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