Are all bivectors in three dimensions simple?

Question

I want to show that all bivectors in three dimensions are simple.

If I understand correctly, a bivector is simply an element from the two-fold exterior product $\bigwedge^2V$ of a vector space $V$, right?

We can define $\wedge(e_i\otimes e_j):=e_i\otimes e_j - e_j\otimes e_i$. Now let $T=t^{ij}e_i\wedge e_j\mapsto t^{ij}(e_i\otimes e_j - e_j\otimes e_i)=(t^{ij}-t^{ji})e_i\otimes e_j$. This is injective. Because the wedge product as a linear map from the tensor product to the exterior product maps all symmetric tensors to 0.

We see that total antisymmetric tensors in this case are represented by skew-symmetric matrices. To show that they are all simple, I would have to show that the rank of any 3x3 skew-symmetric matrix is 1.

But the rank of a skew-symmetric matrix is never one.

I must have made a conceptual mistake somewhere again. Does anybody have a hint for me?

Geometrically, one can use the canonical isomorphism between the two-fold exterior product and $\mathbb{R}^3$ itself to show that any anyisymmetric tensor can be thought of as a vector in 3D, which in turn can be represented by the cross product of two vectors, that are not collinear to each other but orthogonal to that vector.

Answer 1

To show that they are all simple, I would have to show that the rank of any 3x3 skew-symmetric matrix is 1.

This doesn't follow, since as you say the rank of a skew-symmetric matrix can never be $1$. You're conflating $e_i \otimes e_j - e_j \otimes e_i$, which as a tensor has rank $2$ or $0$, with $e_i \wedge e_j$. These are not the same object; one of them lives in $V^{\otimes 2}$ and the other one lives in $\Lambda^2(V)$. In general I don't recommend thinking in terms of antisymmetric tensors; it makes the exterior product look much more complicated than it is.

Anyway, here's a proof of darij's more general claim in the comments. Let $v_1 \in \Lambda^{n-1}(V)$ be a vector, where $\dim V = n$. Choose a nonzero element $\omega \in \Lambda^n(V)$, hence an identification of it with the ground field $k$. Then the exterior product

$$\wedge : V \times \Lambda^{n-1}(V) \to \Lambda^n(V) \cong k$$

is a nondegenerate bilinear pairing. Extend $v_1$ to a basis $v_1, \dots v_n \in \Lambda^{n-1}(V)$. Then it has a unique dual basis $e_1, \dots e_n \in V$ defined by the condition that

$$e_i \wedge v_j = \delta_{ij} \omega \in \Lambda^n(V).$$

Then the $v_i$ must also be the dual basis of the $e_i$ with respect to this pairing. But this dual basis in turn must be

$$v_i = (-1)^{i-1} \frac{\omega}{e_1 \wedge \dots \wedge e_n} e_1 \wedge \dots \wedge \widehat{e_i} \wedge \dots \wedge e_n$$

where the hat denotes that we omit $e_i$, and in particular

$$v_1 = \frac{\omega}{e_1 \wedge \dots \wedge e_n} e_2 \wedge \dots \wedge e_n.$$