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 $A_{p,t,\cA} = \Weyl{\sinh(\pm x^{p} \sinh(t)),\; \sinh(\cA x),\; x^{\cA}}$ in the associative setting and $\Nass{\sinh(\pm x^{p} \sinh(t)),\; \sinh(\cA x),\; x^{\cA}}$ 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 $Z(A_{p,t,\cA}) = \FF[\sinh(\pm x^{p} \sinh(t))]$ for the associative one, proving that $A_{p,t,\cA}$ is an Azumaya algebra over its center and represents a nontrivial class in the Brauer group $\Br(\FF(y))$. 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 $A_{p,t,\cA}$ 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.