A non-semisimple Kitaev lattice model

TL;DR

This paper generalizes Kitaev's topological quantum code model from semisimple to arbitrary finite-dimensional Hopf algebras, expanding its mathematical framework and invariance properties.

Contribution

It introduces a new non-semisimple Kitaev lattice model based on Hopf algebras with pairs in involution, extending the topological invariance of the protected space.

Findings

The model's protected space depends only on the genus of the graph.

Constructs a Yetter-Drinfeld module as a topological invariant.

Provides examples using group algebras and Nichols algebras.

Abstract

The construction of the topologically protected code space of Kitaev's model for fault-tolerant quantum computation is extended from complex semisimple to arbitrary finite-dimensional Hopf algebras admitting pairs in involution. One input of the model are ribbon graphs, that is, the combinatorial data of cellular decompositions of oriented closed surfaces. The other input are certain Hopf bimodules that are closely related to the coefficients in Hopf-cyclic homology. As in previous generalisations of the Kitaev model, a Yetter-Drinfeld module is constructed and shown to be a topological invariant of the surface with boundary that is obtained by "thickening" the graph. The generalisation of the protected space is defined using bitensor products of modules-comodules. Provided that the Hopf bimodule coefficients correspond to pairs in involution, this is shown to depend only on the genus of the graph. As examples, group algebras of finite groups and bosonisations of Nichols algebras are considered.