Affine Lie Algebras, Vertex Operator Algebras and Combinatorial Identities

by Cook, William Jeffrey

Affine Lie algebra representations have many connections with different areas of mathematics and physics. One such connection in mathematics is with number theory and in particular combinatorial identities. In this thesis, we study affine Lie algebra representation theory and obtain new families of combinatorial identities of Rogers-Ramanujan type. It is well known that when $ ilde{g}$ is an untwisted affine Lie algebra and $k$ is a positive integer, the integrable highest weight $ ilde{g}$-module $L(k Lambda_0)$ has the structure of a vertex operator algebra. Using this structure, we will obtain recurrence relations for the characters of all integrable highest-weight modules of $ ilde{g}$. In the case when $ ilde{g}$ is of (ADE)-type and k=1, we solve the recurrence relations and obtain the full characters of the adjoint module $L(Lambda_0)$. Then, taking the principal specialization, we obtain new families of multisum identities of Rogers-Ramanujan type.