Ambidextrous objects and trace functions for nonsemisimple categories

Abstract

15 pagesWe provide a necessary and sufficient condition for a simple object in a pivotal k-category to be ambidextrous. As a consequence we prove that they exist for factorizable ribbon Hopf algebras, modular representations of finite groups and their quantum doubles, complex and modular Lie (super)algebras, the (1,p) minimal model in conformal field theory, and quantum groups at a root of unity

The trace on projective representations of quantum groups

“…The usual construction of quantum invariants do not directly apply to these categories because of the following obstructions: the categories are not semi-simple and have vanishing quantum dimensions. Partial results overcoming these obstructions have been obtained in [9,17,18,19,20,22]. In this paper we generalize some of these results using a new concept called generically semi-simple (loosely meaning the category is graded and semi-simple on a dense portion of the graded pieces).…”

“…which satisfy compatibility conditions (see for example [1,18]). In particular, the left dual and right dual of a morphism f : V → W in C coincide:…”

“…It follows that the trace of Theorem 22 induces a trace on the idea of projective U ξ -module. This trace is unique because from Theorem 4.7.1. of [18] there exists unique nontrivial right trace on the projective ideal of U ξ -modules.…”

“…Next we prove that C sl(2) has a trace and show the modified dimensions only depend on the action of the casimir element. First, the projective cover of the trivial module is self dual, see [15,16] for the even case and [18] for the odd. Therefore, Corollary 3.2.1 of [18] implies the ideal Proj of projective U ξ -modules admits a unique (up to global scalar) nontrivial right trace which we denote by {t V } V ∈Proj .…”

Modified traces for quasi-Hopf algebras

“…Modified traces were introduced in [GPV,GKP2]. They are a generalisation of the categorical trace in a pivotal linear category.…”