Derivations of subalgebras of the Lie algebra of block upper triangular matrices

Abstract

This dissertation studies the derivations of some subalgebras of the Lie algebra of block upper triangular matrices. Specifically, we study the derivations of the Lie algebra of strictly block upper triangular matrices and the Lie algebra of so-called dominant upper triangular (DUT) ladder matrices, which are block upper triangular matrices that take zero on some preset nonconsecutive diagonal blocks. The dissertation consists of six chapters. Chapter 1 provides a brief introduction, back- ground information, and some related literatures to the topics to be studied. In Chapter 2, we introduce the definitions and basic properties of matrices, ladder matrices, and Lie algebras. We also describe some linear transformations between matrix spaces that satisfy certain special properties. These linear transformations will appear in the derivations of Lie algebra to be studied. Chapter 3 provides an explicit description of the derivations of the Lie algebra N of strictly block upper triangular matrices over a field F. In Chapter 4, we completely characterize the derivations of the Lie algebra ML of dominant upper triangular (DUT) ladder matrices over a filed F with char(F) not equals 2. In exploring the results, we obtain some properties of these Lie algebras and their derivations. Chapter 5 discusses the derivations of the Lie algebra [ML , ML] for the so-called strongly dominant upper triangular (SDUT) ladder L over a field F with char(F) not equals 2, 3. The final chapter provides some potential future research directions on those Lie algebras that we study in this dissertation.