  The generalized quantum double lattice realization of 2d topological orders
based on Hopf algebras is discussed in this work. Both left-module and
right-module constructions are investigated. The ribbon operators and the
classification of topological excitations based on the representations of the
quantum double of Hopf algebras are discussed. To generalize the model to a 2d
surface with boundaries and surface defects, we present a systematic
construction of the boundary Hamiltonian and domain wall Hamiltonian. The
algebraic data behind the gapped boundary and domain wall are comodule algebras
and bicomodule algebras. The topological excitations in the boundary and domain
wall are classified by bimodules over these algebras. The ribbon operator
realization of boundary-bulk duality is also discussed. Finally, via the Hopf
tensor network representation of the quantum many-body states, we solve the
ground state of the model in the presence of the boundary and domain wall.
