Let G be a finite group and a G-graded algebra. If is a graded subalgebra of then any polynomial identity of is clearly an identity of , or equivalently . This leads to the question whether the other direction is also true, namely if are G-graded such that , then there is some graded embedding of into .
This question is false in general. An easy exercise show that , hence in particular . If is finite-dimensional, then we can’t have a graded embedding of into .
To avoid such counter examples, we restrict the problem to graded simple algebras. Recall that an ideal in is graded if it is generated by its homogeneous elements. A graded algebra is graded simple if and has no nontrivial graded ideals.
Of course, if is simple, then it is also graded simple. We can construct such examples as follows. Let be an n-tuple for some . Define a G-grading on by setting where is the zero matrix with 1 in the coordinate. Such gradings are called elementary gradings. For example, if and we take the tuple , then the grading is
Other examples of simple graded algebras are the twisted group algebras – if is a nonzero graded ideal, then it contains a nonzero homogeneous elements which is invertible, thus .
These two examples can be combined as follows. Let and be a 2-cocycle. Define to be the algbera with the G-grading
For each 2-cocycle and n-tuple , the algebra 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
Theorem : Let G be a group, an algebraically closed field such that either or is coprime to the order of every finite subgroup of G and a finite-dimensional G-graded F-algebra. Then is graded simple if and only if there is some 2-cocycle and a tuple such that is isomorphic to .
Using the classification above Koshlukov and Zaicev  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  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.
Let G be a finite group, an algebraically closed field of characteristic zero and be two finite-dimensional G-simple algebras. Suppose also that (1) G is abelian, or (2) has an elementary grading. Then there is a graded embedding if and only if .
Is the theorem above still true for G arbitrary finite group and arbitrary finite-dimensional G-simple? Is it true in the special case where and for some 2-cocycle and in both algebras the tuple is trivial (so ).
- Y. Bahturin, M. Zaicev, and S. K. Sehgal, “Finite-dimensional simple graded algebras”,
Sbornik: Mathematics, 199(7):2008, 965-983.
- P. Koshlukov and M. Zaicev, “Identities and isomorphisms of graded simple algebras”,
Linear Algebra and its Applications, 432(12): 2010, 3141-3148.
- E. Aljadeff and D. Haile, “Simple -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.