Generalized trace and modified dimension functions on ribbon categories

TL;DR

This paper develops generalized trace and dimension functions in ribbon categories using topological methods, enabling non-zero invariants in contexts where traditional traces vanish, with applications in representation theory.

Contribution

It introduces new generalized trace and dimension functions on ribbon categories, extending their applicability in representation theory where usual invariants are zero.

Findings

Generalized trace functions are constructed using topological techniques.

Modified dimensions are non-zero in categories where traditional ones vanish.

Applications include categories of modules over Lie algebras and finite groups.

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 non-zero. 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.