Interacting Frobenius algebras are Hopf

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

11 Citations (Scopus)
123 Downloads (Pure)

Abstract

Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the underlying object---the so-called phase group---and investigate the effects of finite dimensionality of the underlying model. We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law.
Original languageEnglish
Title of host publicationProceedings of the 31st annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
Place of PublicationNew York, NY
Number of pages10
DOIs
Publication statusPublished - 17 Dec 2018
Event31st Annual ACM/IEEE Symposium on LOGIC IN COMPUTER SCIENCE (LICS2016): LICS2016 - New York City, United States
Duration: 5 Jul 20168 Jul 2016

Conference

Conference31st Annual ACM/IEEE Symposium on LOGIC IN COMPUTER SCIENCE (LICS2016)
CountryUnited States
CityNew York City
Period5/07/168/07/16

Fingerprint

Frobenius Algebra
Hopf Algebra
Distributive law
Algebra
Concurrent Programming
Quantum Computing
Control Theory
Dimensionality
Computer Science
Generalise
Computer programming
Control theory
Interaction
Computer science
Model

Keywords

  • Frobenius algebras
  • Hopf algebra
  • phase group
  • finite dimensionality

Cite this

Duncan, R., & Dunne, K. (2018). Interacting Frobenius algebras are Hopf. In Proceedings of the 31st annual ACM/IEEE Symposium on Logic in Computer Science (LICS) New York, NY. https://doi.org/10.1145/2933575.2934550
Duncan, Ross ; Dunne, Kevin. / Interacting Frobenius algebras are Hopf. Proceedings of the 31st annual ACM/IEEE Symposium on Logic in Computer Science (LICS). New York, NY, 2018.
@inproceedings{43ab7a648dd0436f8bf1f1e69060309f,
title = "Interacting Frobenius algebras are Hopf",
abstract = "Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the underlying object---the so-called phase group---and investigate the effects of finite dimensionality of the underlying model. We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law.",
keywords = "Frobenius algebras, Hopf algebra, phase group, finite dimensionality",
author = "Ross Duncan and Kevin Dunne",
year = "2018",
month = "12",
day = "17",
doi = "10.1145/2933575.2934550",
language = "English",
isbn = "9781450343916",
booktitle = "Proceedings of the 31st annual ACM/IEEE Symposium on Logic in Computer Science (LICS)",

}

Duncan, R & Dunne, K 2018, Interacting Frobenius algebras are Hopf. in Proceedings of the 31st annual ACM/IEEE Symposium on Logic in Computer Science (LICS). New York, NY, 31st Annual ACM/IEEE Symposium on LOGIC IN COMPUTER SCIENCE (LICS2016), New York City, United States, 5/07/16. https://doi.org/10.1145/2933575.2934550

Interacting Frobenius algebras are Hopf. / Duncan, Ross; Dunne, Kevin.

Proceedings of the 31st annual ACM/IEEE Symposium on Logic in Computer Science (LICS). New York, NY, 2018.

Research output: Chapter in Book/Report/Conference proceedingConference contribution book

TY - GEN

T1 - Interacting Frobenius algebras are Hopf

AU - Duncan, Ross

AU - Dunne, Kevin

PY - 2018/12/17

Y1 - 2018/12/17

N2 - Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the underlying object---the so-called phase group---and investigate the effects of finite dimensionality of the underlying model. We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law.

AB - Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the underlying object---the so-called phase group---and investigate the effects of finite dimensionality of the underlying model. We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law.

KW - Frobenius algebras

KW - Hopf algebra

KW - phase group

KW - finite dimensionality

UR - http://lics.rwth-aachen.de/lics16/

UR - http://arxiv.org/abs/1601.04964

U2 - 10.1145/2933575.2934550

DO - 10.1145/2933575.2934550

M3 - Conference contribution book

SN - 9781450343916

BT - Proceedings of the 31st annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

CY - New York, NY

ER -

Duncan R, Dunne K. Interacting Frobenius algebras are Hopf. In Proceedings of the 31st annual ACM/IEEE Symposium on Logic in Computer Science (LICS). New York, NY. 2018 https://doi.org/10.1145/2933575.2934550