Hopf-Frobenius Algebras and a Simpler Drinfeld Double

Joseph Collins
Ross Duncan

The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, Hopf-Frobenius algebras, starting from a single Hopf algebra which is not necessarily commutative or cocommutative. We provide a few necessary and sufficient conditions for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in the category of finite dimensional vector spaces is a Hopf-Frobenius algebra. In addition, we show that this construction is unique up to an invertible scalar. Due to this fact, Hopf-Frobenius algebras provide two canonical notions of duality, and give us a "dual" Hopf algebra that is isomorphic to the usual dual Hopf algebra in a compact closed category. We use this isomorphism to construct a Hopf algebra isomorphic to the Drinfeld double, but has a much simpler presentation.

In Bob Coecke and Matthew Leifer: Proceedings 16th International Conference on Quantum Physics and Logic (QPL 2019), Chapman University, Orange, CA, USA., 10-14 June 2019, Electronic Proceedings in Theoretical Computer Science 318, pp. 150–180.
Published: 1st May 2020.

ArXived at: http://dx.doi.org/10.4204/EPTCS.318.10 bibtex PDF
References in reconstructed bibtex, XML and HTML format (approximated).
Comments and questions to: eptcs@eptcs.org
For website issues: webmaster@eptcs.org