Decomposition of Group Algebras

This lecture covers examples of decomposition of group algebras into a direct sum of matrix algebras, the group algebra of a finite group as a left module over itself, and the concept of a simple infinite-dimensional algebra, specifically the Weyl algebra over a field of zero characteristic. The lecture also discusses theorems related to simple modules, such as Wedderburn's and Maschke's theorems, and provides examples of group algebras as left modules over themselves. Additionally, it explores the structure of the Weyl algebra, its basis, and its properties as a simple algebra that is not a matrix algebra. The lecture concludes with the study of tensor products in the context of group algebras.