Traces on ideals in pivotal categories

Abstract

Abstract. We construct invariants of C -colored spherical graphs from traces on ideals in a pivotal category C. Then we provide a systematic approach to defining such traces from ambidextrous and spherical traces on a class of objects of C. This extends the notion of an ambidextrous object of a braided category given by the first two authors in a previous work.Mathematics Subject Classification (2010). Primary 18D10; Secondary 16T05.

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).…”

“…The motivation of this paper is to provide the underpinnings for the construction of topological invariants. With this in mind, in this subsection, we recall the notion of re-normalized colored ribbon graph invariants introduced and studied in [19,17,20,21,22]. This subsection is independent of the rest of the paper.…”

“…where the second equality is from Lemma 4(b) of [22] and the final equality holds because V 0 is isomorphic to V * 0 . Therefore, to compute d(V ) we need to give a morphism f : V 0 ⊗ V → V 0 ⊗ V and compute its left and right partial trace.…”

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.…”

“…The original definition of modified traces from [GPV,GKP2] is for general tensor ideals, but here we will restrict our attention to the tensor ideal Proj(C) ⊂ C , (2.4) the full subcategory consisting of all projective objects in C. We note that unless C is semisimple, the categorical trace vanishes identically on Proj(C), see e.g. [GR2,Rem.…”