UC Riverside

Borel and de-Siebenthal classified the maximal connected subgroups of maximal rank

of a connected compact Lie group. This result can be rephrased in terms of automorphisms of the

semisimple Lie algebra and the subalgebra of fixed points. They also give rise in a natural way

to a maximal parabolic subalgebra of an affine Lie algebra. In the case when the automorphism

is non-trivial, we shall see that the parabolic subalgebra is isomorphic to an equivariant map

algebra.

We develop the theory of integrable representations of such Lie algebras. In particular, we

define and study global Weyl modules. These are closely related to the module category of a

commutative associative algebra. In this dissertation, we give a presentation of this algebra, and give

partial results toward a dimension formula for local Weyl modules.