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Interacting Frobenius algebras are Hopf

Duncan, Ross and Dunne, Kevin (2016) Interacting Frobenius algebras are Hopf. In: Proceedings of the 31st annual ACM/IEEE Symposium on Logic in Computer Science (LICS). ACM, New York. (In Press)

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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.