Z/2Z-extensions of Hopf algebra module categories by their base categories

TL;DR

This paper constructs a Z/2Z-graded monoidal category from a self-dual Hopf algebra in a braided monoidal category, unifying various categories in conformal field theory through Hopf algebraic structures.

Contribution

It introduces a new construction of Z/2Z-extensions of Hopf algebra module categories using self-duality and cointegral data, providing a unified framework for related categories.

Findings

Constructs a Z/2Z-graded monoidal category from a self-dual Hopf algebra.

Describes conditions for rigid, braided, and ribbon structures in the constructed category.

Unifies Tambara-Yamagami categories and categories related to symplectic fermions.

Abstract

Starting with a self-dual Hopf algebra H in a braided monoidal category S we construct a Z/2Z-graded monoidal category C = C_0 + C_1. The degree zero component is the category Rep_S(H) of representations of H and the degree one component is the category S. The extra structure on H needed to define the associativity isomorphisms is a choice of self-duality map and cointegral, subject to certain conditions. We also describe rigid, braided and ribbon structures on C in Hopf algebraic terms. Our construction permits a uniform treatment of Tambara-Yamagami categories and categories related to symplectic fermions in conformal field theory.