Superalgebras

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 I^A_odd 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) = Hom_k−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 V_red associated to V from the category A_c of affine k-algebras to the category of sets:

V_red(A_c)= Hom_k−alg(k[V]/I^k[V]_odd, A_c), A_c ∈ A_c

V_red 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 ∈ Hom_k−superalg(k[G], A) be two points of G(A). The product of x and y is defined as:

x · y = _defm_A · x ⊗ y · ∆

where m_A is the multiplication in A and ∆ the comultiplication in k[G]. One can find that x · y ∈ Hom_k−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) = Hom_{k0 − superalg}(k₀[V], A) = Hom_k−superalg(k[V], A), A ∈ A.

We obtain a functor that we still denote by V from the category A_k0 of affine k₀-superalgebras to the category of sets:

V(A_k0) = Hom_{k0−superalg}(k₀[V], A_k0), A ∈ A_k0

thus opening up for consideration of rationality on supervariety.

Advertisement