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Non-commutative algebras, representation theory and special functions
May 23-June 10, 2022

Much activity has recently been deployed in the study of polynomial algebras. Some time ago, in studies of the equations of the quantum inverse scattering method, Sklyanin introduced a quadratic equation that bears his name. Roughly at the same time, the study of Artin-Shelter algebras as natural generalizations of commutative polynomial rings is initiated and actively pursued since. The concept of potential and of Calabi-Yau (CY) algebras have emerged with the latter arising in CY geometry (and string theory). This is being used to develop the theory of non-commutative surfaces with the algebras somehow playing the role of the ”coordinate” ring. A central problem is the characterization and classification of these algebras. As it turns out, the three- dimensional Sklyanin algebras found themselves at the heart of these questions with the study of their representations providing furthermore a determination of the vacua of marginal deformations of N=4 supersymmetric Yang-Mills theories.

In the eighties also, the quadratic algebras that encode the bispectral properties of the polynomials of the Askey scheme have been obtained and given the names of the corresponding polynomials. These algebras have since become ubiquitous: they have been seen to be the symmetry algebras of superintegrable models, the centralizers of the diagonal action in three-fold tensor products of rank 1 Lie, super or quantum algebras or identified as co-ideal subalgebras. Their interpretation as centralizers makes connections with classical invariant theory and algebras from the realm of Schur–Weyl dualities (Hecke and related algebras). These algebras also have emerged in algebraic combinatorics via Leonard pairs in the classification of P-polynomial and Q-polynomial association schemes. The Askey-Wilson algebra which is the emblem of these algebras is related to the (C1∨, C1) DAHA and the Kauffman bracket Skein algebra for the sphere with four punctures. Skein algebras are naturally connected to topological quantum field theories that will be one of the topics of the concentration period on Quantum symmetries. Higher dimensional extensions are being worked on. The notion of algebraic Heun operators and the algebras they generate was derived in this context. This found its way in the description of Ruijsenaars-van Diejen Hamiltonians and in reflection algebras.

Eventually a connection was established between the two large areas of exploration sketched above when the Heun operators were seen to lead to Sklyanin algebras and the Askey-Wilson operators identified within those algebras. In addition, a generalization of the Askey-Wilson algebra, called the Sklyanin-Painlevé algebra was recently introduced to describe the quantization of Painlevé monodromy manifolds. The algebraic interpretation of special functions is thus seen to bring interesting links. The purpose of this concentration period is to foster advances in the active research areas partially evoked here and to bring together the various communities that are looking at these interconnected questions from different viewpoints.