**RESEARCH IN PAIRS**

**Reduction algebras and quantum groups****October 10 - 21, 2016**

Mickelsson algebras (or reduction algebras) were introduced for the study of Harish-Chandra modules of reductive groups. The Mickelsson algebra, related to a real reductive group G, acts on the space of highest weight vectors of its maximal compact subgroup, and each irreducible Harish-Chandra module of the initial reductive group is uniquely characterized by this action. The reduction algebra can be realized as a sub-quotient of the universal enveloping algebra of the Lie algebra of G. A similar construction can be given for any associative algebra A, which contains a universal enveloping algebra (or its q-analog) of a contragredient Lie algebra g.
One of our goals is to establish a close connection between Mickelsson algebras and quantum groups. This includes: from one side a Yang-Baxter type description of various reduction algebras and the use of quantum group techniques for their algebraic description; from another side the application of the methods developed in our investigations of Mickelsson algebras to dynamical quantum groups and related integrable models. Another goal is a study of the representation theory of Mickelsson algebras with applications to classical problems in the theory of Lie groups and in mathematical physics. |