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

TL;DR

This paper introduces locally finite twisted generalized Weyl algebras, associates polynomial Cartan matrices to them, and establishes Serre relations, including a deformation leading to quantum Serre relations, with proofs in specific cases.

Contribution

It extends the theory of TGWAs by defining polynomial Cartan matrices and deriving Serre relations, including deformations related to quantum groups.

Findings

Associated polynomial Cartan matrices to locally finite TGWAs.

Established Serre relations in the context of TGWAs.

Proved the conjecture in type A_2 case.

Abstract

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.