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

TL;DR

This paper introduces a new class of Weyl-type algebras over exponential-polynomial rings, analyzing their structure, representations, and deformation properties, including rigidity and homology, with implications for algebraic and geometric applications.

Contribution

It constructs and studies Weyl-type algebras over exponential-polynomial rings, proving simplicity, representation existence, and deformation rigidity, and explores their Hochschild and cyclic homology.

Findings

Proved the simplicity and existence of faithful irreducible representations.

Established the deformation rigidity of the algebras and computed their Hochschild and cyclic homology.

Demonstrated the associated graded algebra is commutative and related homology to parameter space topology.

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.