nLab super abelian group

Idea

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

Definition

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

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

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

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

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

Super abelian group

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

$$\mathcal{D}_0(a \otimes b) = \mathcal{D}_0(a) \otimes \mathcal{D}_0(b) + \mathcal{D}_1(a) \otimes \mathcal{D}_1(b)$$

$$\mathcal{D}_1(a \otimes b) = \mathcal{D}_0(a) \otimes \mathcal{D}_1(b) + \mathcal{D}_1(a) \otimes \mathcal{D}_0(b)$$

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

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

$$t_{A, B}:(A \otimes B) \to (B \otimes A)$$

such that

$$\mathcal{D}_0(t_{A, B}(a, b)) = \mathcal{D}_0(a) \otimes \mathcal{D}_0(b) - \mathcal{D}_1(a) \otimes \mathcal{D}_1(b)$$

$$\mathcal{D}_1(t_{A, B}(a, b)) = \mathcal{D}_0(a) \otimes \mathcal{D}_1(b) + \mathcal{D}_1(a) \otimes \mathcal{D}_0(b)$$

See also

• abelian group

• graded abelian group

• super commutative monoid

• super module

• super ring

• supercommutative ring