Weyl-Type Algebras over Exponential-Polynomial Rings: Structure, Representations, and Deformations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908696903155712 |
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| author | Rashid, Mohammad H. M. |
| author_facet | Rashid, Mohammad H. M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06479 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weyl-Type Algebras over Exponential-Polynomial Rings: Structure, Representations, and Deformations Rashid, Mohammad H. M. Rings and Algebras 16S32, 17B40, 16E40, 17B56 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. |
| title | Weyl-Type Algebras over Exponential-Polynomial Rings: Structure, Representations, and Deformations |
| topic | Rings and Algebras 16S32, 17B40, 16E40, 17B56 |
| url | https://arxiv.org/abs/2512.06479 |