Bicovariant Differential Geometry of the Quantum Group $SL_h(2)$

TL;DR

This paper develops a bicovariant differential geometry framework for the quantum group $SL_h(2)$, overcoming limitations present in the $SL_q(2)$ case, and derives its fundamental properties.

Contribution

It introduces a consistent bicovariant differential geometry for $SL_h(2)$, which was previously unavailable for $SL_q(2)$ under similar conditions.

Findings

Established a bicovariant differential calculus on $SL_h(2)$

Derived the properties and structure of the differential geometry

Showed the impossibility of such a structure on $SL_q(2)$ under the same conditions

Abstract

There are only two quantum group structures on the space of two by two unimodular matrices, these are the $SL_q(2)$ and the $SL_h(2)$ [9-13] quantum groups. One can not construct a differential geometry on $ SL_q(2)$, which at the same time is bicovariant, has three generators, and satisfies the Liebnitz rule. We show that such a differential geometry exists for the quantum group $SL_h(2)$ and derive all of its properties.