Weyl-Type Algebras over Exponential-Polynomial Rings: Structure, Representations, and Deformations

This paper introduces and studies a class of Weyl-type algebras $A_{p,t,\text{cA}}=\text{Weyl}{e}^{\pm x^p{e}^{tx}},e^{\text{cA}x},x^{\text{cA}}$ constructed over exponential-polynomial rings, where $\text{FF}$ is a field of characteristic zero, $\text{cA}$ is a finitely generated additive subgroup of $\text{FF}$, and $p\in {\mathbb{N}}^n$, $t\in \text{FF}$. 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 ${\mathfrak{g}}_{p,t,\text{cA}}={\text{Der}}_{\mathrm{g}\mathrm{r}}(R_{p,t,\text{cA}})$ 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 $A_{p,t,\text{cA}}$ is classified in terms of the rank of $\text{cA}$, and a Gerstenhaber algebra structure on the Hochschild cohomology is described. Several open problems concerning representation classification and geometric realization are proposed.