Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 085, 18 pages      arXiv:0812.2381
Contribution to the Special Issue on Kac-Moody Algebras and Applications

String Functions for Affine Lie Algebras Integrable Modules

Petr Kulish a and Vladimir Lyakhovsky b
a) Sankt-Petersburg Department of Steklov Institute of Mathematics, Fontanka 27, 191023, Sankt-Petersburg, Russia
b) Department of Theoretical Physics, Sankt-Petersburg State University, 1 Ulyanovskaya Str., Petergof, 198904, Sankt-Petersburg, Russia

Received September 15, 2008, in final form December 04, 2008; Published online December 12, 2008

The recursion relations of branching coefficients kξ(μ) for a module Lg hμ reduced to a Cartan subalgebra h are transformed in order to place the recursion shifts γ Γa h into the fundamental Weyl chamber. The new ensembles FΨ (the ''folded fans'') of shifts were constructed and the corresponding recursion properties for the weights belonging to the fundamental Weyl chamber were formulated. Being considered simultaneously for the set of string functions (corresponding to the same congruence class Ξv of modules) the system of recursion relations constitute an equation M(u) Ξv m(u) μ = δ(u) μ where the operator M(u) Ξv is an invertible matrix whose elements are defined by the coordinates and multiplicities of the shift weights in the folded fans FΨ and the components of the vector m(u) μ are the string function coefficients for Lμ enlisted up to an arbitrary fixed grade u. The examples are presented where the string functions for modules of g = A2(1) are explicitly constructed demonstrating that the set of folded fans provides a compact and effective tool to study the integrable highest weight modules.

Key words: affine Lie algebras; integrable modules; string functions.

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