We propose a definition of a quantised ${\mathrm{𝔰𝔩}}_2$-differential algebra and show that the quantised exterior algebra (defined by Berenstein and Zwicknagl) and the quantised Clifford algebra (defined by the authors) of ${\mathrm{𝔰𝔩}}_2$ are natural examples of such algebras.

The goal of this expository paper is to give a quick introduction to $q$-deformations of semisimple Lie groups. We discuss principally the rank one examples of ${𝒰}_q\left({\mathrm{𝔰𝔩}}_2\right)$, $𝒪\left({\mathrm{SU}}_q\left(2\right)\right)$, $𝒟\left({\mathrm{SL}}_q(2,ℂ)\right)$ and related algebras. We treat quantized enveloping algebras, representations of ${𝒰}_q\left({\mathrm{𝔰𝔩}}_2\right)$, generalities on Hopf algebras and quantum groups, $*$-structures, quantized algebras of functions on $q$-deformed compact semisimple groups, the Peter-Weyl theorem, $*$-Hopf algebras associated to complex semisimple Lie groups and the Drinfeld double, representations...