Numerical generation of semisimple tortile categories coming from quantum groups

Abstract

In this work we set up a general framework for exact computations of the associativity, commutativity and duality morphisms in a quite general class of tortile categories. The source of the categories we study is the work of Gelfand and Kazhdan, Examples of tensor categories, Invent.Mlath. 109 (l992), 595-617. They proved that, associated to the quantized enveloping algebra of any simple Lie group at a primitive prime root of unity, there is a semisimple monoidal braided category with finite number of simple objects. The prime p needs to be greater than the Coxeter number of the corresponding Lie algebra. We show that each of the Gelfand-Kazhdan categories has at least two subcategories which are tortile, and offer algorithms for computing the associativity, commutativity and duality morphisms in any of those categories. A careful choice of the bases of the simple objects and of the product of two such objects rnake the exact computations possible. The algorithms have been implemented in Mathemetica and tested for the categories A₂,p=5, A₃,p=7, A₄.p=7, C₂,p=7, and G₂,p=11.

This work was supported by the Center for Mathematical Computations through NSF grant DMS-9207973.