Abstract
Kitaev’s lattice models are usually defined as representations of the Drinfeld quantum double D(H) = H ⋈ H*op, as an example of a double cross product quantum group. We propose a new version based instead on M(H) = Hcop ⧑ H as an example of Majid’s bicrossproduct quantum group, related by semidualisation or ‘quantum Born reciprocity’ to D(H). Given a finite-dimensional Hopf algebra H, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop ⧑ H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.
Highlights
The Kitaev quantum double models can be understood to describe the moduli space of flat connections on a 2d surface with defect excitations
Given a finite-dimensional Hopf algebra H, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop H
From the point of view of quantum gravity, they are of strong interest as they are directly related to certain 3d TQFTs defined in terms of Hopf algebras
Summary
We first recall the set up for the standard Kitaev model based on the double D(H) but in a manner general enough to apply to M (H). The standard Kitaev model based on the quantum double has D = D(H) = H H∗op acting on A = H, where H is a finite dimensional Hopf algebra (with further properties). We still define T−, L− in the same way for the other orientations but we must be careful in the manner stated to use the coproduct of D when combining these to build Ah(v, p) and Ba(v, p) This much of the setup again makes sense for any bicrossproduct quantum group H1 H2 as in [37] acting on H1∗.
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