UC Riverside

The dissertation is broken into two parts. Part I deals with the following problem: suppose $\g = \g_0 \oplus \g_1$ is a simple $\Z_2$-graded Lie algebra and let $\mathfrak{b}_0$ be a fixed Borel subalgebra of $\g_0$; describe and enumerate the abelian $\mathfrak{b}_0$-stable subalgebras of $\g_1$. The original proof uses a geometric approach; in Part I, we utilize an algebraic method which better describes the corresponding subalgebras. Part II focuses on a generalization of a combinatorial problem related to the representation theory of affine Lie algebras; given an arbitrary partition, we describe an iterative algorithm which at every level generates a ballot number of partitions.