  Matrix Product Operators (MPOs) are tensor networks representing operators acting on 1D systems. They model a wide variety of situations, including communication channels with memory effects, quantum cellular automata, mixed states in 1D quantum systems, or holographic boundary models associated to 2D quantum systems. A scenario where MPOs have proven particularly useful is to represent algebras of non-trivial symmetries. Concretely, the boundary of both symmetry protected and topologically ordered phases in 2D quantum systems exhibit symmetries in the form of MPOs.   In this paper, we develop a theory of MPOs as representations of algebraic structures. We establish a dictionary between algebra and MPO properties which allows to transfer results between both setups, covering the cases of pre-bialgebras, weak bialgebras, and weak Hopf algebras. We define the notion of pulling-through algebras, which abstracts the minimal requirements needed to define topologically ordered 2D tensor networks from MPO algebras. We show, as one of our main results, that any semisimple pivotal weak Hopf algebra is a pulling-trough algebra. We demonstrate the power of this framework by showing that they can be used to construct Kitaev's quantum double models for Hopf algebras solely from an MPO representation of the Hopf algebra, in the exact same way as MPO symmetries obtained from fusion categories can be used to construct Levin-Wen string-net models, and to explain all their topological features; it thus allows to describe both Kitaev and string-net models on the same formal footing. 
Matrix product operator algebras I: representations of weak Hopf   algebras and projected entangled pair states