Algebras with nonlinear commuting relations possessing Heisenberg duality property

Abstract:
Algebras with nonlinear commutation relations (say, Sklyanin and Askey-Wilson algebras) play an important role in theory of integrable systems and special functions. We describe a special class of these algebras possessing a remarkable "Heisenberg duality" (or "semi-classical") property. This property allows one to translate (almost literally) exactly solvable problems from classical to quantum picture. We show that many known algebras describing one- and many-dimensional integrable systems possess this property.

The joint work with Luc Vinet (Université de Montréal, Canada).