Algebras with involutions and the classical groups

Abstract

This thesis deals with the problem of correspondence between semisimple algebraic groups defined over some base field 'k' and semisimple algebras with involutions over 'k'. This fundamental problem was first explored by Weil in 1960 in his paper titled "Algebras with involutions and the classical groups" . The primary result is that over a field of characteristic not equal to 2, almost every semisimple algebraic group with trivial center can be obtained as the connected component of identity in the automorphism group of a semisimple algebra with involution, and conversely, that automorphism group of every semisimple algebra with involution is almost always a semisimple algebraic group with a trivial center. First, we study the classical approach of this problem over a field of characteristic zero, given by Weil in 1960. This approach uses results from the classical theory of algebraic groups and the theory of central simple algebras. In the second part, we study the modern treatment of the problem using the language of Galois cohomology. We prove the Galois descent lemma which enables us to establish a correspondence between twisted forms of an algebra with involution with certain cohomology set. A similar correspondence is true for an algebraic group G defined over a base field. The cohomology set associated with a central simple algebra with involution is the same as that for a classical group G, when the involution is of the same type as the bilinear form whose isometries give G, giving rise to the main result that "classical groups over a base field k" are in one-one correspondence with "central simple algebras with involutions over k."