Twisted generalized Weyl algebras, polynomial Cartan matrices and Serre-type relations

Jonas T. Hartwig
Arxiv ID: 0908.2054Last updated: 6/9/2020
Twisted generalized Weyl algebras (TGWAs) are defined as the quotient of a certain graded algebra by the maximal graded ideal I with trivial zero component, analogous to how Kac-Moody algebras can be defined. In this paper we introduce the class of locally finite TGWAs, and show that one can associate to such an algebra a polynomial Cartan matrix (a notion extending the usual generalized Cartan matrices appearing in Kac-Moody algebra theory) and that the corresponding generalized Serre relations hold in the TGWA. We also give an explicit construction of a family of locally finite TGWAs depending on a symmetric generalized Cartan matrix C and some scalars. The polynomial Cartan matrix of an algebra in this family may be regarded as a deformation of the original matrix C and gives rise to quantum Serre relations in the TGWA. We conjecture that these relations generate the graded ideal I for these algebras, and prove it in type A_2.

PaperStudio AI Chat

I'm your research assistant! Ask me anything about this paper.

Related papers

Commercial Disclosure
© 2023 Paper Studio™. All Rights Reserved.