Structure and Invariants of Weyl-Type Algebras with Hyperbolic Sine Generators

This paper introduces and systematically studies a class of Weyl-type algebras enriched with hyperbolic sine and power generators over a field of characteristic zero, defined as Ap,t, = (± xᵖ (t)),\; ( x),\; x^ in the associative setting and (± xᵖ (t)),\; ( x),\; x^ in a non-associative framework. We establish fundamental structural properties, including the triviality of the center for the non-associative version and the explicit description Z(Ap,t,) = [(± xᵖ (t))] for the associative one, proving that Ap,t, is an Azumaya algebra over its center and represents a nontrivial class in the Brauer group ((y)). Furthermore, we compute the Gelfand--Kirillov dimension for relevant examples and demonstrate its key properties, such as additivity under tensor products and the growth dichotomy. We completely characterize the automorphism group of Ap,t, as a semidirect product of a torus with a discrete group, and provide a sharp isomorphism criterion showing that the parameter t is a complete invariant in the family. The paper concludes with two open problems concerning the GK dimension of non-associative hyperbolic sine algebras and the classification of their deformations, pointing toward future research directions in non-associative growth theory and deformation rigidity.