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We define the concept of quantum section of a line bundle of a homogeneous superspace and we employ it to define the concept of quantum homogeneous projective superspace. We also suggest a generalization of the QDP to the quantum supersetting.

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...