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4

For $k < 2$ and $k > \dim V - 2$ it's always true that a $k$-form $\omega$ both is decomposable and satisfies $\omega \wedge \omega = 0$. For $k = 2$ (and provided the field underlying $V$ does not have characteristic $2$), the condition $\omega \wedge \omega = 0$ is both necessary and sufficient for decomposability. (Proving this is a nice exercise.) ...

1

By definition, the wedge product of $1$-forms $\alpha, \beta$ is $$\alpha \wedge \beta = \alpha \otimes \beta - \beta \otimes \alpha .$$ When $\beta = \alpha$, this is zero. The wedge product of $2$-forms has a different formula, so this does not apply. Indeed, for any finite-dimensional vector space $\Bbb V$ of dimension $\geq 4$, fix a cobasis $(e^a)$ of ...

1

Three vectors $x,y,z$ in a plane are necessarily linearly dependent. Without loss of generality, we may assume $z$ is a linear combination of $x$ and $y$, i.e. $z=\lambda x+\mu y$. Then $$(x\wedge y)z+(y\wedge z)x+(z\wedge x)y=\lambda(x\wedge y)x+\mu(x\wedge y)y+\lambda (y\wedge x)x+\mu(y\wedge y)x+\lambda(x\wedge x)y+\mu(y\wedge x)y.$$ Using skew-symmetry ...

1

I don't know what you mean by the regular representation of a Lie algebra, but the connection is the following. It's a bit confusing because there are a bunch of $3$-dimensional vector spaces that all get identified. Let $V$ be a real inner product space. The inner product induces a canonical isomorphism $V \cong V^{\ast}$ giving an isomorphism ...

0

I'm going to write $\varepsilon^i$ instead of $e^i$ to make the notation clearer: we have $\varepsilon^i(e_j) = \delta_{ij}$. We want to write $B$ as some linear combination of the $\varepsilon^i \otimes \varepsilon^j$. Note that $(\varepsilon^i \otimes \varepsilon^j)(e_p \otimes e_q) = \delta_{ip} \delta_{jq}$ just as for the 1-dimensional case. This means ...

3

If $$\omega = \sum_{i=1}^n\frac{(-1)^{i-1}x_i}{\|x\|^n}\,dx_1 \wedge\cdots \wedge \widehat{dx_i}\wedge \cdots \wedge dx_n,$$then: $$d\omega = \sum_{i=1}^n\sum_{j=1}^n\frac{\partial}{\partial x_j}\left(\frac{(-1)^{i-1}x_i}{\|x\|^n}\right) dx_j \wedge dx_1 \wedge\cdots \wedge \widehat{dx_i}\wedge \cdots \wedge dx_n.$$Now, the only surviving term is when \$j = ...

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