Interacting Frobenius Algebras are Hopf

Research output: Contribution to conferenceSpeech

Abstract

Commutative Frobenius algebras play an important role in both TQFT and CQM; in the first case they correspond to 2d TQFTs, while in the second they are non-degenerate observables. I will consider the case of “special” Frobenius algebras, and their associated group of phases. This gives rise to a free construction from the category of abelian groups to the PROP generated by this Frobenius algebra. Of course a theory with only one observable is not very interesting. I will consider how two such PROPs should be combined, and show that if the two algebras (i) jointly form a bialgebra and (ii) their units are “mutually real”; then they jointly form a Hopf algebra. This gives a free model of a pair of strongly complementary observables. I will also consider which unitary maps must exist in such models.

Fingerprint

Frobenius Algebra
Bialgebra
Commutative Algebra
Hopf Algebra
Abelian group
Algebra
Unit
Model

Keywords

  • higher topological quantum field theory
  • e Frobenius algebras
  • categorical quantum mechanics
  • abelian groups
  • Hopf algebra
  • bialgebra

Cite this

Duncan, R. (2015). Interacting Frobenius Algebras are Hopf. Higher TQFT and categorical quantum mechanics, Vienna, Austria.
Duncan, Ross. / Interacting Frobenius Algebras are Hopf. Higher TQFT and categorical quantum mechanics, Vienna, Austria.
@conference{c995475a928e4f8195912501af49003f,
title = "Interacting Frobenius Algebras are Hopf",
abstract = "Commutative Frobenius algebras play an important role in both TQFT and CQM; in the first case they correspond to 2d TQFTs, while in the second they are non-degenerate observables. I will consider the case of “special” Frobenius algebras, and their associated group of phases. This gives rise to a free construction from the category of abelian groups to the PROP generated by this Frobenius algebra. Of course a theory with only one observable is not very interesting. I will consider how two such PROPs should be combined, and show that if the two algebras (i) jointly form a bialgebra and (ii) their units are “mutually real”; then they jointly form a Hopf algebra. This gives a free model of a pair of strongly complementary observables. I will also consider which unitary maps must exist in such models.",
keywords = "higher topological quantum field theory, e Frobenius algebras, categorical quantum mechanics, abelian groups, Hopf algebra, bialgebra",
author = "Ross Duncan",
year = "2015",
language = "English",
note = "Higher TQFT and categorical quantum mechanics ; Conference date: 19-10-2015 Through 23-10-2015",
url = "http://www.esi.ac.at/activities/events/2015/higher-topological-quantum-field-theory-and-categorical-quantum-mechanics",

}

Duncan, R 2015, 'Interacting Frobenius Algebras are Hopf' Higher TQFT and categorical quantum mechanics, Vienna, Austria, 19/10/15 - 23/10/15, .

Interacting Frobenius Algebras are Hopf. / Duncan, Ross.

2015. Higher TQFT and categorical quantum mechanics, Vienna, Austria.

Research output: Contribution to conferenceSpeech

TY - CONF

T1 - Interacting Frobenius Algebras are Hopf

AU - Duncan, Ross

PY - 2015

Y1 - 2015

N2 - Commutative Frobenius algebras play an important role in both TQFT and CQM; in the first case they correspond to 2d TQFTs, while in the second they are non-degenerate observables. I will consider the case of “special” Frobenius algebras, and their associated group of phases. This gives rise to a free construction from the category of abelian groups to the PROP generated by this Frobenius algebra. Of course a theory with only one observable is not very interesting. I will consider how two such PROPs should be combined, and show that if the two algebras (i) jointly form a bialgebra and (ii) their units are “mutually real”; then they jointly form a Hopf algebra. This gives a free model of a pair of strongly complementary observables. I will also consider which unitary maps must exist in such models.

AB - Commutative Frobenius algebras play an important role in both TQFT and CQM; in the first case they correspond to 2d TQFTs, while in the second they are non-degenerate observables. I will consider the case of “special” Frobenius algebras, and their associated group of phases. This gives rise to a free construction from the category of abelian groups to the PROP generated by this Frobenius algebra. Of course a theory with only one observable is not very interesting. I will consider how two such PROPs should be combined, and show that if the two algebras (i) jointly form a bialgebra and (ii) their units are “mutually real”; then they jointly form a Hopf algebra. This gives a free model of a pair of strongly complementary observables. I will also consider which unitary maps must exist in such models.

KW - higher topological quantum field theory

KW - e Frobenius algebras

KW - categorical quantum mechanics

KW - abelian groups

KW - Hopf algebra

KW - bialgebra

UR - http://www.esi.ac.at/activities/events/2015/higher-topological-quantum-field-theory-and-categorical-quantum-mechanics

M3 - Speech

ER -

Duncan R. Interacting Frobenius Algebras are Hopf. 2015. Higher TQFT and categorical quantum mechanics, Vienna, Austria.