Quantum McKay correspondence and global dimensions for fusion and module-categories associated with Lie groups

Abstract

Global dimensions for fusion categories Ak(G) defined by a pair (G,k), where G is a Lie group and k a positive integer, are expressed in terms of Lie quantum superfactorial functions. The global dimension is defined as the square sum of quantum dimensions of simple objects, for the category of integrable modules over an affine Lie algebra at some level. The same quantities can also be defined from the theory of quantum groups at roots of unity or from conformal field theory WZW models. Similar results are also presented for those associated module-categories that can be obtained via conformal embeddings (they are “quantum subgroups” of a particular kind). As a side result, we express the classical (or quantum) Weyl denominator of simple Lie groups in terms of classical (or quantum) factorials calculated for the exponents of the group. Some calculations use the correspondence existing between periodic quivers for simply-laced Lie groups and fusion rules for module-categories associated with Ak(SU(2)).