Algebras of Tensor Networks, Topological Order, and the Classification of Quantum Phases

Understanding the entanglement patterns within quantum many-body systems is a prominent question in contemporary theoretical physics. In particular, this pursuit has led to the adoption of tensor networks, the central topic of this presentation. We will begin by illustrating how well-known algebraic structures such as weak Hopf algebras – whose representations encompass fusion categories – give rise to one-dimensional tensor networks with exotic properties. We will demonstrate their usefulness in comprehending Kitaev's generalized quantum double models, describing renormalization fixed points in the one dimensional mixed state case, in the two-dimensional pure state case, and in the classification of quantum phases of matter.