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 A₏, ₓ, = (x^p (t) ), \; (x), \; x^{} in the associative setting and (x^p (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 (A₏, ₓ, ) = (x^p (t) ) for the associative one, proving that A₏, ₓ, 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 A₏, ₓ, 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.