We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C * -algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [26]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing a renormalization group flow. Furthermore it is shown that those states of the hierarchy associated with Kitaev's original quantum double models are related to each other by the condensation of topological charges. We conjecture that charge condensation is the physical mechanism underlying the hierarchy in general.
We show that indecomposable exact module categories over the category Rep H of representations of a finite-dimensional Hopf algebra H are classified by left comodule algebras, H -simple from the right and with trivial coinvariants, up to equivariant Morita equivalence. Specifically, any indecomposable exact module category is equivalent to the category of finite-dimensional modules over a left comodule algebra. This is an alternative approach to the results of Etingof and Ostrik. For this, we study the stabilizer introduced by Yan and Zhu and show that it coincides with the internal Hom. We also describe the correspondence of module categories between Rep H and Rep(H * ).
We construct explicit examples of weak Hopf algebras (actually face algebras in the sense of Hayashi [H]) via vacant double groupoids as explained in [AN]. To this end, we first study the Kac exact sequence for matched pairs of groupoids and show that it can be computed via group cohomology. Then we describe explicit examples of finite vacant double groupoids.
For a finite tensor category C and a Hopf monad T : C → C satisfying certain conditions we describe exact indecomposable left C T -module categories in terms of left C-module categories and some extra data. We also give a 2-categorical interpretation of the process of equivariantization of module categories.Recall that, for a given tensor category C, C-module categories, (lax) C-module functors and C-module natural transformations constitute a 2category, that we denote C Mod (respectively, C Mod lax ). We show that the assignment M → M(T ) extends to a 2-monad T on C Mod lax , and there is a 2-equivalence of 2-categories C EqMod ≃ ( C Mod lax ) T , Department of U. Hamburg for the kind hospitality.
We establish a bijective correspondence between gauge equivalence classes of dynamical twists in a finite-dimensional Hopf algebra H based on a finite abelian group A and equivalence classes of pairsA is a family of irreducible representations satisfying certain conditions. Our results generalize the results obtained by Etingof-Nikshych on the classification of dynamical twists in group algebras.
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