Orthogonal Abelian Cartan Subalgebra Decompositions of Classical Lie Algebras Over Finite Commutative Rings

Abstract

Orthogonal decompositions of classical Lie algebras over the complex numbers of types A, B, C and D were studied in the early 1980s and attracted further attention in the past decade, especially in the type A case, due to its application in quantum information theory. In this dissertation, we consider the orthogonal decomposition problem of Lie algebras of type A, B, C and D over a finite commutative ring with identity. We first establish the appropriate definition of orthogonal decomposition under our setting, and then derive some general properties that rely on the finite commutative rings theory. Our goal is to construct interesting orthogonal decompositions of these Lie algebras. We begin with Lie algebras of type A by searching for sufficient conditions for the existence of such an orthogonal decomposition. Our study in the special case when the ring is a finite field provides us important information that leads to the approach we develop in this dissertation. We then apply our results on the orthogonal decomposition of type A Lie algebras to obtain a construction of the orthogonal decomposition of Lie algebras of type C. We also provide methods of constructing orthogonal decompositions for Lie algebras of types B and D.