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

TL;DR

This paper systematically studies Weyl-type algebras with hyperbolic sine generators, establishing their structural properties, automorphism groups, and invariants, and explores their classification and growth characteristics in both associative and non-associative frameworks.

Contribution

It introduces a new class of Weyl-type algebras with hyperbolic sine generators, providing their structural analysis, automorphism groups, and invariants, and characterizes their classification and growth properties.

Findings

Center of associative algebra explicitly described

Algebras are Azumaya over their centers

Automorphism group characterized as a semidirect product

Abstract

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.