I recently read a little about super vector spaces and naturally I have a question and I don't know if I'm correct or if there's something in the definition that I don't quite understand.
My understanding is that a super vector space is a $\mathbb{Z}_2$-graded vector space, that is a vector space $V$ such that $V=V_0\oplus V_1$, $0,1\in \mathbb{Z}_2$ and $\forall x \in V_i$ we denote the parity of $x$ by $\left|x\right|=i$ (so $x\in V_0$ has parity $0$ and $x\in V_1$ has parity $1$).
Also, given a finite dimensional inner product space $(V,\langle\cdot,\cdot \rangle)$, and a subspace $F$ we can construct the subspace $$F^{\perp}=\left\lbrace u \in V, \langle u,v \rangle =0, v\in F\right\rbrace,$$ the subspace orthogonal to $F$. Furthermore, one can readily show that $V=F\oplus F^\perp$. (Not sure if the same can be said about infinite dimensional?). So call $F=V_0$ and $F^\perp=V_1$ then we can make $(V,\langle\cdot,\cdot\rangle)$ into a super vector space by fixing a subspace $F$, and viewing $V$ as the direct sum of $F$ and $F^\perp$ and saying that if $x\in F, \left|x\right|=0$ and if $x\in F^\perp$ then $\left|x\right|=1$.
So is it correct to say that any finite dimensional inner product space can be made into a super vector space ?
Best Answer
Converting my comments into an answer: any vector space $V$ can be made into a super vector space in many different ways, corresponding to any direct sum decomposition $V \cong V_0 \oplus V_1$. This is extra structure in general so it doesn't make sense to say that $V$ "is" a super vector space this way, only that it "can be made" a super vector space this way.
There are two canonical such decompositions, namely $V_0 = V$ (concentrated in even degree) or $V_1 = V$ (concentrated in odd degree). The even one is distinguished because that construction is symmetric monoidal.
The category of super vector spaces is not that interesting, and equivalent to the category of pairs of vector spaces. What is interesting is the symmetric monoidal category of super vector spaces, which is where you define supercommutative algebras and Lie superalgebras and so forth.