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The goal of my the talk (which is based on joint work with David Kazhdan) is to introduce a new family of flat deformations of the symmetric algebras of sl_2-modules, which we refer to as quantum Hankel algebras. Remarkably, all quantum Hankel algebras and their quadratic duals admit some kind of canonical basis, which, hopefully, will help to split symmetric powers of sl_2-modules into the irreducibles.

It turns out that each so constructed basis has a quantum cluster structure. Surprisingly, the initial quantum cluster is related to the so called Q-systems (of type A) studied by Kedem and Di Francesco. Moreover, the members of the initial cluster are q-deformations of Hankel determinants of all sizes, which relates quantum Hankel algebras with the cluster-like approach to orthogonal polynomials recently developed by Gekhtman, Shapiro and Vainshtein.