  We give the definitions of affine algebraic supervariety and affine algebraic
group through the functor of points and we relate them to the other definitions
present in the literature. We study in detail the algebraic supergroup
$SL(m|n)$ and give explicitly the Hopf algebra structure of the algebra
representing its functor of points. At the end we give also the quantization of
$SL(m|n)$ together with its coactions on suitable quantum spaces according to
Manin's philosophy.

Quantum Algebra;Algebraic Geometry