Exactly solvable models for 2+1D topological phases derived from crossed modules of semisimple Hopf algebras

We define an exactly solvable model for 2+1D topological phases of matter on a triangulated surface derived from a crossed module of semisimple finite-dimensional Hopf algebras, the Hopf-algebraic higher Kitaev model. This model generalizes both the Kitaev quantum double model for a semisimple Hopf algebra and the full higher Kitaev model derived from a 2-group, and can hence be interpreted as a Hopfalgebraic discrete higher gauge theory. We construct a family of crossed modules of semisimple Hopf algebras (FC(X)⊗ CE ∂ − → FC(Y ) o CG, .) that depends on four finite groups E,G,X and Y . We calculate the ground-state spaces of the resulting model when G = E = {1} and when Y = {1}, both properly Hopf-algebraic constructions; prove that they are canonically independent of the triangulations; and find a 2+1D TQFT whose state spaces on surfaces give the ground-state spaces. These TQFTs are particular cases of Quinn’s finite total homotopy TQFT and hence the state spaces assigned to surfaces are free vector spaces on sets of homotopy classes of maps from a surface to homotopy finite spaces, in this case obtained as classifying spaces of finite groupoids and finite crossed modules of groupoids. We leave it as an open problem whether the ground-state space of the Hopfalgebraic higher Kitaev model on a triangulated surface is independent of the triangulation for general crossed modules of semisimple Hopf algebras, whether a TQFT always exists whose state space on a surface gives the ground-state space of the model, and whether the ground-state space of the model obtained from E,G,X, Y can always be given a homotopical explanation.