Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 001, 15 pages      arXiv:2403.15968      https://doi.org/10.3842/SIGMA.2025.001

Symplectic Differential Reduction Algebras and Generalized Weyl Algebras

Jonas T. Hartwig a and Dwight Anderson Williams II b
a) Department of Mathematics, Iowa State University, Ames, IA 50011, USA
b) Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA

Received March 27, 2024, in final form December 23, 2024; Published online January 01, 2025

Abstract
Given a map $\Xi\colon U(\mathfrak{g})\rightarrow A$ of associative algebras, with $U(\mathfrak{g})$ the universal enveloping algebra of a (complex) finite-dimensional reductive Lie algebra $\mathfrak{g}$, the restriction functor from $A$-modules to $U(\mathfrak{g})$-modules is intimately tied to the representation theory of an $A$-subquotient known as the reduction algebra with respect to $(A,\mathfrak{g},\Xi)$. Herlemont and Ogievetsky described differential reduction algebras for the general linear Lie algebra $\mathfrak{gl}(n)$ as algebras of deformed differential operators. Their map $\Xi$ is a realization of $\mathfrak{gl}(n)$ in the $N$-fold tensor product of the $n$-th Weyl algebra tensored with $U(\mathfrak{gl}(n))$. In this paper, we further the study of differential reduction algebras by finding a presentation in the case when $\mathfrak{g}$ is the symplectic Lie algebra of rank two and $\Xi$ is a canonical realization of $\mathfrak{g}$ inside the second Weyl algebra tensor the universal enveloping algebra of $\mathfrak{g}$, 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.

Key words: Mickelsson algebras; Zhelobenko algebras; skew affine; quantum deformation; differential operators.

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