  Kitaev's lattice models are usually defined as representations of the
Drinfeld quantum double $D(H)=H\bowtie H^{*\text{op}} $, as an example of a
double cross product quantum group. We propose a new version based instead on
$M(H)=H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$ 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
$H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$. 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.
