Recall that if $A$ is an algebra and $G$ is a group, then a $G$-grading on $A$ is a decomposition $A=\bigoplus A_g$ as a vector space where $A_g A_h \subseteq A_{gh}$ for any $g,h\in G$. Given an algebra $A$, we may ask what are all the possible gradings it admits (and what are the corresponding groups).

Clearly, each algebra can be graded with the trivial group $\{e\}$. More generally, given a group $G$ we can always consider the grading $A_e=A$ and $A_g=0$ for all $e\neq g \in G$. Such gradings, of course, give no relations what so ever between the group $G$ and the algebra $A$ and the question is what “nice” properties of the grading do give some connection.

Consider the matrix algebra $M_2(\mathbb{C})$ and its vector space decomposition

$M_2(\mathbb{C})=\mathbb{C} \overbrace{ \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) }^I$ $\oplus \mathbb{C} \overbrace{ \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) }^X$ $\oplus \mathbb{C} \overbrace{ \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) }^Y$ $\oplus \mathbb{C} \overbrace{ \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right) }^{XY} .$

Clearly, $X,Y$ and $XY$ commute with $I$, but they don’t commute among themselves. On the other hand we do have the identity $XY=-YX$ so each pair of these elements commute up to a $\pm 1$. The decomposition above is actually a ${\mathbb{Z}}/{2\mathbb{Z}} \times {\mathbb{Z}}/{2\mathbb{Z}}$-grading which shows that while the original algebra is not commutative, under this grading it is almost commutative.

This notion of “almost commutativity” was formalized by Regev and Seeman [1] with the following definition: Let G be a finite abelian group and $F$ a characteristic zero field. A G-grading on an $F$-algebra $A$ is said to be regular if there is a commutation function $\theta:G\times G \to F^\times$ such that

1. For any tuple $g_1,\;g_2,\cdots,\;g_n \in G$ there are $a_i \in A_{g_i}$ such that $\prod_1 ^n a_i \neq 0$.
2. For each $g,h\in G$ and any $a_g \in A_g,\;b_h \in A_h$ we have $a_g b_h = \theta(g,h) b_h a_g$.

Other examples for such regular gradings come from the Grassmann algebra and twisted group algebras (see here). For the grassmann algebra case we have a commutation function $\tau$ such that $\tau(1,1)=\tau(1,-1)=\tau(-1,1)=1$ and $\tau(-1,-1)=-1$. For twisted group algebra with cocycle $\alpha$, the commutation function is defined by $\theta(g,h)=\frac{\alpha(g,h)}{\alpha(h,g)}$, and the first condition in the definition is satisfied since all the nonzero homogeneous elements are invertible. In addition, every matrix algebra is isomorphic to some twisted group algebra, so every matrix algebra admits some regular grading (and the regular grading on $M_2(\mathbb{C})$ in the begining is such a case).

An algebra $A$ may admit more than one regular grading. For example, the algebra $A=\mathbb{C}[x]$ admits a regular $\mathbb{Z}/n \mathbb{Z}$-grading for each $n\in \mathbb{N}$, by setting $A_i = span\{ x^m \; \mid \; m\equiv_n i\}$. This is not surprising, since $A$ is already commutative. More generally, the smaller the group is in a regular grading, the “more” commutative the algebra is, and the group can be trivial if and only if the algebra is commutative. Thus, we want to consider regular gradings with the “smallest” group possible.

Let $A$ be some regularly $G$-graded algebra with commutation function $\theta$. Setting $\ker(\theta)=\{h\in G\;\mid\; \theta(h,g)=1 \forall g\in G \}$, we get that the induced $G/\ker(\theta)$ grading on $A$ is still regular with commutation function $\tilde{\theta}$ (and with a smaller group). In addition we have that $\ker(\tilde{\theta})=\{e\}$. A grading which satisfies this condition is called minimal. While an algebra $A$ may admit several minimal regular gradings with non isomorphic groups it was conjectured by Bahturin and Regev [2] that these groups must have the same cardinality. Thus, the size of a group in a minimal regular grading is an invariant of an algebra, which measure, in a sense, how close it is to being commutative.

### Our results [3]:

As the definition of a regular grading is actually a condition on the ideal of identities, we give a classification up to PI-equivalence of all regular graded algebras. In particular, up to a PI-equivalence, every regular graded algebra is a combination of a twisted group algebra and the Grassmann algebra.

Extending the definition to arbitrary groups, we use the classification above and show that the cardinality is indeed an invariant of the algebra. In particular, this invariant is called the exponent of the algebra and appears in PI-theory. The exponent measures in a sense how many multilinear identities of degree n the algebra has when n goes to infinity.

Theorem: Suppose $A$ has a minimal regular grading by a group $G$. Then $|G|$ is an invariant of $A$ and equals $exp(A)$.

### Questions:

The classification of all abelian groups $G$, such that there are algebras $A$ with minimal regular G-grading, is well known. Every such group must either be $G=H\times H$ or $G=H\times H\times C_2$ for some abelian group H (the $C_2$ part comes from the Grassmann algebra).

A group $G$ which has a 2-cocycle $\alpha$ such that $F^\alpha G \cong M_r(k)$ is called central type. The classification of non abelian central type groups is still an open question. By the extended definition of regular grading, every central type group admits a minimal regular grading on some algebra $A$.

Is there a classification of groups admitting minimal regular gradings, if we are given the classifiction of groups of central type?

### Remarks:

The commutation functions for abelian groups also appear in a generalization of Lie algebras called Lie color algebras. In particular, Lie superalgebra are Lie color algebra corresponding to the commutation function of the Grassmann algebra (with the cyclic group of order 2). These “commutation functions” also appear (with similar definitions) in Hopf Algebras.

Before the definition we first recall one of Jordan’s theorems. It states that if $G$ is any finite group which can be embedded in $GL_n(\mathbb{C})$, then it is almost abelian in the following sense. There is a function $f:\mathbb{N} \to \mathbb{N}$ such that if $G$ can be embedded in $GL_n(\mathbb{C})$, then it has a normal abelian subgroup $H$ such that $[G:H].

A dual question that we may ask is the following: if $G$ grades $M_n(\mathbb{C})$ instead of embedded in $GL_n(\mathbb{C})$, is it almost abelian? If so, can it be generalized to arbitrary algebras?

As before, we can always grade algebras in a trivial way $A_e = A$ and lose all the structure of $G$. We thus consider only nondegenerate gradings, namely gradings which satisfy the first condition of regular gradings:

• For all $g_1,...,g_n \in G$ there are $a_i \in A_{g_i}$ such that $a_1 \cdots a_n \neq 0$.

While $M_n(\mathbb{C})$ has a “natural” size, namely its dimension $n^2$ which can be used to bound the abelian part in $G$, we would also like to consider infinite dimensional algebras, so the trivial choice of dimension as the size of the algebra is insufficient.

As before, the nondegenerate condition is actually a condition on the polynomial identities of the algebra. Thus, a natural choice for the “size” of the algebra comes from its identities, and is given by its exponent $exp(A)$ (which is only a function of the algebra and not of the grading). In particular, for matrix algebras we have that $exp(M_n(F))=n^2$.

### Our results [4]:

We prove the following theorem:

Theorem: There is a function $f:\mathbb{N}\to \mathbb{N}$ which satisfies the following. If $A$ is a PI algebra graded nondegenerately by an arbitrary group $G$, then $G$ contains an abelian subgroup $H$ such that $[G:H].

We note here that if $G$ has an abelian subgroup of bounded index, then it also has a normal abelian subgroup of bounded index. Also, we can choose $f$ to be $f(n)=n^K$ for $K$ big enough.

A quick sketch of the proof is as follows. First, considering the theorem only with finite groups, it is known (though the proof is very complex) that every graded algebra is graded PI equivalent to either a finite dimensional algebra, or a Grassmann envelope of a finite dimensional algebra. Thus we transfer the problem to the case of finite dimensional algebras.

Then, using structure theorems on finite dimensional algebras and their PI identities, we can reduce to the case where the algebra is graded simple, and even more to the case where the algebra is just a group algebra. Finally, we use a result due to D. Gluck which bounded the minimal index of an abelian subgroup of $G$ as a function of its representations.

The part of infinite groups relies mainly on the following fact. Suppose that $G$ grades a PI algebra $A$ nondegenerately. Let $f(x_1,...,x_n)$ be any nonzero multilinear polynomial identity and suppose that the monomial $x_1 \cdots x_n$ appears with coefficient 1. If $g_1 ,..., g_n$ is any tuple of elements in $G$, then we can find $a_i\in A_{g_i}$ such that $\prod a_i \neq 0$ (by definition of nondegenerate grading). Putting them in the identity $f$ we must have that $\prod a_i \in A_{\prod g_i}$ is not zero, so there must be another monomial in the same homogeneous component. In particular there is some permutation $\sigma\in S_n$ such that $g_1 \cdots g_n =g_{\sigma(1)} \cdots g_{\sigma(n)}$. Groups which satisfy this condition for any $n$-tuple of elements are called permutable groups. It is known that all finitely generated permutable group are abelian by finite, so in particular any finitely generated group which grades nondegenerately a PI algebra, is residually finite.

Finally, the move from finitely generated groups to arbitrary groups is a general theorem proved by Isaacs and Passman.

### Bibliography:

1. A. Regev and T. Seeman, $\mathbb{Z}_2$-Graded tensor products of PI-algebras, Journal of Algebra, Volume 291(1):(2005), 274-296.
DOI: 10.1016/j.jalgebra.2005.01.049.
2. A. Regev and Y. Bahturin, Graded tensor products, Journal of Pure and Applied Algebra 213(9): (2009), 1643-1650.
DOI 10.1016/j.jpaa.2008.12.010
3. E. Aljadeff and O. David, On regular $G$-gradings, to appear in Transactions of the American Mathematical Society.
arXiv: arXiv:1212.0343v2 [math.RA]
4. E. Aljadeff and O. David, On group gradings on PI algebras,
arXiv: arXiv:1403.0200 [math.RA]