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Interacting Frobenius Algebras are Hopf

Duncan, Ross (2015) Interacting Frobenius Algebras are Hopf. In: Higher TQFT and categorical quantum mechanics, 2015-10-19 - 2015-10-23, Erwin Schrödinger Institute.

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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.