Twisted Generalized Weyl algebras
by
Volodymyr Mazorchuk
Department of mathematics, Gothenburg University
Coauthors: Lyudmyla Turowska (Gothenburg University), Mariya Ponomarenko (Kyiv University)

The language of (rank 1) Generalized Weyl algebras is very convenient to study all possible generalizations and quantum deformations of the algebra A=U(sl(2)), however, in the situation of higher rank the behaviour of generalized Weyl algebras can be compared with the behaviour of the tensor product of several copies of A. The definition of twisted generalized Weyl algebras was an attempt to formulate an appropriate language for the study of algebras related to U(\mathfrakg), where \mathfrakg is an arbitrary simple Lie algebra. In this talk we are going to discuss the following:

1) Two examples of algebras, canonically associated with the Lie algebra sl(n), which can be realized as twisted generalized Weyl algebras (and not as generalized Weyl algebras). They are: Orthogonal Gelfand-Zetlin algebras (these are associative algebras canonically associated with the Gelfand-Zetlin formal construction of simple finite-dimensional sl(n)-modules) and Mickelsson step algebras, i.e certain algebras associated with the reduction process on the smaller subalgebra and acting on the space of primitive vectors in each sl(n)-module.

2) Shapovalov form on twisted generalized Weyl algebras and its application to the construction of simple weight modules. Shapovalov form can also be used to prove that some algebras, obtained by a more general procedure are in fact twisted generalized Weyl algebras. In many cases the set of simple modules constructed using the Shapovalov form turns to be the complete set of simple weight modules.

Date received: December 7, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafv-02.