Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model

We provide a systematic treatment of boundaries based on subgroups $K\subseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $\Xi\subseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra dependent on a choice of transversal $R$. We provide decomposition formulae for irreducible representations of $D(G)$ pulled back to $\Xi$. We also provide explicitly the monoidal equivalence of the category of $\Xi$-modules and the category of $G$-graded $K$-bimodules and use this to prove that different choices of $R$ are related by Drinfeld cochain twists. Examples include $S_{n-1}\subset S_n$ and an example related to the octonions where $\Xi$ is also a Hopf quasigroup. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=\{e\}$ vertically and show how these could be used in a quantum computer using the technique of lattice surgery.