Simple Algebras and Involutions

Abstract

The first part of this chapter contains basic results about finite-dimensional simple algebras due to Wedderburn, Dickson, Albert, R. Brauer, E. Noether, and others. This is a classical part of (linear) algebra. In many respects, it is similar to the algebraic theory of quadratic forms. In both cases, the theory is dominated by a few basic structure theorems. Subsequently, we develop Albert’s theory of involutions on simple algebras. There are many interesting connections between the theory of quadratic and hermitian forms on the one hand and the theory of simple algebras and involutions on the other. We want to mention the most important ones, though not all will be pursued in this book:

• 1) With every quadratic form a simple algebra — its Clifford algebra — is associated in a functorial way. This algebra reflects many important properties of the quadratic form (see chapter 9).

• 2) It will be shown in this chapter that the classification of involutions on simple algebras is almost identical with the classification of hermitian forms over division algebras.

• 3) A particularly important part of the theory of hermitian forms is concerned with hermitian forms over group rings and group algebras. This theory is characterized by an attractive combination of methods from many parts of mathematics: representation theory, simple algebras, orders, quadratic forms, algebraic number theory, algebraic K-theory, and even algebraic topology.

And most basically:

• 4) The very basis of the notion of hermitian forms is a ring with involution, and it is only natural that properties of this ring and involution are reflected in the pertinent theory of hermitian forms.