On regular G-grading

Let A be an associative algebra over an algebraically closed field F of characteristic zero and let G be a finite abelian group. Regev and Seeman introduced the notion of a regular G-grading on A, namely a grading A= {\Sigma}_{g in G} A_g that satisfies the following two conditions: (1) for every integer n>=1 and every n-tuple (g_1,g_2,...,g_n) in G^n, there are elements, a_i in A_{g_i}, i=1,...,n, such that a_1*a_2*...*a_n != 0. (2) for every g,h in G and for every a_g in A_g,b_h in A_h, we have a_{g}b_{h}=theta(g,h)b_{h}a_{g}. Then later, Bahturin and Regev conjectured that if the grading on A is regular and minimal, then the order of the group G is an invariant of the algebra. In this article we prove the conjecture by showing that ord(G) coincides with an invariant of A which appears in PI theory, namely exp(A) (the exponent of A). Moreover, we extend the whole theory to (finite) nonabelian groups and show that the above result holds also in that case.


Introduction and statement of the main results
Group gradings on associative algebras (as well as on Lie and Jordan algebras) have been an active area of research in the last 15 years or so. In this article we will consider group gradings on associative algebras over an algebraically closed field F of characteristic zero. The fact that a given algebra admits additional structures, namely graded by a group G, provides additional information which may be used in the study of the algebra itself , e.g. in the study of group rings, twisted group rings and crossed products algebras in Brauer theory (indeed "gradings" on central simple algebras is an indispensable tool in Brauer theory, as it provides the isomorphism of Br(k) with the second cohomology group H 2 (G k ,k × ). Here k is any field, G k is the absolute Galois group of k andk × denotes the group of units of the separable closure of k). In addition, and more relevant to the purpose of this article, G-gradings play an important role in the theory of polynomial identities. Indeed, if A is a PI-algebra which is G-graded, then one may consider the T -ideal of G-graded identities (see Subsection 2.1), denoted by Id G (A), and it turns out in general, that it is easier to describe G-graded identities than the ordinary ones for the simple reason that the former ones are required to vanish on G-graded evaluations, rather than on arbitrary evaluations. Nevertheless, two algebras A and B which are G-graded PI-equivalent are PI-equivalent as well, that is, Id G (A) = Id G (B) ⇒ Id(A) = Id(B).
Recall that a G-grading on an algebra A is a vector space decomposition A particular type of G-gradings which is of interest was introduced by Regev and Seeman in [13], namely regular G-gradings where G is a finite abelian group. Let us recall the definition.
Definition 1 (Regular Grading). Let A be an associative algebra over a field F and let G be a finite abelian group. Suppose A is G-graded. We say that the G-grading on A is regular if there is a commutation function θ : G × G → F × such that 1. For every integer n ≥ 1 and every n-tuple (g 1 , g 2 , . . . , g n ) ∈ G n , there are elements a i ∈ A gi , i = 1, . . . , n, such that n 1 a i = 0. 2. For every g, h ∈ G and for every a g ∈ A g , b h ∈ A h , we have a g b h = θ g,h b h a g .
Remark 2. One of our main tasks in this article is to extend the definition above to groups which are not necessarily abelian and prove the main results in that general context. For clarity we will continue with the exposition of the abelian case and towards the end of the introduction we will discuss extensions to the nonabelian setting. It seems to us that the extension to the nonabelian case is rather natural in view of the abelian case. In those cases below where the statement in the general case is identical to the abelian case, we will make a note indicating it. As for the proofs (sections 2 and 3), in case the result holds for arbitrary groups, we present the general setting only, possibly with some remarks concerning the abelian case.
A key property of the "envelope operation" is the following equality of Z /2Z-graded T -ideals of identities (and hence, also of the corresponding ungraded T -ideals of identities).
IdZ /2Z (E ⊗(E ⊗B))) = IdZ /2Z (B). (1) We refer to the "envelope operation" as being involutive. It is well known that applying this operation, one can extend the solution of the Specht problem and proof of "representability" from affine to nonaffine PI-algebras (see [10], [3]). Interestingly, the property satisfied by the Grassmann algebra we just mentioned follows from the fact that the Z /2Z-grading on E is regular, and indeed in Theorem 6 we show that a similar property holds for arbitrary regular graded algebras. In order to state the result precisely we introduce the notion of G-envelope of two algebras A and B where G is a finite abelian group.
Definition 5 (G-envelope). Let A, B be two G-graded algebras. We denote by A ⊗B the G-graded algebra defined by A ⊗B g = A g ⊗ B g .
The following result generalizes equation (1). The proof is presented in section §2.
Theorem 6. Let A be a regularly G-graded algebra with commutation function θ, and let B, C be two G-graded algebras.

LetÃ = ⊗
|G|−1 A be the envelope of |G| − 1 copies of A, thenÃ is regularly G-graded and Our main goal in this article is to investigate the general structure of (minimal) regular gradings on associative algebras over an algebraically closed field of characteristic zero and in particular to give a positive answer to conjecture 2.5 posed by Bahturin and Regev in [5].
It is easy to see that a given algebra A may admit regular gradings with nonisomorphic groups and even with groups of distinct orders. Therefore, in order to put some restrictions on the possible regular gradings on an algebra A, Bahturin and Regev introduced the notion of regular gradings which are minimal. A regular G-grading on an algebra A with commutation function θ is said to be minimal if for any e = g ∈ G there is g ′ ∈ G such that θ(g, g ′ ) = 1.
Given a regular G-grading on an algebra A with commutation function θ, one may construct a minimal regular grading with a homomorphic imageḠ of G. To see this, let H = {h ∈ G | θ(h, g) = 1 : for all g ∈ G}.
One checks easily that θ is a skew symmetric bicharacter and hence H is a subgroup of G. Consequently, the commutation function θ on G induces a commutation functionθ onḠ = G/H. Moreover, the induced regularḠ-grading on A is minimal.
In this article we consider the problem of uniqueness of a minimal regular G-grading on an algebra A (assuming it exists). It is not difficult to show that an algebra A may admit nonisomorphic minimal regular G-gradings. Furthermore, an algebra A may admit minimal regular gradings with nonisomorphic abelian groups. However, it follows from our results (as conjectured by Bahturin and Regev) that the order of the group is uniquely determined. In fact, the order of any group which provides a minimal regular grading on an algebra A coincides with a numerical invariant of A which arises in PI-theory, namely the PI-exponent of the algebra A (denoted by exp(A)). In order to state the result precisely we need some terminology which we recall now.
Given a regular G-grading on an F-algebra A we consider the corresponding commutation matrix M A defined by M A g,h = θ(g, h), g, h ∈ G (see [5]). The commutation matrix encodes properties of θ. For instance, a regular grading is minimal if and only if there is only one row of ones in M A (resp. with columns).
Next we recall the definition of exp(A). For any positive integer n we consider the n!-dimensional F-space P n , spanned by all monomials of degree n on n different variables {x 1 , . . . , x n } and let c n (A) = dim F (P n /(P n ∩ Id(A)). This is the n-th coefficient of the codimension sequence of the algebra A. It was shown by Giambruno and Zaicev (see [6], [7]) that the limit lim n→∞ c n (A) 1/n exists and is a nonnegative integer. The limit is denoted by exp(A).
We can now state the main result of the paper in case the gradings on A are given by abelian groups. 3. In fact, det(M A G ) = ± |G| |G|/2 . Remark 8. Some of the results stated in Theorem 7 were conjectured in [5]. Specifically, as mentioned above, Bahturin and Regev conjectured that the order of a group which provides a minimal regular gradings on an algebra A is uniquely determined. Moreover, they conjectured that if M A G and M A H are the commutation matrices of two minimal regular gradings on A with groups G and H respectively, then det(M A G ) = det(M A H ) = 0.

Not necessarily abelian groups
Suppose now that G is an arbitrary finite group and let A be a G-graded algebra. As above, A is an associative algebra over an algebraically closed field F of characteristic zero. We denote by A g the corresponding g ∈ G-homogeneous component.

Definition 9.
We say that the G-grading on A is regular if the following two conditions hold.
Remark 10. In the special case where the elements g, g ′ ∈ G commute we write θ g,g ′ instead of θ ((g,g ′ ), (12)) . In particular we will often use the notation θ g,g . Note that if G is abelian, then θ ((g1,...,gn),σ) is determined by θ gi,gj , Typical examples of regularly graded algebras (G arbitrary) are the well known group algebras FG, and more generally, any twisted group algebra F α G where α is a 2-cocycle on G with values in F × . Indeed, this follows easily from the fact that each homogeneous component is 1-dimensional and every nonzero homogeneous element is invertible.
Additional examples can be obtained as follows.
1. If A is a regularly G-graded algebra then E ⊗ A has a natural regular Z /2Z × G-grading where E is the infinite dimensional Grassmann algebra.

Let
A be a regularly G-graded algebra and suppose the group G contains a subgroup H of index 2. Then we may view A as a Z /2Z ∼ = G/H-graded algebra and we let be the Grassmann envelope of A and consider the following G-grading on it. For any g ∈ H, we put We claim the grading is regular. Indeed, let (g 1 , . . . , g n ) ∈ G n and let σ ∈ Sym(n) be a permutation such that g 1 · · · g n = g σ(1) · · · g σ(n) . Then for elements z g1 , . . . , z gn where z gi ∈ E(A) gi , we have where τ is the commutation function of the infinite Grassmann algebra. For future reference we denote the commutation function which corresponds to the regular G-grading on E(A) by τ θ.
Following the discussion in the abelian case we define now nondegenerate gradings for arbitrary finite groups as the counterpart of minimal gradings. Let A be an associative algebra and suppose it has a regular G-grading with commutation function θ. We say that the grading is nondegenerate if for every g = e in G, there is an element g ′ ∈ C G (g) (the centralizer of g in G) such that θ (g,g ′ ) = 1.
Remark 11. It turns out (see Lemma 37) that g → θ g,g is a homomorphism from G to {±1} and therefore its kernel H = {g ∈ G : θ g,g = 1} is a subgroup of G (of index ≤ 2). In case H = G, there is a cohomology class [α] ∈ H 2 (G, F × ) such that F α G has commutation function θ. Then, the nondegeneracy of the G-grading on A corresponds to α being a nondegenerate 2-cocycle. Groups G which admit nondegenerate 2-cocycle are called "central type". It is a rather difficult problem to classify central type groups. It is known, using the classification of finite simple groups(!), that any central type group must be solvable. It seems to be an interesting problem to classify finite groups which admit nondegenerate commutation functions (modulo the classification of central type groups).
Our main results in the general case are extensions of the results appearing in Theorem 7.
Remark 13. In case G is abelian, the commutation function θ is a skew symmetric bicharacter (see Definition 24). In this case, it is a well known theorem of Scheunert [14] that θ arises from a 2-cocycle (as mentioned in the previous remark). The notion of a bicharacter was considerably generalized to cocommutative Hopf algebras (and hence in particular to group algebras) (see [4]). Furthermore, whenever the bicharacter is skew symmetric, the theorem of Scheunert can be extended to that case. However, it should be noted that already for group algebras FG where G is a nonabelian group, the linear extension of β(g, h) = f (g, h)/f (h, g) for a 2-cocycle f : G × G → F × is not in general a skew symmetric bicharacter on FG.
Our results should be viewed or interpreted as to overcome this problem by considering commutation functions which satisfy certain natural necessary conditions in case they arise from 2-cocycles on G, and then Lemma 32 provides a generalization of Scheunert's theorem to that context, namely every such commutation function on G ideed arises from a 2-cocycle on G.

Commutation matrix
As for the commutation matrix and its characteristic values, we need to fix some notation. Suppose A has a regular G-grading and let θ A be the corresponding commutation function. Suppose first that θ A g,g = 1 for every g ∈ G.
In that case we know that the commutation function corresponds to an element [α] ∈ H 2 (G, F × ). With this data we consider the corresponding twisted group algebra B = F α G which is regularly G-graded with commutation function θ A . It is well known that the algebra F α G is spanned over F by a set of invertible homogeneous elements {U g } g∈G that satisfy U g U h = α(g, h)U gh for every g, h ∈ G.
Let us construct the corresponding commutation matrix. For every pair (g, h) ∈ G 2 we consider the element h for every g, h ∈ G and we note that this element does not depend on the choice of the basis {U g : g ∈ G}.
Next we consider the general case. Let ψ : G → F × be the map determined by ψ(g) = θ g,g . The function ψ will be shown to be a homomorphism with its image contained in {±1} and we set H = ker(ψ) = {g ∈ G : θ g,g = 1}. Applying the construction above we may define a G-grading on E(A) where the Z /2Z-grading on A is defined by A = A H ⊕ A G\H . The commutation function τ · θ A of E(A) satisfies (τ θ A ) g,g = 1 for every g ∈ G. As in the previous case the function τ θ A corresponds to a cohomology class [α] ∈ H 2 (G, F × ) were α is a representing 2-cocycle. We let B = F α G be the corresponding twisted group algebra with commutation function θ B = τ θ A and for every g, h ∈ G we consider the element The commutation matrix is defined by h . We will usually write τ g,h instead of τ ψ(g),ψ(h) . Note that if ψ ≡ 1, then H = G and τ | G ≡ 1, so we have that θ B = θ A as in the first case.
Theorem 14. Let A be an associative algebra over an algebraically closed field of characteristic zero. Suppose A admits a nondegenerate regular G-grading and let θ be the corresponding commutation function. Let M G be the commutation matrix constructed above. Then M 2 G = |G| · Id. As a consequence we extend Theorem 7(2,3) for arbitrary nondegenerate regular gradings (see Subsection 3.1, Corollary 47). In case θ g,g = 1 for every g ∈ G, we have that the elements and so the commutation matrix may be viewed as a matrix in M r 3 (F ). In that case we obtain the following corollary.

Preliminaries, examples and some basic results
In the first part of this section we recall some general facts and terminology on G-graded PI-theory which will be used in the proofs of the main results (we refer the reader [3] for a detailed account on this topic). In the second part of this section we present some additional examples of regular gradings on finite and infinite dimensional algebras. Finally, we present properties of regular gradings and prove Theorem 6.

Graded polynomial identities
Let W be a G-graded PI-algebra over F and I = Id G (W ) be the ideal of G-graded identities of W . These are polynomials in F X G , the free G-graded algebra over F generated by X G , that vanish upon any admissible evaluation on W . Here X G = g∈G X g and X g is a set of countably many variables of degree g. An evaluation is admissible if the variables from X g are replaced only by elements of W g . It is known that I is a G-graded T -ideal, i.e. closed under G-graded endomorphisms of F X G .
We recall from [3] that the T -ideal I = Id G (W ) is generated by multilinear polynomials and so it does not change when passing toF, the algebraic closure of F, in the sense that the ideal of identities of WF overF is the span (overF) of the T -ideal of identities of W over F.

Additional examples of regular gradings
We present here some more examples (in addition to the ones presented in the introduction). The following example corresponds to the grading determined by the symbol algebra (1, 1) n .
Example 16. Let M n (F) be the matrix algebra over the field F, and let G = Z /nZ × Z /nZ. For ζ a primitive n-th root of 1 we define Let us check the G-grading is regular. For any two basis elements we have that and hence the second condition in the definition of a regular grading is satisfied. The first condition in the definition follows at once from the fact that the elements X and Y are invertible. Finally we note that since ζ is a primitive n-th root of unity, the regular grading is in fact minimal.
Example 17. For any n ∈ N and c ∈ F × , we can define a regular Z /nZ-grading on A = F[x] / x n −c by setting A k = F · x k . Clearly, the commutation function here is given by θ h,g = 1 for all g, h ∈ Z /nZ.
Example 18. For any algebra A we have the trivial G = {e}-grading by setting A e = A. In this case the grading is regular if and only if A is abelian and nonnilpotent.
Example 19. We present an algebra with a nondegenerate regular G-grading where G is isomorphic to the dihedral group of order 8.
Consider the presentation x, y : x 4 = y 2 = e, yxy −1 = x 3 of the group G.
It is well known that there is a (unique) nonsplit extension The map π is determined by π(u) = x and π(v) = y. Note that the extension is nonsplit on any nontrivial subgroup of G, that is, if {e} = H ≤ G, then the restricted extension y, x 2 y} and so we can consider the corresponding Grassmann envelope E(F αG G). We show that E(F αG G) is regularly G-graded and moreover the grading is nondegenerate. Clearly the natural G-grading on the twisted group algebra F αG G is regular and hence the corresponding G-grading on E(F αG G) is also regular. Let θ be the corresponding commutation function. To see that the G-grading on E(F αG G) is nondegenerate, note that since the cocycle α H is nontrivial on every subgroup H = {e} of G, the group π −1 (K) is isomorphic to the quaternion group of order 8 and hence the twisted group subalgebra F αK K of F αG G is isomorphic to M 2 (F). This shows that the nondegeneracy condition (see Definition 21) is satisfied by any nontrivial element of K. For elements g in G \ K we have that θ g,g = −1 and we are done. We will return to this example at the end of the paper.

The commutation function θ and the commutation matrix
We now turn to study some properties of the commutation function θ. We start with some notation. Let G be a group andḡ = (g 1 , ..., g n ) ∈ G n .
The conditions in the following lemma correspond to the properties of T-ideal, namely (1)-closed to multiplication (2)-closed to substitution and (3)-closed to addition.
Lemma 20. Let G be a group and A a regularly G-graded algebra with commutation function θ. Then θ satisfies the following conditions.

Proof.
1. This is an immediate consequence of the associativity of the product in A.
Next, we define G-commutation functions. We remind the reader that if g, h ∈ G commute we may denote by θ g,h the scalar θ ((g,h),(1,2)) .
Definition 21. Let θ be a function from the pairsḡ = (g 1 , ..., g n ) ∈ G n , σ ∈ Sym(ḡ) with values in F × . We say that θ is a G-commutation function if it satisfies conditions (1, 2, 3) from the last lemma. The function θ is said to be nondegenerate if for any e = g ∈ G there is some h ∈ C G (g) such that θ g,h = 1.
In Lemma 37 below we show that each G-commutation function is in fact the commutation function of some regularly G-graded algebra. By the definition, we get that a regular grading is nondegenerate if and only if the commutation function is nondegenerate.
Lemma 22. Let θ be a G-commutation function. Then the following hold.

For every commuting pair
For any fixed g ∈ G, the functions h → θ g,h and h → θ h,g are characters on C G (g).

If
3. By the conditions in Lemma 20 we get that if g ∈ G and h 1 , h 2 ∈ C G (g), then Similarly we have that θ h1h2,g = θ h1,g θ h2,g .
If G is abelian, then the commutation function θ (ḡ,σ) is defined by its values on pairs θ g,h . In that case we get that C G (g) = G for all g ∈ G and the conditions in Lemma 20 follow from those in the last lemma. We recall the definition of such functions.
Definition 24 (Bicharacter). Let η : G × G → F × be a map where G is a group and F × is the group of units of the field F. We say that the map η is a bicharacter of G if for any A bicharacter is said to be nondegenerate if for any e = g ∈ G there is an element h ∈ G such that θ(g, h) = 1.
Remark 25. In general, if θ is a commutation function on a finite group G, then for any commuting elements g, h ∈ G we have ord( θ(g, h) ) | gcd( ord(g), ord(h) ), so θ(g, h) is contained in the group of roots of unity of order |G| in F × . In fact, as it will be shown below, this holds for any θ (ḡ,σ) . Also, we have that θ(g, g) = θ(g, g) −1 so θ(g, g) ∈ {±1} for every g ∈ G.
We present now two lemmas which summarize properties of the commutation function and the "G-envelope operation". The proof of the first lemma follows directly from the definitions and is left to the reader.
Lemma 26. Suppose that A, B are G, H-regulary graded algebra with commutation functions θ and η respectively. Then the following hold.
Furthermore, the corresponding commutation function is θ.
In particular n 1 A is regularly G-graded for any n ∈ N.
3. If N ≤ G is a subgroup, then A N = g∈N A g is a regularly N -graded algebra with commutation function θ | N -the restriction to tuples in N .
If the groups G, H are abelian, then the commutation matrix which corresponds to the cases considered in the lemma are calculated as follows: (1)  In the nonabelian case we have a similar connection between the commutation matrices, though the ring over which the matrices are defined may differ. More details are presented in the end of Subsection 3.1.
Suppose A is regularly G-graded and let θ be the corresponding commutation function. Given a multilinear polynomial f (x g1,1 , ..., Lemma 27. Let A be a regularly G-graded algebra with commutation function θ and let B be any Proof. By multilinearity of f we only check that f vanishes on a spanning set. For any a i ∈ A gi and b i ∈ B gi we get that then the first term is always zero. Since the grading on A is regular, we can find a i such that a i = 0, Before we proceed with the proof of Theorem 6 we recall that for any G-graded algebra over a field of characteristic zero F, the T -ideal of G-graded identities is generated by multilinear polynomials which are strongly homogeneous, namely polynomials of the form Proof. of Theorem 6 1. This is immediate since f θ = f when θ ≡ 1. 2.Ã ⊗A is the product of |G| copies of A. This is a regularly G-graded algebra with commutation function θ |G| ≡ 1. We now use the associativity of the envelope operation and part (1) to conclude that Id G (Ã ⊗(A ⊗B)) = Id G (B).
3. This follows immediately from the previous lemma.

Main Theorem
Our main objective in this section is to prove Theorem 7. The first step is to translate the definition of "regular grading" into the language of graded polynomial identities.
Lemma 28. Let A be an algebra over F, G a finite group and A = g∈G A g a G-grading on A.
Then the grading is regular if and only if the following conditions hold.

There is a function
Proof. The proof is clear. Indeed, condition (1) (resp. (2)) of the lemma is equivalent to the first (resp. second) condition in the definition of a regular grading.
As mentioned above, the conditions in Lemma 20 correspond to the properties of the T-ideal Id G (A), where (1), (2), (3) correspond to closure under multiplication, closure under endomorphisms and closure under addition respectively. Here is the precise statement.
Then the following hold.
1. The vector space I is a T -ideal.
2. The G-grading on F XG /I is regular with commutation function θ.
In particular, any G-commutation function is a commutation function of some regular algebra.
Proof. The proof is based on translating the conditions of 20 into the language of T -ideals. We give here only an outline of the proof and leave the details to the reader.
1. By definition I is closed under addition. To see I is closed under the multiplication of arbitrary polynomials, it is sufficient to show it is closed under multiplication by x g,j for any g ∈ G and j ∈ N which is exactly condition (1) in Lemma 20. Next we show the ideal I is closed under endomorphisms. Notice that if s ∈ S is multilinear and ϕ ∈ End(F X G ), one can decompose ϕ = ϕ 1 • ϕ 2 such that ϕ 2 sends each x gj ,ij to a sum of multilinear monomials, all disjoint from each other, and ϕ 1 sends each x g,j to some x g ′ ,j ′ . It now follows from condition (2) in Lemma 20 that ϕ 2 (s) = s l for some s l ∈ S multilinear, and that ϕ 1 (S) ⊆ S. This completes the proof.
Definition 30. Let θ be a G-commutation function. The algebra F XG /I defined in the previous proposition is called the θ-relatively free algebra.
Let A be a G-graded algebra. Let π : G →Ḡ be a surjective homomorphism and let A = ḡ∈Ḡ Aḡ be the induced grading on A byḠ (that is Aḡ = π(g)=ḡ A g ). Clearly, for multilinear polynomial f we have f (xḡ 1 ,1 , ..., xḡ n ,n ) ∈ IdḠ(A) if and only if f (x g1,1 , ..., x gn,n ) ∈ Id G (A) for every g i ∈ G with π(g i ) =ḡ i and so, in the particular case where π : G → {e}, we obtain the aforementioned fact that algebras which are G-graded PI-equivalent, are also PI-equivalent. This simple but important fact will enable us to replace the algebra A by a more tractable G-graded algebra B (satisfying the same G-graded identities as A) from which it will be easier to deduce the invariance of the order of the group which provides a nondegenerate regular grading on A.
For the rest of this section, unless stated otherwise, we assume that F is algebraically closed and char(F) = 0. Proof. Clearly, it is sufficient to consider multilinear polynomials. Letḡ = (g 1 , ..., g n ) ∈ G n . Applying binomial G-graded identities of A (see Lemma 28), a poly- Thus, the statement f (x g1,1 , ..., x gn,n ) ∈ Id G (A) is equivalent to a condition on the commutation function θ A and the result follows.
As mentioned above, we wish to replace any regularly G-graded algebra A with commutation function θ A by a better understood regularly G-graded algebra B with commutation function θ B = θ A .
We first deal with the case where θ g,g = 1 for all g ∈ G (we remind the reader that in general θ g,g = ±1 for all g ∈ G). Here the algebra B will be isomorphic to a suitable twisted group algebra B = F α G, where α is a 2-cocycle on G with values in F × . Recall that B = F α G is isomorphic to the group algebra FG as an F-vector space and if {U g : g ∈ G} is an F-basis of F α G, then the multiplication is defined by the rule U g U h = α(g, h)U gh for every g, h ∈ G. It is well known that up to a G-graded isomorphism, the twisted group algebra F α G depends only on the cohomology class of α ∈ H 2 (G, F × ) and not on the representative α. In order to construct the 2-cocycle α = α θ , we show that the commutation function θ = θ A (with θ g,g = 1, for all g ∈ G) determines uniquely an element in Hom(M (G), F × ), where M (G) denotes the Schur multiplier of the group G. Then applying the Universal Coefficient Theorem, we obtain an element in H 2 (G, F × ) which by abuse of notation we denote again by θ.
Here is the precise statement and its proof.
Lemma 32. Let θ be a G-commutation function such that θ g,g = 1 for all g ∈ G. Then there is a 2-cocycle α ∈ Z 2 (G, F × ) such that the commutation function of B = F α G is θ.
Recall that from the Universal Coefficient Theorem we get that for any group G we have an exact sequence where M (G) is the Schur multiplier of G. Note that since F is assumed to be algebraically closed, we have that Ext 1 (G ab , F × ) = 0 and hence in that case, the map π is an isomorphism. Thus, our task is to find a suitable element in Hom(M (G), F × ) and then show that its inverse image in H 2 (G, F × ) satisfies the required property.
To start with, we fix a presentation of M (G) via the Hopf formula: Let F be the free group F = y g | g ∈ G and define ϕ : F → G by ϕ(y g ) = g. Setting R = ker(ϕ) we have the exact sequence
Let A = F XG /I be the θ-relatively free algebra defined in Proposition 29. Then A is regularly G-graded with commutation function θ. If we can find elements a g ∈ A g , g ∈ G, which are invertible, then we can define a group homomorphismψ : F → A × induced by the map y g → a g . Notice that the image of any commutator in [R, F ] is mapped to 1 (because R is mapped to A e which is in the center) while y g1 · · · y gn y −1 g σ(n) · · · y −1 g σ(1) is mapped to θ(ḡ, σ)1. The induced map ϕ : M (G) → F × is the required map. In general A might not have such invertible elements, so we need to construct new elements.
Let S = x g,1 x g −1 ,2 g∈G ⊆ F X G . Note that it is sufficient to show that the elements in S represent nonzero divisors in A since in that case, the localized algebra A ′ = AS −1 will still be regularly G-graded with commutation function θ and in addition each x g,1 will be invertible (notice that x g −1 ,2 x g,1 = θ(g −1 , g)x g,1 x g −1 ,2 and θ(g −1 , g) = θ(g, g) −1 = 1, so x g,1 is right and left invertible).
Suppose that there is some 0 = f ∈ A such that x g,1 x g −1 2 · f ≡ 0. We can assume that f is homogeneous (i.e. its monomials have the same G-homogeneous degree), and by standard methods (since the field is infinite) we can assume that every variable x h,i appears with the same total degree in each monomial of x g,1 x g −1 2 · f and therefore this is true also in f . Finally, using the binomial identities we can assume that f is a monomial. Now, by assumption x g,1 x g −1 2 · f cannot be a monomial with different variables, so we need to show there is no general monomial identities (i.e. with possibly repeated variables).
Let a 1 x g,i a 2 x g,i a 3 · · · a n x g,i a n+1 ∈ Id G (A) be a monomial identity where x g,i does not appear in the monomials a i . Then using linearization we get that the polynomial f = σ∈Sn a 1 z g,σ(1) a 2 z g,σ(2) a 3 · · · a n z g,σ(n) a n+1 .
is an identity as well. We claim that the monomials in f are equal modulo identities of A. In order to see this suppose that a, b, c are monomials and denote by h the degree of a. Let g be some element in G. Then where the middle equation is true since θ g,g = 1 and the first and third equalities are true because monomials of degree e are in the center. We therefore have ax g,1 bx g,2 c ≡ ax g,2 bx g,1 c. Applying this equivalence we have that σ∈Sn a 1 y g,σ(1) a 2 y g,σ(2) a 3 · · · a n y g,σ(n) a n+1 ≡ n!a 1 y g,1 a 2 y g,2 a 3 · · · a n y g,n a n+1 .
Finally, we see that if we repeat this process for every pair g ∈ G, i ∈ N such that x g,i has total degree greater than 1 in our monomial identity, we obtain a monomial identity with distinct variables -contradiction.
Suppose that A has a nondegenerate regular grading with commutation function θ such that θ(g, g) = 1 for all g ∈ G. Let B = F α G as constructed in the last lemma. Clearly, the twisted group algebra B is regularly G-graded and the commutation function is θ. Invoking Lemma 31 we have the following Corollary. Our goal is to extract the cardinality of G from Id(B). By Maschke's theorem, we know that any twisted group algebra B = F α G is a direct sum of matrix algebras. We wish to show that the commutation function θ is nondegenerate if and only if B is simple, or equivalently dim(Z(B)) = 1. It is easily seen that the center Z(B) is spanned by elements of the form σ∈G λ σ U σgσ −1 where g ∈ G and λ σ ∈ F. We call a conjugacy class that contribute a nonzero central element a ray class. The determination of the ray classes and their corresponding central elements is well known (for example see [11], section 2). The next lemma gives the condition for a conjugacy classes to be a ray class and Lemma 38 will generalize this idea to the Z /2Z-simple case.
Lemma 34. Let g ∈ G and choose some set of left coset representatives {t i } k 1 of C G (g) in G. For any 2-cocycle α ∈ Z 2 (G, F × ) the following conditions are equivalent: In addition, if there are λ i ∈ F, i = 1, ..., k, not all zero, In particular we get that a = 1 λ1 b is central in F α G. Proof. Suppose first that (1) holds. Let w ∈ G. Then for every i ∈ {1, ..., k}, there are τ (i) ∈ {1, ..., k}, h i ∈ C G (g) and c i ∈ F × such that U w U ti = c i U t τ (i) U hi . Note that τ = τ w is a permutation of {1, ..., k}. Then we have that and so U w a = aU w . Since the set {U w : w ∈ G}, spans F α G, we get that a is central. On the other hand if a is central and h ∈ C G (g), then there is some so we must have that c = 1 and we get that (2) ⇒ (1).
By the last lemma, each ray class contributes only one central element element up to a scalar multiplication, which we call a ray element. In addition, ray elements from different ray classes are linearly independent. Thus we get that dim(Z(B)) is the number of ray classes. We can complete now the proof of Theorem 7 in case the commutation function satisfies θ g,g = 1 for every g ∈ G. Indeed, in [12] Regev showed that exp(M k (F)) = dim (M k (F)) = k 2 and since the exponent of an algebra depends only on its ideal of identities, we have from Lemma 31 that if A has a regular G-grading such that θ g,g = 1 for all g ∈ G, then |G| is an invariant of A (as an algebra and independent of the grading).
Corollary 36. Let A be an algebra over an algebraically closed field F of characteristic zero. If G is a finite group such that A has a nondegenerate regular G-grading with θ g,g = 1 for all g ∈ G then |G| = exp(A).
We move on to the general case where θ g,g can be −1. Let H = {g ∈ G | θ g,g = 1}. We are to show that H is a subgroup of G of index 1 or 2. Then, if the index is 1, we are in the previous case where θ g,g = 1 for all g ∈ G whereas in the second case, we will find a twisted group algebra for the group G such that its Grassmann envelope will be PI-equivalent to A.
Let E = E 1 ⊕ E −1 be the infinite dimensional Grassmann algebra over the field F, where E 1 and E −1 are the even and odd components of E. As noted above, this grading on E is a regular C 2 -grading with commutation function τ : Lemma 37. Let θ be a G-commutation function. Then there is a 2-cocycle α ∈ Z 2 (G, F × ) and a subgroup H ≤ G such that for is a regularly G-graded algebra with commutation function θ.
Proof. Let ψ : G → {±1} be the map ψ(g) = θ g,g . We claim that ψ is a homomorphism. To see this, let A be the θ-relatively free algebra (see Proposition 29). For h, g ∈ G, let θ gh,gh be the (unique) scalar such that x g,1 y h,1 x g,2 y h,2 = θ gh,gh x g,2 y h,2 x g,1 y h,1 in A (we use y h, * instead of x h, * for clarity). From Remark 23 we see that monomials of total degree e are central and hence we have It follows that θ gh,gh = θ g,g θ h,h , and hence letting H = ker(ψ) we have either H = G or [G : H] = 2. The case where H = G is the case considered above so we can assume that H = G. In this case, roughly speaking, we apply first the Grassmann envelope operation to "turn the −1's (in the image of θ g,g ) into +1's", then use the previous case to find some B = F α G, and finally apply the Grassmann envelope operation once again in order to return to the original identities.
Let us pause for a moment and summarize what we have so far. By the previous lemma we have constructed an algebraB which has a regular G-grading whose commutation function coincides with a given commutation function θ and hence, if θ is the commutation function of a regularly G-graded algebra A, we have in fact constructed a regularly G-graded algebraB with the same commutation function. It follows from Lemma 31 that Id G (A) = Id G (B), Id(A) = Id(B) and hence exp(A) = exp(B). The main point for constructing the algebraB is that in case the grading is nondegenerate, it enables us to show that ord(G) = exp(B). For this we need to further analyze the algebraB (constructed in Lemma 37).
Note that the algebras B andB in the lemma above satisfy θ g1,g2 = τ ψ(g1),ψ(g2)θg1,g2 . In particular if h ∈ H and g ∈ C G (h) then θ g,h =θ g,h . Since the grading onB is nondegenerate then for every h ∈ H there is some g ∈ C G (h) with θ h,g = 1, and from what we just said, this is also true for B.
Lemma 38. Let G be a finite group and H a subgroup of index 2. Let B = F α G be a twisted group algebra such that for every e = h ∈ H there is some Proof. Suppose first that the twisted group algebra F β H, where β = α | H , is simple. Let 0 = I be a Z 2 -graded ideal of B and denote I 0 = I ∩ B H and I 1 = I ∩ B G\H , so I = I 0 ⊕ I 1 . Observe that since I 0 is an ideal in F α H, it is either 0 or F α H. On the other hand, taking any U g where g / ∈ H, we have U g · I 0 ⊆ I 1 and since U g is invertible in F α G we have equality. It follows that I 0 = F α H for otherwise I = 0. We now have so we see that I = B. This proves that B is Z 2 -simple in that case.
If B is simple, then it must also be Z 2 -simple, so assume that neither B nor F β H are simple, or equivalently both α and β are degenerate 2-cocycles. This means that there is h 0 ∈ H such that U h0 U h = U h U h0 for all h ∈ C H (h 0 ), and similarly there is g 0 ∈ G such that U g U g0 = U g0 U g for all g ∈ C G (g 0 ). Note that by the assumption on B we must have g 0 / ∈ H. Let {t i } , {s i } be left coset representatives of C H (h 0 ) and C G (g 0 ) respectively. By Lemma 34 we have a = U ti U h0 U −1 ti ∈ Z(F β H) and b = U si U g0 U −1 si ∈ Z(F α G). If s i / ∈ H then s i g 0 ∈ H is a representative of the same left coset of C G (g 0 ) as s i , so we may assume that s i ∈ H for all i.
By the assumption on B, there is some g 1 ∈ C G (h 0 ) such that U g1 U h0 U −1 g1 = cU h0 with c = 1, and in particular g 1 / ∈ H by the choice of h 0 . It is easily seen that is again a set of left coset representatives of C H (h 0 ) in H (H is normal in G and g 1 ∈ C G (h 0 )). We now have that Let h ∈ H be such that hg 1 = g 0 . Then and we get a contradiction. Thus, we must have that either F α G or F β H are simple. In both cases the algebra F α G is Z 2 ∼ = G /H-simple and the lemma is proved.
Lemma 39. Let G be a finite group and H a subgroup of index 2. Let B = F α G and letB = (E 1 ⊗ B H )⊕ E −1 ⊗ B G\H be the Grassmann envelope of B. We denote by θ andθ the commutation functions of B andB respectively. If the regular G-grading onB is nondegenerate, then B is a Z 2simple algebra.
Proof. By nondegeneracy of the grading, we have for any e = h ∈ H an element g ∈ C G (h) such that θ g,h = 1 inB. But the Grassmann envelope operation does not change this property so it holds for the G-graded algebra B. Now use the previous lemma.
The fact that the algebra B = F α G is finite dimensional over F (F is algebraically closed of characteristic zero) and Z 2 -simple almost determines the structure of B.
Proof. This is well known. See for instance Lemma 6 in [9].
In our case, the algebra B satisfies an additional condition, namely dim(B 1 ) = dim(B 2 ) = |H| so if B is of the second type above we must have n = 2m.
We can now complete the proof of part 1 of Theorem 7.
Corollary 41. Let A be an algebra over an algebraically closed field F of characteristic 0. For every finite group G, if A has a nondegenerate regularly G-graded structure then |G| = exp(A).
Proof. We know that there is a simple Z 2 -graded algebra B = F α G (where B is one of the three types mentioned in the corollary above) such that the algebraB = E(B) satisfies Id G (B) = Id G (A). In We close this section with some additional corollaries of Lemma 37 and Lemma 39. Let us denote the Grassmann envelope of the algebra B in Corollary 40 (types (2) and (3) respectively) as follows: Corollary 42. Suppose that A has a nondegenerate regular G-grading for some finite group G. Then one of the following holds. It is well known that the families considered in the corollary above are mutually exclusive. Furthermore, different integers n or m yield algebras which are PI-nonequivalent. Indeed, algebras within the same type are PI-nonequivalent as their exponent is different. Next, any algebra of type one satisfies a Capelli polynomial whereas any algebra of type 2 or 3 does not. Finally, the exponent of any algebra of type 2 is an exact square whereas this is not the case for any algebra of type 3. Corollary 43. Suppose that A has a nondegenerate regular G-grading for some finite group G. Then there is a unique algebra C ∈ U such that A and C are PI-equivalent.
From the results above we can now derive easily a consequence on the commutation matrix M A for a regularly G-graded algebra A with commutation function θ.
The complete proof of Theorem 7 (parts 2 and 3) is presented in the next section.
Recall that for nondegenerate regularly G-graded algebras of type 1 we have θ g,g = 1 for all g ∈ G, whereas for type 2 and 3 half of the entries on the diagonal of M A are 1's and half are −1's. From the definition of the commutation matrix (see Subsection 1.1.1) we see that M A g,g = θ g,g U e . This clearly implies the following corollary.

The commutation matrix
It is easy to exhibit algebras with nonisomorphic nondegenerate regular G-gradings for some group G as well as examples of algebras with minimal regular gradings with nonisomorphic groups. For instance the algebra M 4 (F) admits (precisely) two nonisomorphic minimal gradings with the group Z /4Z× Z /4Z = g, h . These gradings are determined by bicharacters θ 1 and θ 2 , where θ 1 (g, h) = ζ 4 and θ 2 (g, h) = ζ 3 4 . On the other hand the algebra M 2 (F) admits a (unique) nondegenerate grading with the Klein 4-group and hence the algebra M 4 (F) ∼ = M 2 (F) ⊗ M 2 (F) admits a nondegenerate regular grading with the group ( Z /2Z) 4 . We therefore see that in general the entries of commutation matrices which correspond to different nondegenerate regular gradings on an algebra A may be distinct. However, the last corollary shows that the trace of the commutation matrices remains invariant.
Our goal in this section is to extend Corollary 44 and show that any two such matrices corresponding to nondegenerate gradings are conjugate (Theorem 7). We will follow the notation from Subsection 1.1.1. In particular we have B = F α G, H = ker (g → θ g,g = ψ(g)) (a subgroup of G of index ≤ 2) and A is PI-equivalent to the Grassmann envelope of B with respect to the Z /2Z-grading B = B H ⊕ B G\H .
Before we consider nondegenerate gradings, let us analyze briefly the degenerate case. If G is abelian, the commutation matrix is given by M A g,h = θ(g, h)U e . Hence, since the grading is not minimal, there exists g = e such that θ(g, h) = θ(h, g) = 1 for all h ∈ G and so M A is not invertible. The next proposition shows that this is true in the nonabelian case as well.
Proposition 45. Let A be a regularly G-graded algebra with a degenerate grading. Then M A is not invertible.
Proof. Let B = F α G be the twisted group algebra which corresponds to the G-graded algebra A and denote by θ B and θ A the corresponding commutation functions. We note that for commuting elements g 1 , g 2 ∈ G we have θ B g1,g2 = −θ A g1,g2 if θ A g1,g1 = θ A g2,g2 = −1 and θ B g1,g2 = θ A g1,g2 otherwise. Since the field F is algebraically closed of characteristic zero, B is a direct sum of matrix algebras over F. Fix a representation ρ : B → M n (F). The grading on A is degenerate so there is some e = h ∈ G such that θ A h,g = 1 for all g ∈ C G (h) and in particular θ A h,h = 1. We thus have θ B h,g = 1 for all g ∈ C G (h). As a consequence, applying Lemma 34, the element Let v ∈ g∈G B (a vector of size G with entries in B) where v g = λ g U g for some λ g ∈ F, and consider Clearly, we may choose the λ h 's such that h∈G τ g,h λ h U h = λ 1 U e + λ 2 z for all g ∈ G. This element is central so we have M A v g = λ 1 U e + λ 2 z. But the center of M n (F) is F · I, so there is some c ∈ F such that ρ(z) = c · I and hence ρ( M A v g ) = (λ 1 + cλ 2 ) · I. We see that if we choose λ 1 , λ 2 not both zero such that λ 1 + cλ 2 = 0 we have that ρ(M A v) = 0 for some v = 0. Moreover, we note that the nonzero entries of v are invertible in B. Let ρ i be the distinct representations of B and let e i ∈ B be such that ρ i (e j ) = δ i,j · I. For each i let v i be a vector corresponding to ρ i as constructed above and let v = v i e i ∈ g∈G B. Then . Furthermore, taking g ∈ G such that v i g = 0, we know that v i g is invertible and so ρ i (v i g ) = 0. This implies that v = 0. On the other hand we get for each i and so M A v = 0. We conclude that M A is not invertible from the left, and similar computations show that it is not invertible from the right.
Now we consider the case where the grading is nondegenerate.
Proposition 46. Let A be a nondegenerate regularly G-graded algebra, then Proof. Recall that for any fixed g ∈ G, the function θ A (·,g) : C G (g) → F × is a character, and since the grading is nondegenerate, this character is nontrivial for g = e.
For fixed a, c ∈ G, set N = C G (a −1 c) and choose a set of left coset representatives Notice that since ψ : is the commutation function of the Grassmann algebra with the Z 2 -grading. In addition, the character θ A (·, a −1 c) : N → F × is nontrivial if and only if a = c and so we get that This completes the proof of the proposition.
In the next discussion we use the notation of abelian groups, namely M A ∈ M |G| (F) with M A g,h = θ(g, h). This can be generalized to the nonabelian (i.e. not necessarily abelian) in the following way. We may view B = F α G as a direct sum of matrices, and then also M |G| (F α G) is isomorphic to a direct sum of matrices. Alternatively, we may factor through a representation ρ : B → M t (F) of B and then extend it to ρ : M |G| (F α G) → M |G|t (F). In any case, the matrix M A can be viewed as a matrix in M k (F) for some k large enough.
It follows from the last proposition that the commutation matrix M A satisfies the polynomial p(x) = x 2 − n = (x − √ n)(x + √ n) where n = |G| = 0. Hence, the corresponding minimal polynomial is either (x − √ n),(x + √ n) or p(x). In each case the minimal polynomial has only simple roots and hence the matrix M A is diagonalizable.
Let α + and α − denote the multiplicities of the eigenvalues √ n and − √ n respectively. Then we have α + + α − = n and α + − α − = tr(M A ) √ n . In our case, M A has only 1's on the diagonal (the first type of regular algebras), or half 1's and half −1's (the second and third type of regular algebras). Moreover, by Corollary 44 we know that this depends only on the algebra A and not on the grading.
Thus, for algebras of the first type (in Corollary 42) we have that n = exp(A) = |G| is a square and tr(M A ) = n. In that case, the equalities above take the form and hence For algebras of the second or third type (in Corollary 42) we have n = exp(A), which is either 2m 2 or (2m) 2 for some m, and tr(M A ) = 0. Then, here, the corresponding equalities are In case n = 2m 2 we have α + = α − = n 2 = m 2 and so the characteristic polynomial is and the minimal polynomial is (x − √ n)(x + √ n) = x 2 − n.
Finally, an easy computation of the free coefficient of the characteristic polynomial in each one of the cases considered above yields that det(M A G ) = ± |G| |G|/2 . This proves part 3 of Theorem 7 and hence the entire theorem is now proved.
As promised, we now compute the commutation matrix for the G-regular algebras constructed in Lemma 26, where G is an arbitrary finite group.
In case (2), the algebras A, B and A⊕B have the same commutation function θ. Thus, the cocycles corresponding to A, B, A ⊕ B are isomorphic (up to a coboundary) and therefore the corresponding twisted group algebras of A, B and A ⊕ B are isomorphic. With this identification of twisted group algebras we get that the commutation matrices of A, B and A ⊕ B are the same.
In case (3) we consider A N = g∈N A g for some subgroup N of G. Let α N be the restriction of α to N × N . Then α N is the cocycle corresponding to the algebra A N and there is a natural graded embedding F αN N ֒→ F α G. Let M ′ be the restriction of M A to the coordinates in N × N , then the entries of M ′ are in F αN N and this submatrix is actually M AN .
In cases (1) and (4) we have algebras A, B with commutation functions θ A , θ B and cocycles α, β which are defined on groups G and H respectively. In case the groups G and H are abelian, the matrix M A⊗B is just M A ⊗ M B . For the general case, let α ⊗ β ∈ Z 2 (G × H, F × ) be the cocycle defined by (α⊗β)((g 1 , h 1 ), (g 2 , h 2 )) = α(g 1 , g 2 )β(h 1 , h 2 ). Clearly, α⊗β represents the regular algebra A⊗ B (with commutation function θ A ⊗ θ B ). Furthermore, since F α G⊗ F β H ∼ = F α⊗β (G× H), we can extend this product to a "matrix tensor product". In other words, if ϕ : F α G ⊗ F β H → F α⊗β (G × H) is an isomorphism, then M A⊗B is determined by M A⊗B (g1,h1),(g2,h2) = ϕ(M A g1,g2 ⊗ M B h1,h2 ). In case (4), a similar computation shows that there is a isomorphism ψ : F α G ⊗F β G → F α·β G and then M A ⊗B (which is defined over F αβ G) is determined by M A ⊗B g,h = ψ(M A g,h ⊗ M B g,h ). Finally, we note that there is a natural embedding G ∼ =G = {(g, g) | g ∈ G} ≤ G × G. Hence may view A ⊗B as (A ⊗ B)G and with this identification M A ⊗B is the restriction of M A⊗B toG.

Nondegenerate skew-symmetric Bicharacters
If the group G is abelian then any G-commutation function θ is defined by the skew-symmetric bicharacter θ(g, h) = θ g,h for every g, h ∈ G (the commutation of two elements).
Our goal in this section is to present a classification of the the pairs (G, φ) where G is a (finite) abelian group and φ is a nondegenerate skew-symmetric bicharacter defined on G. In fact, this classification is known and can be found in [1]. Nevertheless for the reader convenience and completeness of the article, we recall the main results here.
In what follows, we present three types of regularly graded algebras. It turns out that the bicharacters which correspond to some special cases of these gradings are irreducible and generate all possible skew-symmetric bicharacters. Then Define a Z 2n × Z 2n -grading on M 2n,n (E) by M 2n,n (E) (k,l) = U k V l ⊗ E (−1) l , where E = E 1 ⊕ E −1 is the usual grading on the Grassmann algebra. This induces a minimal regular grading on M 2n,n (E) with commutation function θ determined by We consider the bicharacters which correspond to (some special cases of) the gradings just described.