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:

  1. Y. Bahturin, M. Zaicev, and S. K. Sehgal, “Finite-dimensional simple graded algebras”, 
    Sbornik: Mathematics, 199(7):2008, 965-983.
    DOI: 10.1070/SM2008v199n07ABEH003949.
  2. 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
  3. 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].
  4. 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