Structural and Classification Theorems for Weyl-Type Algebras over Expolynomial Rings
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908707007234048 |
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| author | Rashid, Mohammad H. M |
| author_facet | Rashid, Mohammad H. M |
| contents | This paper introduces and systematically studies Weyl-type, Witt-type, and non-associative algebras defined over expolynomial rings -- commutative rings generated by exponential functions $e^{αx}$, exponentials of exponentials $e^{\pm x^p e^{t}}$, and power functions $x^α$ for $α$ in an additive subgroup $\cA$ of a characteristic zero field $\FF$. We establish several fundamental structural results: scalar extensions preserve both the algebraic structure and simplicity; intermediate subalgebras associated with subgroups $\ZZ \subseteq \cB \subseteq \cA$ remain simple; the algebra of graded derivations is isomorphic to a semidirect product $\Weyl{e^{\pm x^p e^{t}},\; e^{\cA x},\; x^{\cA}} \rtimes \FF^n$; tensor products over disjoint variable sets decompose naturally into larger algebras; and a complete isomorphism criterion is given, showing that isomorphism depends precisely on the orbit of the parameter $p$ under the automorphism group of $\cA$ and the equality of the deformation parameter $t$. These theorems generalize classical results on Weyl and Witt algebras, provide new families of simple algebras, and offer a foundation for further research in deformation theory, representation theory, and cohomology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06497 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structural and Classification Theorems for Weyl-Type Algebras over Expolynomial Rings Rashid, Mohammad H. M Rings and Algebras Quantum Algebra Representation Theory 13N10, 17B66, 17B35, 16S32 This paper introduces and systematically studies Weyl-type, Witt-type, and non-associative algebras defined over expolynomial rings -- commutative rings generated by exponential functions $e^{αx}$, exponentials of exponentials $e^{\pm x^p e^{t}}$, and power functions $x^α$ for $α$ in an additive subgroup $\cA$ of a characteristic zero field $\FF$. We establish several fundamental structural results: scalar extensions preserve both the algebraic structure and simplicity; intermediate subalgebras associated with subgroups $\ZZ \subseteq \cB \subseteq \cA$ remain simple; the algebra of graded derivations is isomorphic to a semidirect product $\Weyl{e^{\pm x^p e^{t}},\; e^{\cA x},\; x^{\cA}} \rtimes \FF^n$; tensor products over disjoint variable sets decompose naturally into larger algebras; and a complete isomorphism criterion is given, showing that isomorphism depends precisely on the orbit of the parameter $p$ under the automorphism group of $\cA$ and the equality of the deformation parameter $t$. These theorems generalize classical results on Weyl and Witt algebras, provide new families of simple algebras, and offer a foundation for further research in deformation theory, representation theory, and cohomology. |
| title | Structural and Classification Theorems for Weyl-Type Algebras over Expolynomial Rings |
| topic | Rings and Algebras Quantum Algebra Representation Theory 13N10, 17B66, 17B35, 16S32 |
| url | https://arxiv.org/abs/2512.06497 |