|
Statement of Research Interest
Introduction My current research concerns the representation theory of Lie algebras. Since their introduction over a century ago, Lie groups and Lie algebras have been exploited in and connected with a wide array of mathematical pursuits, including algebraic geometry, di erential geometry, topology and knot theory, foundations of quantum mechanics, and quantum eld theory. My primary focus is on nite{dimensional representations of in nite{dimensional Lie algebras, in particular of Kac{Moody algebras and some of their generalizations.
A representation of some algebraic or combinatorial structure A (such as a group, ring, algebra, quiver) is (in most cases) a vector space V and a homomorphism from A to End(V ), where End(V ) is endowed with the same algebraic structure as A. More conceptually, a representation is some vector space (e.g., a space of states, for many physicists) on which A is allowed to act in some way that is compatable with the internal structure of A. In one sense a representation of A provides a concrete realization of A, loosely analogous to the way that matrices are `realizations' of linear transformations or embeddings are `realizations' of manifolds. When one studies a representation of A, one is interested in the vector space on which A is acting and the structures which are preserved by this action, but also in the manifestation of A as a set of transformations. 我要提问
|