### Abstract

Original language | English |
---|---|

Number of pages | 26 |

Journal | Electronic Proceedings in Theoretical Computer Science |

Early online date | 10 Jun 2019 |

Publication status | E-pub ahead of print - 10 Jun 2019 |

Event | 16th International Conference on Quantum Physics and Logic 2019 - Chapman University, Orange, United States Duration: 10 Jun 2019 → 14 Jun 2019 Conference number: 16th |

### Fingerprint

### Keywords

- quantum mathematics
- quantum computation
- quantum software

### Cite this

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**Hopf-Frobenius algebras and a simpler Drinfeld double.** / Collins, Joseph; Duncan, Ross.

Research output: Contribution to journal › Article

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 -