Characters of affine Kac-Moody algebras

Abstract

Kac-Moody algebras G(A) of rank r are Lie algebras associated with n X n generalised Cartan matrices A. If n = r then Q{A) is a complex simple finite-dimensional Lie algebra with finite Weyl group W, but if n = r + 1 then Q(A) is a complex infinite- dimensional affine Lie algebra with affine Weyl group W. This thesis is concerned with explicit calculations based on the use of W. Manipulating the Weyl-Kac character formula for highest weight modules provides a means of expanding Weyl orbit sums in terms of irreducible characters. These expan- sions are inverted to obtain analytic weight multiplicity generating functions for level 1 and 2 modules for all affine algebras of rank 1 and 2. The orbit-character expansions and weight multiplicity generating functions are then used to obtain branching rule multiplicities for some affine embeddings. On the other hand, the Weyl-Kostant-Liu character formula provides a means of expressing irreducible characters of an affine algebra in terms of irreducible characters of a simple finite-dimensional algebra. The key step is the identification of coset repre- sentatives {W : W} for each of the seven infinite series of affine Kac-Moody algebras indexed by their rank r. The proof is given in detail for A^\ while for the other affine algebras the results are expressed as conjectures which have been extensively verified by a computer program. Young diagrams are used to specify the action of the coset representatives on arbitrary weights as required in the character formula. This allows the computation of the irreducible characters to be done independently of the rank of the affine algebra. Since the weight multiplicities of finite-dimensional modules of the classical simple Lie algebras are polynomial in the rank this establishes that the weight multiplicities of irreducible highest weight modules of the seven infinite series of affine Kac-Moody algebras are also polynomial in the rank. Illustrative examples are given.

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