$h$-Deformation as a Contraction of $q$-Deformation

TL;DR

This paper demonstrates that $h$-deformation structures can be derived as a singular limit of $q$-deformation through a similarity transformation, connecting different quantum group deformations.

Contribution

It introduces a method to obtain $h$-deformation from $q$-deformation via a singular limit, unifying their structures and extensions.

Findings

Derived $ ext{GL}_h(2)$ from $q$-deformation

Obtained differential structure for $h$-deformation

Extended to inhomogeneous and $ ext{U}_h( ext{sl}(2))$ structures

Abstract

We show that $h$-deformation can be obtained, by a singular limit of a similarity transformation, from $q$-deformation; to be specefic, we obtain $\GL_h(2)$, its differential structure, its inhomogenous extension, and $\Uh{\sl(2)}$ from their $q$-deformed counterparts.