Structural and Classification Theorems for Weyl-Type Algebras over Expolynomial Rings

Abstract

This paper introduces and systematically studies Weyl-type, Witt-type, and non-associative algebras defined over expolynomial rings -- commutative rings generated by exponential functions $e^{\alpha x}$, exponentials of exponentials $e^{\pm x^p e^{t}}$, and power functions $x^{\alpha}$ for $\alpha$ 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.