Does the Lie bracket operation factoring though exterior product have any meaning?

Question

I just started learning about Lie algebras.

Since the Lie bracket is bilinear and alternating, I recall that this allows us to factor the bracket operation $L \times L \to L$ though the exterior product $\bigwedge^2 (L) \to L$.

I may be jumping the gun, but does this have any meaning? Will this somehow show up later? Or is it just a meaningless observation?

Answer 1

This is a basic but indeed important observation; it naturally occurs in the Chevalley-Eilenberg complex computing the homology of the Lie algebra $$\dots\;\;\Lambda^3L\;\;\stackrel{d_3}\to\;\;\Lambda^2L\;\;\stackrel{d_2}\to\;\;\Lambda^1L(=L)\;\;\stackrel{d_1=0}\to\;\;\Lambda^0L(=0)$$

Here $d_2(x\wedge y)=-[x,y]$ is (up to sign) the map you have in mind. $d_3(x\wedge y\wedge z)=x\wedge [y,z]+y\wedge [z,x]+z\wedge [x,y]$. The homology $H_1(L)$ is $\mathrm{Ker}(d_1)/\mathrm{Im}(d_2)=L/[L,L]$, etc.