Weyl-Type and Witt-Type Algebras with Exponential Generators:Structure, Automorphisms, and Representation Theory

TL;DR

This paper introduces and studies a new class of non-commutative algebras generated by exponential and power functions, analyzing their structure, automorphisms, and representation theory, and connecting them to classical algebraic theories.

Contribution

It defines Weyl-type and Witt-type algebras with exponential generators and establishes their automorphism groups, Galois descent properties, and representation-theoretic classifications.

Findings

Automorphism group isomorphic to a semi-direct product involving ^{ imes} and ^{ imes}

Finite-dimensional simple modules do not exist for the Weyl-type algebra

Classification of irreducible weight modules and structure of category 

Abstract

This paper introduces and systematically studies a new class of non-commutative algebras -- Weyl-type and Witt-type algebras -- generated by differential operators with exponential and generalized power function coefficients. We define the expolynomial ring $R_{p,t,\mathcal{A}} = \mathbb{F}\bigl[ e^{\pm x^{p} e^{t}},\; e^{\mathcal{A} x},\; x^{\mathcal{A}} \bigr]$ associated to an additive subgroup $\mathcal{A} \subset \mathbb{F}$, and investigate its Ore extension $A_{p,t,\mathcal{A}} = R_{p,t,\mathcal{A}}[\partial; \delta]$ (Weyl-type) and its derivation algebra $\mathfrak{g}_{p,t,\mathcal{A}} = \operatorname{Der}_{\mathbb{F}}(R_{p,t,\mathcal{A}})$ (Witt-type). Our main results establish: (1) the automorphism group of $R_{p,t,\mathcal{A}}$ is isomorphic to $(\mathbb{F}^{\times})^{2r+1} \rtimes \operatorname{GL}(2r+1,\mathbb{Z})$; (2) a Galois descent theorem showing that fixed-point subalgebras under finite Galois actions recover the original Weyl-type algebra; (3) the non-existence of finite-dimensional simple modules for $A_{p,t,\mathcal{A}}$; (4) the Zariski density of isomorphism classes in moduli spaces as transcendental parameters vary; (5) the stability of simplicity under generic quantum deformation; and (6) a complete representation-theoretic framework including the classification of irreducible weight modules, the construction of Harish--Chandra modules with BGG-type resolutions, and the structure of category $\mathcal{O}$. These results unify and extend classical theories of Weyl algebras, Witt algebras, and generalized Weyl algebras, while opening new directions in deformation theory, non-commutative geometry, and the representation theory of infinite-dimensional algebras.