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

TL;DR

This paper develops structural theorems for Weyl-type and Witt-type algebras over expolynomial rings, generalizing classical results and establishing conditions for simplicity, isomorphism, and derivation structures.

Contribution

It introduces a comprehensive framework for Weyl-type algebras over expolynomial rings, including new simplicity and isomorphism criteria, extending classical algebraic theories.

Findings

Scalar extensions preserve algebraic structure and simplicity.

Subalgebras associated with subgroups remain simple.

Derivation algebra is a semidirect product involving the Weyl algebra.

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.