The Kitaev model and aspects of semisimple Hopf algebras via the graphical calculus



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Abstract

The quantum double models are parametrized by a finite-dimensional semisimple Hopf algebra (over $\mathbb{C}$). I will introduce the graphical calculus of these Hopf algebras and sketch how it is equivalent to the calculus of two interacting symmetric Frobenius algebras. Since symmetric Frobenius algebras are extended 2D TQFTs, this suggests that there is a canonical way to 'lift' a compatible pair of 2D TQFTs to a 3D TQFT. Time permitting, I will also showcase how to rederive graphically parts of the Larson-Sweedler theorem, giving various equivalent characterizations of semisimplicity, thereby generalizing these results to arbitrary Hopf monoids in traced symmetric monoidal categories.