On the Rosenberg-Zelinsky sequence in abelian monoidal categories

TL;DR

This paper explores the structure of invertible bimodules over Frobenius algebras in abelian monoidal categories, focusing on the Rosenberg-Zelinsky sequence and conditions for surjectivity related to orbifold constructions in conformal field theory.

Contribution

It investigates conditions under which the Rosenberg-Zelinsky sequence becomes surjective via Morita equivalence in the context of Frobenius algebras.

Findings

Identifies conditions for surjectivity of the homomorphism in the Rosenberg-Zelinsky sequence.

Connects algebraic structures to orbifold constructions in conformal field theory.

Provides criteria for Morita equivalence to achieve surjectivity.

Abstract

We consider Frobenius algebras and their bimodules in certain abelian monoidal categories. In particular we study the Picard group of the category of bimodules over a Frobenius algebra, i.e. the group of isomorphism classes of invertible bimodules. The Rosenberg-Zelinsky sequence describes a homomorphism from the group of algebra automorphisms to the Picard group, which however is typically not surjective. We investigate under which conditions there exists a Morita equivalent Frobenius algebra for which the corresponding homomorphism is surjective. One motivation for our considerations is the orbifold construction in conformal field theory.