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

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.