### Abstract

### Workshop

Workshop | Higher TQFT and categorical quantum mechanics |
---|---|

Country | Austria |

City | Vienna |

Period | 19/10/15 → 23/10/15 |

Internet address |

### Fingerprint

### Keywords

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

### Cite this

*Interacting Frobenius Algebras are Hopf*. Higher TQFT and categorical quantum mechanics, Vienna, Austria.

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**Interacting Frobenius Algebras are Hopf.** / Duncan, Ross.

Research output: Contribution to conference › Speech

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 -