Let k be an algebraically closed field. Given a superalgebra A we will denote with A0 the even part, with A1 the odd part and with IAodd the ideal generated by the odd part.
A superalgebra is said to be commutative (or supercommutative) if
xy = (−1)p(x)p(y)yx, ∀ homogeneous x, y
where p denotes the parity of an homogeneous element (p(x) = 0 if x ∈ A0, p(x) = 1 if x ∈ A1).
Let’s denote with A the category of affine superalgebras that is commutative superalgebras such that, modulo the ideal generated by their odd part, they are affine algebras (an affine algebra is a finitely generated reduced commutative algebra).
Define affine algebraic supervariety over k a representable functor V from the category A of affine superalgebras to the category S of sets. Let’s call k[V] the commutative k-superalgebra representing the functor V,
V (A) = Homk−superalg(k[V], A), A ∈ A
We will call V (A) the A-points of the variety V. A morphism of affine supervarieties is identified with a morphism between the representing objects, that is a morphism of affine superalgebras.
We also define the functor Vred associated to V from the category Ac of affine k-algebras to the category of sets:
Vred(Ac)= Homk−alg(k[V]/Ik[V]odd, Ac), Ac ∈ Ac
Vred is an affine algebraic variety and it is called the reduced variety associated to V. If the algebra k[V] representing the functor V has the additional structure of a commutative Hopf superalgebra, we say that V is an affine algebraic supergroup.
Let G be an affine algebraic supergroup. As in the classical setting, the condition k[G] being a commutative Hopf superalgebra makes the functor group valued, that is the product of two morphisms is still a morphism. In fact let A be a commutative superalgebra and let x, y ∈ Homk−superalg(k[G], A) be two points of G(A). The product of x and y is defined as:
x · y = defmA · x ⊗ y · ∆
where mA is the multiplication in A and ∆ the comultiplication in k[G]. One can find that x · y ∈ Homk−superalg(k[G], A), that is:
(x · y)(ab) = (x · y)(a)(x · y)(b)
The non commutativity of the Hopf algebra in the quantum setting does not allow to multiply morphisms(=points). In fact in the quantum (super)group setting the product of two morphisms is not in general a morphism.
Let V be an affine algebraic supervariety. Let k0 ⊂ k be a subfield of k. We say that V is a k0-variety if there exists a k0-superalgebra k0[V] such that k[V] ≅ k0[V] ⊗k0 k and
V(A) = Homk0 − superalg(k0[V], A) = Homk−superalg(k[V], A), A ∈ A.
We obtain a functor that we still denote by V from the category Ak0 of affine k0-superalgebras to the category of sets:
V(Ak0) = Homk0−superalg(k0[V], Ak0), A ∈ Ak0
thus opening up for consideration of rationality on supervariety.