nLab super ring

Contents

Context

Algebra

Super-Algebra and Super-Geometry

Contents

Idea

A super ring is an \mathbb{Z}-super algebra.

Definition

A super ring is a ring RR with decomposition functions 𝒟 0:RR\mathcal{D}_0:R \to R and 𝒟 1:RR\mathcal{D}_1:R \to R, such that

  • for all a:Ra:R, a=𝒟 0(a)+𝒟 1(a)a = \mathcal{D}_0(a) + \mathcal{D}_1(a)
  • for all a:Ra:R, and b:Rb:R, 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:Ra:R, and b:Rb:R, 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:Ra:R, and b:Rb:R, 𝒟 0(ab)=𝒟 0(a)𝒟 0(b)𝒟 1(a)𝒟 1(b)\mathcal{D}_0(a \cdot b) = \mathcal{D}_0(a) \cdot \mathcal{D}_0(b) - \mathcal{D}_1(a) \cdot \mathcal{D}_1(b)
  • for all a:Ra:R, and b:Rb:R, 𝒟 1(ab)=𝒟 0(a)𝒟 1(b)+𝒟 1(a)𝒟 0(b)\mathcal{D}_1(a \cdot b) = \mathcal{D}_0(a) \cdot \mathcal{D}_1(b) + \mathcal{D}_1(a) \cdot \mathcal{D}_0(b)
  • for all a:Ra:R, and b:Rb:R, 𝒟 0(1)=1\mathcal{D}_0(1) = 1
  • for all a:Ra:R, and b:Rb:R, 𝒟 1(1)=1\mathcal{D}_1(1) = 1
  • for all a:Ra:R, 𝒟 0(𝒟 0(a))=𝒟 0(a)\mathcal{D}_0(\mathcal{D}_0(a)) = \mathcal{D}_0(a)
  • for all a:Ra:R, 𝒟 0(𝒟 1(a))=0\mathcal{D}_0(\mathcal{D}_1(a)) = 0
  • for all a:Ra:R, 𝒟 1(𝒟 0(a))=0\mathcal{D}_1(\mathcal{D}_0(a)) = 0
  • for all a:Ra:R, 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 decompostion functions im(𝒟 0)\im(\mathcal{D}_0) and im(𝒟 1)\im(\mathcal{D}_1) are rings and there exists an abelian group isomorphism i:Vim(𝒟 0)im(𝒟 1)i:V \cong \im(\mathcal{D}_0) \otimes \im(\mathcal{D}_1), where F:RingAbF:Ring \to Ab is a forgetful functor and F(A)F(B)F(A) \otimes F(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.

See also

Last revised on May 20, 2022 at 20:11:14. See the history of this page for a list of all contributions to it.