Lie Algebras 14 -- Cartan's criterion for semisimplicity

[@ouni2022]

Thanks! There are some videos missing in the playlist - could you please add them?

[@nathansmith2168]

The definition of nondegeneracy in the theorem statement at 5:10 is incorrect. It should be that if x is nonzero then there exists some y such that K(x,y) is nonzero.

[@milenamarquez]

@21:51 I am sure I am missing something huge, but would somebody tell me why J is a solvable ideal of I perp if it commutes with it? (J can't be a center of I perp, as J isn't contained in I perp in the first place, I and I perp being a direct sum.)

[@theelk801]

we are so back

[@simonb9372]

I don't know how to solve the 2 HW exercices and the fact that [I,I^T]=0

[@arnouth5260]

Take i in I and j in I^T, now choose some random x in L. We have k([i,j],x) = k(i,[j,x]). Because I^T is an ideal, we have [j,x] is in I^T, but then k(i,[j,x]) = 0. Because we are working within a semisimple Lie algebra we have that k is non-degenerate so [i,j] = 0.

[@YitzharVered]

Thumbnail much?

[@A_doe_wasting_her_life]

what does that mean

[@aug3842]

@@A_doe_wasting_her_lifeit says ‘simisimple’

[@Juniper-111]

simisimple?

[@kapoioBCS]

Simi 😅