Symplectic differential reduction algebras and skew-affine generalized Weyl algebras

For a map φ\!:U(g)→ A of associative algebras, U(g) the universal enveloping algebra of a (complex) finite-dimensional reductive Lie algebra, the representation theory of A is intimately tied to the representation theory of the A-subquotient known as the reduction algebra for (A,g, φ). Herlemont and Ogievetsky studied differential reduction algebras for the general linear Lie algebra gl(n) as the algebra of h-deformed differential operators formed from realizations of gl(n) in the N-fold tensor product of the nth Weyl algebra. In this paper, we further the study of differential reduction algebras by presenting the symplectic differential reduction algebra D(sp(4)), by generators and relations, and showing its connections to Bavula's generalized Weyl algebras (GWAs). In doing so, we determine a new class of GWAs we call skew-affine GWAs, of which D(gl(2)) and D(sp(4)) are examples. We conjecture that the differential reduction algebra of the orthosymplectic Lie superalgebra osp(1|2n) is a twisted generalized Weyl algebra (TGWA) and that the relations for D(sp(2n)) yield solutions to the dynamical Yang-Baxter equation (DYBE).