Quantum McKay correspondence and global dimensions for fusion and module-categories associated with Lie groups. (English) Zbl 1295.81091
Summary: Global dimensions for fusion categories \(\mathcal{A}_k(G)\) 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 \(\mathcal{A}_k(\mathrm{SU}(2))\).
MSC:
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 33E99 | Other special functions |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Online Encyclopedia of Integer Sequences:
The classical Lie superfactorial of type Dr SO(2r): When a Lie group G is simply laced, the classical Lie superfactorial sf_G is the product of s! where s belongs to the multiset E of exponents of G. Here G=Dr.The classical Lie superfactorial of types E6, E7, E8.
The product of factorials s! where s belongs to the multiset of exponents of the Lie groups G=Br or G=Cr. Also 2^r times the classical Lie superfactorial of type Br SO(2r+1). Also 2^{r(r-1)} times the Lie superfactorial of type Cr Sp(2r).
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