|M.Sc Student||Karasik Igor|
|Subject||Crossed Products and Their Corresponding Central|
|Department||Department of Mathematics||Supervisor||PROF. Eli Aljadeff|
Let G be a finite group. It is known that every G-grading on a matrix algebra A=Mn(k) over an algebraically closed field F, is a composite of two kinds of gradings: fine and elementary.
In this work we will consider a special kind of elementary gradings on A where the defining n-tuple of the grading is given precisely by an ordering of all elements of the group G. In particular G is of order n. We show that if K is a subfield of F and K/k is a G-Galois extension, then the skew group algebras KtG, is a G-graded k-form of the matrix algebra Mn(k) with the above elementary G-grading. For these gradings we recall the G-graded polynomial identities and the corresponding G-graded relatively free algebra. Furthermore, using the skew group algebra description, we present the G-graded relatively free algebra as the algebra of generic elements of KtG.
The main objective of the thesis is to use the algebra of generic elements of KtG in order to give a T-generating set of the G-graded central polynomials of A with the above G-grading, where G is an arbitrary finite group. A similar result was obtained by A. Brandao in case G is cyclic. We emphasize that our approach is different from the one of A. Brandao.