Structure of representations of infinite dimensional Lie algebras and conformal field theory

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The main objectives of this project are explicit constructions of representations, related to the fermionic formulas for characters of representations of Virasoro or other W-algebras and affine Lie algebras. The project also uses the ideas encountered in fermionic constructions, to give formulas for graded multiplicities of irreducible modules in the tensor product of finite-dimensional Lie algebra modules, or integrable modules in the fusion product of affine algebra modules (coinvariants or conformal blocks). Some of the specific goals are: The study of fusion products of representations of finite-dimensional simple Lie algebras; the inductive limit of the fusion product as the number of factors becomes infinite in a stabilized regime, which is expected to give new realizations of integrable modules of affine Lie algebras; combinatorial identities for q-series which result from these constructions; semi-infinite constructions of arbitrary highest weight representations of affine Lie algebras modules; and applications of semi-infinite (c.f. Feigin and Styoanovskii's approach) constructions in solutions of the fractional quantum Hall effect. This research is at the interface of combinatorial representation theory and mathematical physics. The constructions are guided by conformal field theory and exactly solvable models in statistical mechanics. The results are important in the representation theory of Lie algebras and affine Lie algebras. Combinatorial questions such as dimensions of weight spaces in irreducible representations, multiplicities of irreducible components in tensor products of Lie-algebra modules are related to the counting of certain matrix elements or conformal blocks. The physical applications include the study of wave functions in the quantum Hall effect, and partition functions in statistical mechanical systems at criticality.