Module Information

Combining the reconstruction of a fascinating historical tale with the development of more and more refined technical tools the topic of this module provides an opportunity for undergraduate students to follow the emergence of Abstract Algebra along one of its most central lines and to learn to appreciate the unifying power of modern mathematics.

The theory of associative algebras grows historically out of the study of hypercomplex number systems in the 19th century and occupies a central position in modern Abstract Algebra. We start with the historic background, introduce a multitude of examples and then develop carefully the basics of this theory up to some classifying structure theorems. Following the strategy in the book of Farenick (see Essential Reading) we mainly study the finite-dimensional case, thus keeping everything accessible with tools from Linear Algebra. Focusing on algebras of linear operators we concentrate on those examples which are most relevant for applications outside Abstract Algebra. With the introduction of von Neumann algebras and C*-algebras towards the end of the course we finally see some important mathematical developments of the 20th century emerge from these origins which still play an important role in present day research.