Koszul Duality

I have a f.d. semisimple Lie algebra L. What is it's structure? I could give you a basis and structure constants and then you could check that that works.



Or I could give you this chain complex where the differential is as follows






This chain complex is generated in degree 1 so that is all I had to give you. If you compute  , you get Jacobi Identity. This is Chevalley-Eilenberg.

So the data of a Lie algebra is the same as a semifree differential graded-commutative algebra generated in degree 1.

Aside:
You can change 1 to 1-n to get a Lie n-algebra, but then you would need to specify the higher brackets because of the other generators. If you replace it with just n, you get a n-Lie algebra. Removing all restrictions on generators whatsover lands you in   .

Now let's flip the arrows and go the other way.

I have a free graded Lie algebra structure on something called SC^*. If you see the condition for it to be a complex, you see it forces C to be a commutative algebra.

Again we could play with where it is generated to get the rest of the E_n type structures instead of just Comm.