I know there is a lot of topics about pseudovectors (see 1, 2, 3), but I am not really satisfied with the answers in this topics, let me explain why.
Let $V$ is some vector space such that $\dim V = 3$, then in all topics above offer think about pseudovectors as about elements of $\wedge^2 V$, but for me pseudovector is element of $\star(\wedge^2 V)$ where $\star$ is Hodge star operator. So for me $a \wedge b$ is $2$-vector and $\star(a \wedge b)$ is pseudovector. The problem is after using hodge star on $2$-vector we can't remember that this is actually pseudovector.
So, i am interesting about next notion of pseudovector. Let $(V, \mathcal{O})$ is some vector space with fixed orientation and $\operatorname{inv} : (V, \mathcal{O}) \to (V, -\mathcal{O})$ is natural operator which reverse orientation, then pseudovector $p$ is such object that
- Can be naturally constructed from vector $v \in (V,\mathcal{O})$, $p = p(v)$
- Satisfied to identity $\operatorname{inv}(p(v)) = -p(\operatorname{inv}(v))$
- Codomen of Hodge star restricted on $(\wedge^2 V, \wedge^2 \mathcal{O})$, is actually pseudovectors of $(V, \mathcal{O})$ not vectors.
Does such notion exists? Thank you.