Hopf-Frobenius Algebras and a Simpler Drinfeld Double
Abstract
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.
Keywords
Cite
@article{arxiv.1905.00797,
title = {Hopf-Frobenius Algebras and a Simpler Drinfeld Double},
author = {Joseph Collins and Ross Duncan},
journal= {arXiv preprint arXiv:1905.00797},
year = {2020}
}
Comments
In Proceedings QPL 2019, arXiv:2004.14750