Symplectic Differential Reduction Algebras and Generalized Weyl Algebras
Jonas T. Hartwig, Dwight Anderson Williams
TL;DR
The paper addresses the problem of constructing and presenting a differential reduction algebra for the symplectic Lie algebra $\mathfrak{sp}(4)$ via a canonical oscillator realization inside the tensor product algebra $A_2\otimes U(\mathfrak{sp}(4))$. It develops a localization framework to dynamical scalars and employs the extremal projector to obtain the differential reduction algebra $D(\mathfrak{sp}(4))$, proving it has a finite presentation with a well-organized $R$-basis. Furthermore, it demonstrates that $D(\mathfrak{sp}(4))$ is a generalized Weyl algebra and, in particular, a rank-two skew-affine GWA, thereby enabling transfer of GWA techniques to study its representations. This work links reduction-algebra constructions with GWA theory, offering a pathway to classifying irreducible weight modules and contributing to noncommutative algebra and representation theory in the symplectic setting.
Abstract
Given a map $Ξ\colon U(\mathfrak{g})\rightarrow A$ 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 $(A,\mathfrak{g},Ξ)$. Herlemont and Ogievetsky described differential reduction algebras for the general linear Lie algebra $\mathfrak{gl}(n)$ as algebras of deformed differential operators. Their map $Ξ$ 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 $Ξ$ 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.