Algebras, Quaternions and Quaternionic Symplectic Groups

Abstract

In this chapter we begin by studying algebras over a field, with their groups of units providing many interesting groups. In particular, we study division algebras and their linear algebra. Then we introduce the quaternions which form the only non-commutative example of a real division algebra. There is an associated family of compact connected matrix groups defined using the quaternions, the quaternionic symplectic groups which provide another infinite family of compact simply connected matrix groups.