Representations of $\mathbb{Z}/2$ as super vector spaces?

Question

I heard someone say that representations of $\mathbb{Z}/2$ "are" super vector spaces. As far as I understand, super vector spaces are $\mathbb{Z}/2$-graded vector spaces, so my question is whether there is a canonical way of obtaining a $\mathbb{Z}/2$-graded vector space from a representation of $\mathbb{Z}/2$ (and vice versa).

Answer 1

Here's my best guess: let's write $\Bbb Z_2 = \{0,1\}$. A representation of $\Bbb Z_2$ consists of any assignment $\rho(1) = A:V \to V$ for which $A^2 = I$. Given such a linear transformation $A$, we have $$ V = \ker (A - I) \oplus \ker(A + I) $$ We would then take $V_0 = \ker(A - I)$ and $V_1 = \ker(A + I)$, in the notation of the wiki page.

Conversely, for any decomposition $V = V_0 \oplus V_1$, we could define a map $A$ by extending the definition $$ Av = \begin{cases} v & v \in V_0\\ -v & v \in V_1 \end{cases} $$