Hot answers tagged

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 = ...


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 ...



Only top voted, non community-wiki answers of a minimum length are eligible