Interacting Frobenius Algebras are Hopf

• Published in: Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science pp. 535 - 544
• Main Authors: Duncan, Ross, Dunne, Kevin
• Format: Conference Proceeding
• Language: English
• Published: New York, NY, USA ACM 05.07.2016
• Series: ACM Conferences
• Subjects: Software and its engineering > Software notations and tools > Formal language definitions; Theory of computation > Formal languages and automata theory; Theory of computation > Logic; Theory of computation > Models of computation > Computability; Theory of computation > Semantics and reasoning > Program semantics
• ISBN: 9781450343916; 1450343910
• DOI: 10.1145/2933575.2934550

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 [9] have shown that, given a suitable distribution 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 [9] 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, and recover the system of Bonchi et al as a subtheory in the prime power dimensional case. However the more general theory does not arise from a distributive law.