then the universal central simple G-graded algebra Q(U G,c ) is independent (up to a non-graded isomorphism) of the cocycle c. In fact we will show that for groups G on the list the automorphism group of G acts transitively on the set of classes of nondegenerate cocycles.
To begin let G be any central type group (not necessarily on Λ) and let c be a nondegenerate two-cocycle on G with coefficients in C × . Let ϕ be an automorphism of G and let ϕ(c) be the two-cocycle defined by ϕ(c)(σ, τ ) = c(ϕ −1 (σ), ϕ −1 (τ )). It is clear that ϕ(c) is also nondegenerate.
Theorem 18. If G is a group on the list Λ and c and c ′ are nondegenerate cocycles on G with values in C × , then there is an automorphism ϕ of G such that ϕ(c) is cohomologous over C to c ′ .
Proof. Let G be a group on the list Λ. Because G is necessarily nilpotent we can assume that G lies on Λ p for some prime p. In the course of the proof whenever we refer to basis elements {u g } g∈G in the twisted group algebra C c G, we assume they satisfy u g u h = c(g, h)u gh .
The strategy is as follows: For each group G ∈ Λ p we fix a set of generators Φ. Then we consider the family of sets of generators Φ ′ = ϕ(Φ) that are obtained from Φ via an automorphism ϕ of G. We say that Φ and Φ ′ are equivalent. Next, we exhibit a certain nondegenerate cohomology class α ∈ H 2 (G, C × ) by setting a set of equations (denoted by E G ) satisfied by elements {u g } g∈Φ in C c 0 G, where c 0 is a two-cocycle representing α. We say that the cohomology class α (or by abuse of language, the two-cocycle c 0 ) is of "standard form" with respect to Φ. It is indeed abuse of language since in general the equations do not determine the two-cocycle c 0 uniquely but only its cohomology class. Finally we show that for any nondegenerate two-cocycle c with values in C × there is a set of generators Φ ′ = ϕ(Φ) with respect to which c is of standard form. The desired automorphism of the group G will be determined by compositions of "elementary" automorphisms, that is automorphisms that are defined by replacing some elements of the generating set. We start with Φ (0) = Φ and denote the updated generating sets by Φ (r) = {g (r) 1 , . . . , g (r) n }, r = 1, 2, 3, . . . . Elements in the generating set that are not mentioned remain unchanged, that is g . In all steps it will be an easy check that the map defined is indeed an automorphism of G.
Fixing a primitive p n -th root of unity ε, where n ≥ max(r k ), 1 ≤ k ≤ m, we say a two-cocycle c 0 is of standard form with respect to Φ if there are elements {u g } g∈Φ in C c 0 G that satisfy (u γ 2k−1 , u γ 2k ) = ε p n−r k for all 1 ≤ k ≤ m, all other commutators of u γ i ′ s are trivial. (1) Note that [c 0 ] is determined by (1). Given any nondegenerate two-cocycle c it is known that there is a set of generators Φ ′ = ϕ(Φ) with respect to which c is of standard form (see, for example, Aljadeff and Sonn [5, Theorem 1.1]). Hence we are done in this case.
Fix a primitive p n -th root of unity ε. There exist a two-cocycle c 1 ∈ Z 2 (G 1 , C × ) and elements {u g } g∈{π,σ} in C c 1 G 1 that satisfy Moreover the class [c 1 ] ∈ H 2 (G 1 , C × ) is uniquely determined by (2) 2 ) (here G ab denotes the abelianization of G) and hence there exist a twococycle c 0 ∈ Z 2 (G, C × ) and elements {u g } g∈Φ in C c 0 G that satisfy equations (2), (1) and u π u γ i = u γ i u π , u σ u γ i = u γ i u σ . We denote this set of equations by E G . The class [c 0 ] ∈ H 2 (G, C × ) is uniquely determined by E G and is easily seen to be nondegenerate. We say that a class [c 0 ] is of standard form with respect to Φ if there are elements {u g } g∈Φ in C c 0 G that satisfy E G . Now, let c ∈ Z 2 (G, C × ) be any nondegenerate two-cocycle, and let u g be representatives of elements of G in C c G. As explained above we set We are to exhibit a sequence of automorphisms such that their composition applied to Φ (0) yields a generating set with respect to which the cocycle c is of standard form. First, note that we may assume (by passing to an equivalent cocycle, if necessary) that u p n π (0) = u p n σ (0) = 1. Let α ∈ C × be determined by u σ (0) u π (0) u −1 σ (0) = αu p s +1 π (0) . It follows that u σ (0) u p n−1 π (0) u −1 σ (0) = α p n−1 u p n−1 π (0) and hence, α p n−1 is a p-th root of unity. Next, observe that since c is nondegenerate, u σ (0) cannot commute with u p n−1 π (0) (for otherwise, u p n−1 π (0) is contained in the center of the algebra) and hence α is a primitive p n -th root of unity. Now, replacing π (0) by a suitable prime to p power of π (0) (that is π (1) = (π (0) ) l , l prime to p), we may assume that α = ε. Leaving σ (0) and all other generators unchanged, that is σ (1) = σ (0) , γ (1) 2m } of G, equivalent to Φ, such that the elements u π (1) , u σ (1) in C c G 1 satisfy (2). Assume (u γ (1) i , u π (1) ) = ξ i = 1 (clearly ξ i is a root of unity of order p r (≤ ord(γ (1) i ) ≤ p s )). Now, by the nondegeneracy of c, the root of unity ζ determined by the equation = ζu π (1) is of order p s and hence ξ i = ζ l for some l. Observe that ord(((σ (1) ) p n−s ) l ) ≤ p r and hence we may put γ i (σ (1) ) −lp n−s and get (u γ (2) i , u π (1) ) = 1. Performing this process for all 1 ≤ i ≤ 2m (and leaving π (1) and σ (1) unchanged) we get a set of generators Φ (2) such that (u γ (2) i , u π (2) ) = 1 for all i.
Finally, we may proceed as in Case I and obtain a generating set Φ (4) , equivalent to Φ, with respect to which c is of standard form.
We first exhibit a construction of a field K and a cocycle β ∈ Z 2 (G 1 , K × ) such that the algebra D ∼ = K β G 1 is K-central simple (in fact a K-central division algebra), which is analogous to that in Natapov [15]. Let K = Q(s, t) be the subfield of C generated by algebraically independent elements s and t. Let L = K(v π )/K be a cyclotomic extension where v 2 n+1 π = −1. The Galois action of Gal(L/K) ∼ = Z 2 n × Z 2 = σ, τ on L is given by The algebra D is the crossed product D = (L/K, σ, τ ) determined by the following where v σ and v τ represent σ and τ in D. It is easy to see that D is isomorphic to a twisted group algebra of the form K β G 1 .
As in the previous case, a construction of a K-central division algebra of the form K β G 1 , analogous to that in Natapov [15], yields a 4 × 4 matrix algebra isomorphic to C c 1 G 1 with a basis {u g } g∈G 1 such that u π i σ j τ k = u i π u j σ u k τ and u 4 π = u 2 σ = u 2 τ = 1, u σ u π = εu 3 π u σ , u τ u π = −u π u τ , u τ u σ = u σ u τ , where ε is a 4-th root of unity. Clearly, the class [c 1 ] is determined by (5).
Recall from the introduction that for a group G of central type we let ind(G) denote the maximum over all nondegenerate cocycles c of the indices of the simple algebras Q(U G,c ). We have seen that if G is not on the list, then the universal central simple algebra Q(U G ) is not a division algebra and so the index of G is strictly less than ord(G) 1/2 . In fact it is shown in Aljadeff and Natapov [4] that the groups on the list are the only groups responsible for the index of Q(U G ). Here is the precise result.
Theorem 21. Let P be a p-group of central type and let Q(U P,c ) be the universal central simple P -graded algebra for some nondegenerate two-cocycle c on P . Then there is a sub-quotient group H of P on the list Λ, such that Q(U P,c ) ∼ = M p r (Q(U H,α ) DeMeyer and Janusz prove in [9, Corollary 4] that for an arbitrary group of central type G and a nondegenerate two-cocycle c ∈ Z 2 (G, C × ) any Sylow-p subgroup G p of G is of central type, and furthermore the restriction of c to G p is nondegenerate. Using this result Geva in his thesis proved that for any subfield k of C the twisted group algebra k c G is (non-graded) isomorphic to p| ord(G) k c G p . It follows that for each prime p dividing the order of G there is a p-group H p on the list Λ which is isomorphic to a sub-quotient of G and such that F s G ∼ = p M p r (F αp H p ). Combining this with the preceding Corollary we obtain: Corollary 23. Let G be a group of central type, and for each prime p, G p be a Sylow-p subgroup of G. Consider all central type groups H p that are isomorphic to sub-quotients of G p and are members of Λ p . Then ind(G) ≤ p sup(ord(H p ) 1/2 ).
It follows from Proposition 17 that for any group G of central type not on the list and any G-grading on M n (C), the universal central simple algebra is not a division algebra. That means we can find nonidentity polynomials p(x ig ) and q(x ig ) over the field of definition Q(µ) such that p(x ig )q(x ig ) is a graded identity. In fact there will exist a nonidentity polynomial p(x ig ) over Q(µ) such that p(x ig ) r is a graded identity for some positive integer r. This is clearly equivalent to saying that under every substitution the value of p in M n (C) is a nilpotent matrix. We will refer to such a polynomial as a nilpotent polynomial.
We present an explicit example of a nilpotent polynomial. We let S 3 be the permutation group on three letters, and C 6 = z be a cyclic group of order 6. Let σ = (123) and τ = (12) be generators of S 3 and define an action of S 3 on C 6 by τ (z) = z −1 and σ(z) = z. We consider the group G = S 3 ⋉ C 6 . Note that G is not nilpotent and hence not on the list Λ.
It follows that the polynomial f 3 (x τ , x σ , x y ) is a polynomial identity for M 6 (C) while f and f 2 are not.
We end this example by expressing f 3 explicitly as a Q(ω)−linear combination of elementary identities: If G is on the list Λ and c is a nondegenerate two-cocycle on G we have seen that the algebra Q(U G,c ) is a division algebra of degree n = ord(G) 1/2 . In the next result we calculate the index of Q(U G,c ) ⊗ Q(µ) C, where Q(µ) is the field of definition for the graded identities.
Proposition 24. If G is on the list Λ and c is a nondegenerate two-cocycle on G then the index of Q(U G,c ) ⊗ Q(µ) C is n/ ord(G ′ ). In particular Q(U G,c ) ⊗ Q(µ) C is a division algebra if and only if G is abelian of symmetric type.
Proof. By Aljadeff and Haile [1] (see the discussion at the beginning of section two) the subalgebra F s G ′ of Q(U G,c ) = F s G is a cyclotomic extension of F . It follows that the index of Q(U G,c ) ⊗ Q(µ) C is at most n/ ord(G ′ ). We proceed to prove the opposite inequality. By Natapov [15, proof of Theorem 6] there is a subfield K of C containing Q(µ) and a nondegenerate cocycle β on G with values in K such that the algebra K β G ⊗ Q(µ) C has index exactly n/ ord(G ′ ). By a specialization argument it follows that the algebra Q(U G,β ) has index at least n/ ord(G ′ ). By Corollary 19 Q(U G,β ) is isomorphic to Q(U G,c ) and hence the index of Q(U G,c ) is at least n/ ord(G ′ ).