Inner products on super vector spaces Let $V=V^0\oplus V^1$  be a super vector space (https://en.wikipedia.org/wiki/Super_vector_space)

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*Is there a special definition of an inner product on $V$ other than just an inner product on the underlying vector space (forgetting the grading)?


*Does the supertrace have a role in the induced inner product on the space $End(V)$ just like the trace has a role in the non graded case?
 A: I am missing well written references for this (sign) sensitive topic! But here is what I learned from a few talks I have attended:
An inner product on the super vector space $V = V^0 \oplus V^1$ should be a morphism $B:V\otimes V \to \mathbb{K}$ of super vector spaces. The standard convention is to have $\mathbb{K} = \mathbb{K}^{1|0}$ to be purely even. (But I have seen people use $\mathbb{K}^{0|1}$ or $\mathbb{K}^{1|1}$ instead.)
Then additionally one can require $B$ to be an even morphism, meaning that it vanishes on odd elements of $V\otimes V$, implying that homogenous vectors of opposite parity are orthogonal to each other.
Now $B$ is called symmetric if $B \circ c = B$ where $c:V\otimes V \overset{\sim}{\to} V\otimes V$ is the braiding which on homogenous elements $v,w$ is given by $v\otimes w \mapsto (-1)^{|v||w|}w\otimes v$.
(edited:)
Assume now $V$ is finite dimensional. Then $\mathrm{str}:\mathrm{End}(V) \to \mathbb{K}$ can be defined to be the unique linear map vanishing on superommutators with normalisation $\mathrm{str}(\mathrm{id}) = \mathrm{sdim} V = \dim V^0 - \dim V^1$.
Then \begin{align*}
\mathrm{End}(V)\otimes \mathrm{End}(V) &\to \mathbb{K}\\ A\otimes B &\mapsto \mathrm{str}(AB) \end{align*} defines an even, symmetric, non-degenerate bilinear form.
A: In addition to the other answer, a very good reference on that matter is the book of Yuri Ivanovich Manin "Gauge Field Theory and Complex Geometry", specifically Chapter 3 "Introduction to superalgebra", even more specifically Section 5 "Scalar products" of that chapter.
