nLab super abelian group

Contents

Context

Algebra

Super-Algebra and Super-Geometry

Group theory

Contents

Idea

A super abelian group is an \mathbb{Z}-super module, or a .

Definition

/2\mathbb{Z}/2\mathbb{Z}-graded abelian group

A /2\mathbb{Z}/2\mathbb{Z}-graded is an abelian group GG with decomposition functions 𝒟 0:GG\mathcal{D}_0:G \to G and 𝒟 1:GG\mathcal{D}_1:G \to G, such that

  • for all a:Ga:G, a=𝒟 0(a)+𝒟 1(a)a = \mathcal{D}_0(a) + \mathcal{D}_1(a)
  • for all a:Ga:G, and b:Gb:G, 𝒟 0(a+b)=𝒟 0(a)+𝒟 0(b)\mathcal{D}_0(a + b) = \mathcal{D}_0(a) + \mathcal{D}_0(b)
  • for all a:Ga:G, and b:Gb:G, 𝒟 1(a+b)=𝒟 1(a)+𝒟 1(b)\mathcal{D}_1(a + b) = \mathcal{D}_1(a) + \mathcal{D}_1(b)
  • for all a:Ga:G, 𝒟 0(𝒟 0(a))=𝒟 0(a)\mathcal{D}_0(\mathcal{D}_0(a)) = \mathcal{D}_0(a)
  • for all a:Ga:G, 𝒟 0(𝒟 1(a))=0\mathcal{D}_0(\mathcal{D}_1(a)) = 0
  • for all a:Ga:G, 𝒟 1(𝒟 0(a))=0\mathcal{D}_1(\mathcal{D}_0(a)) = 0
  • for all a:Ga:G, 𝒟 1(𝒟 1(a))=𝒟 1(a)\mathcal{D}_1(\mathcal{D}_1(a)) = \mathcal{D}_1(a)

As a result, the image of the two decomposition functions im(𝒟 0)\im(\mathcal{D}_0) and im(𝒟 1)\im(\mathcal{D}_1) are abelian groups and there exists a group isomorphism i:Vim(𝒟 0)im(𝒟 1)i:V \cong \im(\mathcal{D}_0) \otimes \im(\mathcal{D}_1), where ABA \otimes B is the tensor product of abelian groups.

The elements of im(𝒟 0)\im(\mathcal{D}_0) are called even elements or bosonic elements, and the elements of im(𝒟 1)\im(\mathcal{D}_1) are called odd elements or fermionic elements.

Super abelian group

The tensor product of /2\mathbb{Z}/2\mathbb{Z}-graded abelian groups ABA \otimes B for a:Aa:A, b:Bb:B is defined as the following:

𝒟 0(ab)=𝒟 0(a)𝒟 0(b)+𝒟 1(a)𝒟 1(b)\mathcal{D}_0(a \otimes b) = \mathcal{D}_0(a) \otimes \mathcal{D}_0(b) + \mathcal{D}_1(a) \otimes \mathcal{D}_1(b)
𝒟 1(ab)=𝒟 0(a)𝒟 1(b)+𝒟 1(a)𝒟 0(b)\mathcal{D}_1(a \otimes b) = \mathcal{D}_0(a) \otimes \mathcal{D}_1(b) + \mathcal{D}_1(a) \otimes \mathcal{D}_0(b)

This plus the group homomorphisms 𝒟 0\mathcal{D}_0 and 𝒟 1\mathcal{D}_1 result in the category of /2\mathbb{Z}/2\mathbb{Z}-graded abelian groups to be a monoidal category.

A super abelian group is an object of the category of /2\mathbb{Z}/2\mathbb{Z}-graded abelian groups with the braiding for the tensor product ABA \otimes B:

t A,B:(AB)(BA)t_{A, B}:(A \otimes B) \to (B \otimes A)

such that

𝒟 0(t A,B(a,b))=𝒟 0(a)𝒟 0(b)𝒟 1(a)𝒟 1(b)\mathcal{D}_0(t_{A, B}(a, b)) = \mathcal{D}_0(a) \otimes \mathcal{D}_0(b) - \mathcal{D}_1(a) \otimes \mathcal{D}_1(b)
𝒟 1(t A,B(a,b))=𝒟 0(a)𝒟 1(b)+𝒟 1(a)𝒟 0(b)\mathcal{D}_1(t_{A, B}(a, b)) = \mathcal{D}_0(a) \otimes \mathcal{D}_1(b) + \mathcal{D}_1(a) \otimes \mathcal{D}_0(b)

See also

Last revised on May 11, 2022 at 07:53:25. See the history of this page for a list of all contributions to it.