Low-Dimensional Quaternionic Matrix Groups

Abstract

We focus on several properties of the Lie groups Sp(n) and SLn(H). We discuss their Lie algebras, the exponential map from the Lie algebras to the groups, as well as when this map is surjective. Since quaternionic multiplication is not commutative, the process of calculating the exponential of a matrix in Sp(n) or SLn(H) is more involved than the process of calculating the exponential of a matrix over the real or complex numbers. We develop processes by which this calculation may be reduced to a simpler problem, and provide an example to illustrate this. Additionally, we discuss properties of these groups such as centers, maximal tori, normalizers of the maximal tori, Weyl groups, and Clifford Algebras.