Adjoint Orbits of Borel Subgroup on Maximal Nilpotent Subalgebra of Types A and C.

Abstract

We describe the adjoint orbits of a Borel subgroup on a maximal nilpotent algebra of Types A and C. In Chapter 1, we provide a historical background and motivation behind the study. We highlight the important works in the subject that lead to the formulation of the problem. In Chapter 2, we present the necessary mathematical prerequisites. In Chapter 3, we let B_n; U_n and N_n be the set of n x n nonsingular, unit and nilpotent upper triangular matrices respectively. This section describes a novel approach to classify the B_n-similarity orbits in N_n. The Belitskii's canonical form of A in N_n under B_n-similarity is in Un where is the subpermutation such that A in Bn Bn. Using graph representations and U_n-similarity actions stablizing Un, we obtain new properties of the Belitskii's canonical forms and present an efficient algorithm to nd the Belitskii's canonical forms in N_n. As consequences, we list all Belitskii's canonical forms for n = 7; 8. We give the proof of the main theorem for subalgebra of type A cases. We would like to point out that the main results of Chapter 3 in this dissertation appear in [21]. In Chapter 4, we consider the adjoint action of Borel subgroup of the symplectic Lie group Sp_2n on the maximum nilpotent subalgebra n of the Lie algebra sp_2n to study adjoint orbits in the type C case. Our goals are to describe elementary adjoint actions in n in terms of the positive root system, give a redefi ned version of Belitskii's algorithm and use this algorithm to describe the corresponding canonical forms on the lattice of positive roots. Main results are proved and supportive examples are provided. In Chapter 5, we close this dissertation by discussing several possible directions for future research and by applying the main results.