# Hopf-Frobenius algebras and a simpler Drinfeld double

Research output: Contribution to journalArticle

### Abstract

The ZX-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of $\dag$-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 the necessary and sufficient condition for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in FVect is a Hopf-Frobenius algebra. Hopf-Frobenius algebras provide a notion 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 that is defined on $H \otimes H$ rather than $H \otimes H^*$.
Original language English 26 Electronic Proceedings in Theoretical Computer Science 10 Jun 2019 E-pub ahead of print - 10 Jun 2019 16th International Conference on Quantum Physics and Logic 2019 - Chapman University, Orange, United StatesDuration: 10 Jun 2019 → 14 Jun 2019Conference number: 16th

Algebra

### Keywords

• quantum mathematics
• quantum computation
• quantum software

### Cite this

@article{51db01e295aa4eb2a88ca801ee38c3ca,
title = "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 $\dag$-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 the necessary and sufficient condition for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in FVect is a Hopf-Frobenius algebra. Hopf-Frobenius algebras provide a notion 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 that is defined on $H \otimes H$ rather than $H \otimes H^*$.",
keywords = "quantum mathematics, quantum computation, quantum software",
author = "Joseph Collins and Ross Duncan",
year = "2019",
month = "6",
day = "10",
language = "English",
journal = "Electronic Proceedings in Theoretical Computer Science",
issn = "2075-2180",

}

In: Electronic Proceedings in Theoretical Computer Science, 10.06.2019.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Hopf-Frobenius algebras and a simpler Drinfeld double

AU - Collins, Joseph

AU - Duncan, Ross

PY - 2019/6/10

Y1 - 2019/6/10

N2 - The ZX-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of $\dag$-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 the necessary and sufficient condition for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in FVect is a Hopf-Frobenius algebra. Hopf-Frobenius algebras provide a notion 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 that is defined on $H \otimes H$ rather than $H \otimes H^*$.

AB - The ZX-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of $\dag$-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 the necessary and sufficient condition for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in FVect is a Hopf-Frobenius algebra. Hopf-Frobenius algebras provide a notion 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 that is defined on $H \otimes H$ rather than $H \otimes H^*$.

KW - quantum mathematics

KW - quantum computation

KW - quantum software

UR - https://arxiv.org/abs/1905.00797

UR - http://forthcoming.eptcs.org/

M3 - Article

JO - Electronic Proceedings in Theoretical Computer Science

JF - Electronic Proceedings in Theoretical Computer Science

SN - 2075-2180

ER -