PIRSA:17080011 Conference Quantum Foundations Quantum Information

DOI10.48660/17080011

Conference/School: Hopf Algebras in Kitaev's Quantum Double Models: Mathematical Connections from Gauge Theory to Topological Quantum Computing and Categorical Quantum Mechanics

Reutter, D. (2017). Frobenius algebras, Hopf algebras and 3-categories. Perimeter Institute. https://pirsa.org/17080011

Reutter, David. Frobenius algebras, Hopf algebras and 3-categories. Perimeter Institute, Aug. 03, 2017, https://pirsa.org/17080011 

@misc{ pirsa_PIRSA:17080011,
  doi = {10.48660/17080011},
  url = {https://pirsa.org/17080011},
  author = {Reutter, David},
  keywords = {Quantum Foundations, Quantum Information},
  language = {en},
  title = {Frobenius algebras, Hopf algebras and 3-categories},
  publisher = {Perimeter Institute},
  year = {2017},
  month = {aug},
  note = {PIRSA:17080011 see, \url{https://pirsa.org}}
}

Abstract

It is well known that commutative Frobenius algebras can be represented as topological surfaces, using the graphical calculus of dualizable objects in monoidal 2-categories. We build on related ideas to show that the interacting Frobenius algebras of Duncan and Dunne, which have a Hopf algebra structure, arise naturally in a similar way, by requiring a single 3-morphism in a 3-category to be invertible. We show that this gives a purely geometrical proof of Mueger's version of Tannakian reconstruction of Hopf algebras from fusion categories equipped with a fibre functor. We also relate our results to the theory of lattice code surgery.

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