New algebraic structures in quantum integrability: towards 3D – NASQI3D

Exactly solvable (or integrable) models have played a major role in our understanding of the scaling limit of statistical mechanical models, in particular in the study of phase transitions. They also are linked with rich algebraic structures such as quantum groups and the Yang-Baxter equation, as well as with knot theory and topology. However, despite some interesting progress, most applications to this date are restricted to the case of two-dimensional models, hinting at a certain "rigidity" of the underlying structures.

Motivated by recent works, this project aims at revisiting integrability beyond the scope of traditional algebraic structures, with a focus on higher-dimensional models. A central role will be played by the Onsager algebra and by braided tensor categories. Such structures have indeed proved to have deep relations with integrable models, and may allow to overcome some of the limitations imposed by the usual framework of quantum groups, in particular on the spatial dimensions of the considered models. Among the project's applications, we will try constructing three-dimensional versions of exactly solvable models and of Onsager-type algebra. Other applications include studying conformal invariance on the lattice, as well as new algebraic approaches to some Markov chains.