Symplectic Differential Reduction Algebras and Generalized Weyl Algebras

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2025
• Main Authors: Hartwig, Jonas T., Williams II, Dwight Anderson
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2025
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/212890
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Symplectic Differential Reduction Algebras and Generalized Weyl Algebras. Jonas T. Hartwig and Dwight Anderson Williams II. SIGMA 21 (2025), 001, 15 pages

Given a map Ξ: () → of associative algebras, with () the universal enveloping algebra of a (complex) finite-dimensional reductive Lie algebra g, the restriction functor from -modules to ()-modules is intimately tied to the representation theory of an -subquotient known as the reduction algebra with respect to (, , Ξ). Herlemont and Ogievetsky described differential reduction algebras for the general linear Lie algebra () as algebras of deformed differential operators. Their map Ξ is a realization of () in the -fold tensor product of the -th Weyl algebra tensored with (()). In this paper, we further the study of differential reduction algebras by finding a presentation in the case when is the symplectic Lie algebra of rank two, and Ξ is a canonical realization of inside the second Weyl algebra tensor the universal enveloping algebra of g, suitably localized. Furthermore, we prove that this differential reduction algebra is a generalized Weyl algebra (GWA), in the sense of Bavula, of a new type we term skew-affine. It is believed that symplectic differential reduction algebras are all skew-affine GWAs; then their irreducible weight modules could be obtained from standard GWA techniques.