Weyl-Type Algebras over Exponential-Polynomial Rings: Structure, Representations, and Deformations (2512.06479v1)

Abstract: This paper introduces and studies a class of Weyl-type algebras (A_{p,t,\cA} = \Weyl{e^{\pm x^{p} e^{t x}},\; e^{\cA x},\; x^{\cA}}) constructed over exponential-polynomial rings, where (\FF) is a field of characteristic zero, (\cA) is a finitely generated additive subgroup of (\FF), and (p \in \mathbb{N}^n), (t \in \FF). We investigate their structural properties, proving simplicity, establishing faithful infinite-dimensional irreducible representations, and demonstrating the failure of the Noetherian property. A natural filtration by exponential order is introduced, with the associated graded algebra shown to be commutative. We also examine the corresponding Witt-type Lie algebra (\mathfrak{g}{p,t,\cA} = \Der{\mathrm{gr}}(R_{p,t,\cA})) and prove the vanishing of its second cohomology group with adjoint coefficients, implying rigidity under formal deformations. Furthermore, we construct explicit deformation quantizations of the underlying exponential-polynomial rings, compute Hochschild and cyclic homology groups, and relate them to the topology of the parameter space. The deformation rigidity of (A_{p,t,\cA}) is classified in terms of the rank of (\cA), and a Gerstenhaber algebra structure on the Hochschild cohomology is described. Several open problems concerning representation classification and geometric realization are proposed.