Jonas Torbjorn Hartwig
Universidade de Gotheborg, Sweden / IME-USP, Brazil
Twisted generalized Weyl algebras and Serre-type relations.
Abstract: The twisted generalized Weyl algebras (TGWAs) constitute a large class of algebras which include many interesting examples such as quantized Weyl algebras, Mickelsson step algebras, Gelfand-Zetlin algebras etc. We define the notion of a locally finite TGWA and show that one can associate a generalized Cartan matrix to any such algebra so that corresponding Serre-type relations hold. All examples studied so far, including the above mentioned, are "trivial" in the sense that they are of type (A_1)^n. We give a new construction which associates a locally finite TGWA to any symmetric generalized Cartan matrix. We conjecture that the corresponding quantum Serre relations give, together with other relations, a complete presentation of these algebras, and prove it in type A_2. This gives a new link between TGWAs and (quantized) enveloping algebras of Lie algebras.