Differential Operator Representations of Affine Kac-Moody Lie Algebras A (1) 2 and A (2) 2

Abstract

Kac-Moody Lie algebras are infinite-dimensional Lie algebras whose representation theory is very similar to those of simple Lie algebras. Kac-Moody Lie algebras are of three types, namely finite type, indefinite type, and affine type based on the characteristics of the generalized Cartan matrix. Here we work with affine Kac-Moody algebras which are algebraic structures with determinant value of its generalized Cartan matrix zero. In this paper, we give a brief description of the root systems and the Dynkin diagrams of untwisted and twisted Kac-Moody algebras. Finally, we have calculated the irreducible representations of two affine untwisted and twisted Kac-Moody Lie algebras, namely A (1) 2 and A (2) 2 by first-order differential operators. Our results have potential implementations in two-dimensional conformal field theory, string theory, supersymmetric gauge theory, and many others.