Generalized trace and modified dimension functions on ribbon categories

Abstract

Abstract. In this paper we use topological techniques to construct generalized trace and modified dimension functions on ideals in certain ribbon categories. Examples of such ribbon categories naturally arise in representation theory where the usual trace and dimension functions are zero, but these generalized trace and modified dimension functions are nonzero. Such examples include categories of finite dimensional modules of certain Lie algebras and finite groups over a field of positive characteristic and categories of finite dimensional modules of basic Lie superalgebras over the complex numbers. These modified dimensions can be interpreted categorically and are closely related to some basic notions from representation theory.

Some remarks on the unrolled quantum group of sl(2)

“…The proof follows from results of [20,24]. Here we explain this proof without recalling the definitions given in these papers: In [24] we show that if α ∈ C \ 1 2 Z then V α is an ambidextrous object in C .…”

“…Here we explain this proof without recalling the definitions given in these papers: In [24] we show that if α ∈ C \ 1 2 Z then V α is an ambidextrous object in C . In [20] we show that an ambidextrous object J leads to the existence of a unique (up to a constant) trace t on the ideal I J generated by J. When J is simple then the trace is uniquely determined by the assignment t J (f ) = c f , where c is a constant.…”

“…The subcategory Proj is an ideal (see also [20]): it is closed under retracts (i.e. if W ∈ Proj and α : X → W and β : W → X satisfy β • α = Id X , then X ∈ Proj) and if X is in C and Y is in Proj then X ⊗ Y is in Proj.…”

Renormalized Hennings Invariants and 2 + 1-TQFTs

“…To explain this work more precisely we will recall some past results. Our construction relies on the modified trace defined in [28,29,31]. In nonsemisimple representation theory the categorical trace can vanish for many modules.…”

The trace on projective representations of quantum groups

“…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.…”