Twisted Generalized Weyl Algebras, Polynomial Cartan Matrices and Serre-Type Relations

Abstract

A twisted generalized Weyl algebra (TGWA) is 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 article 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.

Key Words:

• Generalized Weyl algebra
• Quantum group
• Serre relation

2000 Mathematics Subject Classification:

• 16S32 (17B37)

ACKNOWLEDGMENTS

The author was supported by the Netherlands Organization for Scientific Research (NWO) in the VIDI-project “Symmetry and modularity in exactly solvable models.” The author would like to thank L. Turowska for commenting on an early version of this article, and J. Palmkvist and J. Öinert for interesting discussions.

Notes

Communicated by T. Lenagan.