lecture notes5

Suppose x s and y l then xy z x yz 0 lemma

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: cal) of the Killing form of L is an ideal. The definition of the kernel of κ is: S = {x ∈ L | κ(x, z ) = 0 for all z ∈ L} Proof. Suppose x ∈ S and y ∈ L. Then κ([xy ], z ) = κ(x, [yz ]) = 0. Lemma 5.2.3. Every abelian ideal in L is contained in the kernel of its Killing form. ￿ 18 MATH 223A NOTES 2011 LIE ALGEBRAS Proof. Suppose J ⊆ L is an abelian ideal, x ∈ J and y ∈ L. Then ad x ad y sends L into J and ad x ad y (J ) ⊆ ad x(J ) = 0. So, (ad x ad y )2 = 0. Since nilpotent endomorphisms have zero trace, κ(x, y ) = Tr(ad x ad y ) = 0 showing that J ⊆ S . ￿ Theorem 5.2.4 (Cartan). Suppose that L is a fi...
View Full Document

This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.

Ask a homework question - tutors are online