The quantum Cartan algebra associated to a bicovariant differential calculus

TL;DR

This paper introduces a quantum Cartan algebra linked to bicovariant differential calculus on quantum groups, exemplified by the $SU_q(2)$ case, expanding the algebraic tools for quantum geometry.

Contribution

It constructs a Cartan Hopf algebra with representations via Cartan calculus operators for bicovariant differential calculi on quantum groups, including a detailed example on $SU_q(2)$.

Findings

Defined a quantum Cartan algebra for bicovariant calculi

Established representations using Cartan calculus operators

Provided an explicit example for $SU_q(2)$

Abstract

We associate to any (suitable) bicovariant differential calculus on a quantum group a Cartan Hopf algebra which has a left, respectively right, representation in terms of left, respectively right, Cartan calculus operators. The example of the Hopf algebra associated to the $4D_+$ differential calculus on $SU_q(2)$ is described.