Inspired by the quantum McKay correspondence, we consider the classical ADE Lie theory as a quantum theory over sl2. We introduce anti-symmetric characters for representations of quantum groups and investigate the Fourier duality to study the spectral theory. In the ADE Lie theory, there is a correspondence between the eigenvalues of the Coxeter element and the eigenvalues of the adjacency matrix. We formalize related notions and prove such a correspondence for representations of Verlinde algebras of quantum groups: this includes generalized Dynkin diagrams over any simple Lie algebra g at any level k. This answers a recent comment of Terry Gannon on an old question posed by Victor Kac in 1994.

The mathematical picture language project that we began in 2016 has already yielded interesting results. We also point out areas of mathematics and physics where we hope that it will prove useful in the future.

We give a transparent algebraic formulation of our pictorial approach to the reflection positivity (RP), that we introduced in a previous paper. We apply this quantization to the 2+1 Levin–Wen model to obtain 1+1 anyonic/quantum spin chain theory on the boundary, possibly entangled in the bulk. The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin–Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.

The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states on the parafermion algebra, including finding a distinction that depends on whether the order of the parafermion algebra is square free. We characterize the extremal nature of product states (an inverse de Finetti theorem).

We explore the tunneling behavior of a quantum particle on a finite graph in the presence of an asymptotically large potential with two or three potential wells. The behavior of the particle is primarily governed by the local spectral symmetry of the graph around the wells. In the case of two wells the behavior is stable in the sense that it can be predicted from a sufficiently large neighborhood of the wells. However in the case of three wells we are able to exhibit examples where the tunneling behavior can be changed significantly by perturbing the graph arbitrarily far from the wells.