Embeddings of Graded Simple Algebras

Let G be a finite group and $A$ a G-graded algebra. If $B$ is a graded subalgebra of $A$ then any polynomial identity of $A$ is clearly an identity of $B$ , or equivalently $Id_G(A) \subseteq Id_G(B)$ . This leads to the question whether the other direction is also true, namely if $A,B$ are G-graded such that $Id_G(A) \subseteq Id_G(B)$ , then there is some graded embedding of $B$ into $A$ .

This question is false in general. An easy exercise show that $Id_G(A\oplus B)=Id_G (A) \oplus Id_G(B)$ , hence in particular $Id_G(A\oplus A)=Id_G (A)$ . If $A$ is finite-dimensional, then we can’t have a graded embedding of $A\oplus A$ into $A$ .

To avoid such counter examples, we restrict the problem to graded simple algebras. Recall that an ideal in $A$ is graded if it is generated by its homogeneous elements. A graded algebra $A$ is graded simple if $A\cdot A\neq 0$ and $A$ has no nontrivial graded ideals.

Of course, if $A$ is simple, then it is also graded simple. We can construct such examples as follows. Let $(g_1,...,g_n)\in G^n$ be an n-tuple for some $n\in \mathbb{N}$ . Define a G-grading on $A=M_n(k)$ by setting $A_g = span_k \{ E_{i,j} \mid g_i ^{-1} g_j =g \}$ where $E_{i,j}$ is the zero matrix with 1 in the $(i,j)$ coordinate. Such gradings are called elementary gradings. For example, if $G=C_2=<\sigma>$ and we take the tuple $(e,\sigma)$ , then the grading is

$\left(M_{2}(k)\right)_{1}=\left\{ \left(\begin{array}{cc} a & 0\\ 0 & b \end{array}\right):a,b\in k\right\} \;;\;\left(M_{2}(k)\right)_{-1}=\left\{ \left(\begin{array}{cc} 0 & a\\ b & 0 \end{array}\right):a,b\in k\right\}.$

Other examples of simple graded algebras are the twisted group algebras – if $I$ is a nonzero graded ideal, then it contains a nonzero homogeneous elements which is invertible, thus $I=A$ .

These two examples can be combined as follows. Let $\bar g=(g_1,...,g_n)\in G^n$ and $\alpha \in Z^2(G,k^\times)$ be a 2-cocycle. Define $k^\alpha G\otimes M_{\bar g}(k)$ to be the algbera $k^\alpha G\otimes M_n(k)$ with the G-grading

$(k^\alpha G\otimes M_n(k))_g = span\{ U_h\otimes E_{i,j} \mid \;g_i ^{-1}hg_j = g\}.$

For each 2-cocycle $\alpha$ and n-tuple $(g_1,...,g_n)\in G^n$ , the algebra $k^\alpha G\otimes M_{\bar g}(k)$ is G-simple.

Bahturin, Seghal and Zaicev classified all finite-dimensional graded simple algebra over an algebraically closed field, and showed that under certain conditions, all the graded simple algebras are of the form $k^\alpha G\otimes M_{\bar g}(k)$

Theorem [1]: Let G be a group, $k$ an algebraically closed field such that either $char(F)=0$ or $char(k)$ is coprime to the order of every finite subgroup of G and $A$ a finite-dimensional G-graded F-algebra. Then $A$ is graded simple if and only if there is some 2-cocycle $\alpha\in Z^2(G,k^\times)$ and a tuple $\bar g = (g_1,...,g_n)\in G^n$ such that $A$ is isomorphic to $k^\alpha G \otimes M_{\bar g}(k)$ .

Using the classification above Koshlukov and Zaicev [2] showed that if the conditions of the theorem are true and G is abelian, then two finite-dimensional graded simple algebras are isomorphic if and only if they have the same graded identities. Later on Aljadeff and Haile [3] showed that if the field has characteristic zero and G is finite (not necessarily abelian), then again two finite-dimensional graded simple algebras are isomorphic if and only if they are graded PI equivalent.

Our results[4]:

Let G be a finite group, $k$ an algebraically closed field of characteristic zero and $A,B$ be two finite-dimensional G-simple algebras. Suppose also that (1) G is abelian, or (2) $A$ has an elementary grading. Then there is a graded embedding $B \hookrightarrow_G A$ if and only if $Id_G(A)\subseteq Id_G(B)$ .

Questions:

Is the theorem above still true for G arbitrary finite group and $A,B$ arbitrary finite-dimensional G-simple? Is it true in the special case where $A=kG\otimes M_n(k)$ and $B=k^\alpha G\otimes M_r(k)$ for some 2-cocycle $\alpha$ and in both algebras the tuple is trivial (so $A_g=U_g \otimes M_n(k)$ ).

Bibliography:

Y. Bahturin, M. Zaicev, and S. K. Sehgal, “Finite-dimensional simple graded algebras”,
Sbornik: Mathematics, 199(7):2008, 965-983.
DOI: 10.1070/SM2008v199n07ABEH003949.
P. Koshlukov and M. Zaicev, “Identities and isomorphisms of graded simple algebras”,
Linear Algebra and its Applications, 432(12): 2010, 3141-3148.
DOI: 10.1016/j.laa.2010.01.010
E. Aljadeff and D. Haile, “Simple $G$ -graded algebras and their polynomial identities”, to appear in Transactions of the American Mathematical Society.
arXiv: arXiv:1107.4713 [math.RA].
O. David , “Graded embeddings of finite dimensional simple graded algebras” , Journal of Algebra, 367: 2012, 120-141.
DOI: 10.1016/j.jalgebra.2012.06.005