nLab super module




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A more general version of super vector space.


/2\mathbb{Z}/2\mathbb{Z}-graded RR-modules

Given a commutative ring RR, an /2\mathbb{Z}/2\mathbb{Z}-graded RR-module is a RR-module VV with decomposition functions 𝒟 0:VV\mathcal{D}_0:V \to V and 𝒟 1:VV\mathcal{D}_1:V \to V, such that

  • for all v:Vv:V, v=𝒟 0(v)+𝒟 1(v)v = \mathcal{D}_0(v) + \mathcal{D}_1(v)
  • for all a:Ra:R, b:Rb:R, v:Vv:V, and w:Vw:V, 𝒟 0(av+bw)=a𝒟 0(v)+b𝒟 0(w)\mathcal{D}_0(a v + b w) = a \mathcal{D}_0(v) + b \mathcal{D}_0(w)
  • for all a:Ra:R, b:Rb:R, v:Vv:V, and w:Vw:V, 𝒟 1(av+bw)=a𝒟 1(v)+b𝒟 1(w)\mathcal{D}_1(a v + b w) = a \mathcal{D}_1(v) + b \mathcal{D}_1(w)
  • for all v:Vv:V, 𝒟 0(𝒟 0(v))=𝒟 0(v)\mathcal{D}_0(\mathcal{D}_0(v)) = \mathcal{D}_0(v)
  • for all v:Vv:V, 𝒟 0(𝒟 1(v))=0\mathcal{D}_0(\mathcal{D}_1(v)) = 0
  • for all v:Vv:V, 𝒟 1(𝒟 0(v))=0\mathcal{D}_1(\mathcal{D}_0(v)) = 0
  • for all v:Vv:V, 𝒟 1(𝒟 1(v))=𝒟 1(v)\mathcal{D}_1(\mathcal{D}_1(v)) = \mathcal{D}_1(v)

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 RR-modules and there exists a linear 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 modules.

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 modules

The tensor product of /2\mathbb{Z}/2\mathbb{Z}-graded RR-modules 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 linearity of the 𝒟 0\mathcal{D}_0 and 𝒟 1\mathcal{D}_1 functions result in the category of /2\mathbb{Z}/2\mathbb{Z}-graded RR-modules to be a monoidal category.

A super module is an object of the category of /2\mathbb{Z}/2\mathbb{Z}-graded RR-modules 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 11:47:56. See the history of this page for a list of all contributions to it.