Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Generalised Kitaev models from Hopf monoids: topological invariance, examples and relation to Kuperberg invariants

Quantum double models were introduced by Kitaev to obtain a realistic model for a topological quantum computer. They are based on two ingredients: a directed ribbon graph and a finite-dimensional semisimple Hopf algebra. The ground state of these models is a topological invariant of a surface, i.e. only depends on the homeomorphism class of the oriented surface but not the ribbon graph. Meusburger and Voß generalised part of the construction from Hopf algebras to pivotal Hopfmonoids in symmetric monoidal categories and described associated mapping class group actions. We explain the construction of the ground state for involutive Hopf monoids and why it is topological invariant. We explicitly describe this construction for Hopf monoids in Set, Cat and SSet. After that, we relate the mapping class group actions in the model to Kuperberg invariants of 3-manifolds. The latter invariants were introduced by Kuperberg based on involutive Hopf algebras and generalised to involutive Hopf monoids by Kashaev and Virelizier.