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 [1]: 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 [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, 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
.
Questions:
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
).
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
-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