Crossed product algebras: representation, Brauer group and application.

Abstract

In this master thesis we study a special kind of central simple algebras called crossed product algebras. We will show how to construct them and study some of their properties, mainly their representation theory and divisibility. In particular, Brauer groups are employed to study the divisibility of crossed product algebras. A special kind of crossed product algebra called cyclic algebra has relatively simple structure and is particularly interesting. We compute the representation and Brauer group, of cyclic algebras and construct a new cyclic algebra with higher degree by tensoring two cyclic algebras with lower degree. Motivated by a coding theory problem, we next discuss the existence of unitary involution over crossed product algebras. We will show that the Hermitian conjugation of the representation matrix can be translated into the assumption of this involution. These generalize the results in [51. Finally we discuss how to find unitary elements in crossed product algebras.