M.Sc Student | Karasik Igor |
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Subject | Crossed Products and Their Corresponding Central Polynomials |

Department | Department of Mathematics |

Supervisor | Professor Eli Aljadeff |

Full Thesis text |

Let G be a finite group. It is known
that every G-grading on a matrix algebra A=M_{n}(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 K_{t}G, is a G-graded
k-form of the matrix algebra M_{n}(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 K_{t}G.

The main objective of the thesis is
to use the algebra of generic elements of K_{t}G 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.