A Hennings TQFT Construction for Quasi-Hopf Algebras

TL;DR

This paper extends the Hennings TQFT construction from ribbon Hopf algebras to ribbon quasi-Hopf algebras, addressing non-associativity and integral theory complexities, with potential applications to Dijkgraaf-Witten TQFT.

Contribution

It develops a framework for constructing TQFTs from ribbon quasi-Hopf algebras, handling non-trivial coassociators and integrals, expanding the scope of topological quantum field theories.

Findings

Successfully extended Hennings TQFT to quasi-Hopf algebras

Addressed non-associativity in tangle category representations

Connected the construction to Dijkgraaf-Witten TQFT

Abstract

We extend the construction of the Hennings TQFT for ribbon Hopf algebras to the case of ribbon quasi-Hopf algebras as defined by Drinfeld. Calculations proceed in a similar fashion to the ordinary Hopf algebra case, but also require the handling of the non-trivial coassociator in the triple tensor product of the algebra as well as several special elements. The main technical difficulties we encounter are representing tangle categories in the non-associative setting, and the definition and use of integrals and cointegrals in the non-coassociative case. We therefore discuss the integral theory for quasi-Hopf algebras, using work of Hausser and Nill. A motivating example for this work is the Dijkgraaf-Pasquier-Roche algebra which is believed to be related to the Dijkgraaf-Witten TQFT.