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Quantum Hankel algebras, clusters, and canonical bases
Algebra & Discrete Mathematics| Speaker: | Arkady Berenstein, University of Oregon |
| Location: | 2112 MSB |
| Start time: | Fri, Dec 12 2008, 2:10PM |
Description
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.