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The Killing form is invariant and nondegenerate on a complex semisimple Lie algebra
Statement
Let be a complex semisimple Lie algebra, and let be its Killing form from The Killing form of a semisimple Lie algebra. Then
and is nondegenerate.
Facts & Assumptions
Given: A finite-dimensional complex semisimple Lie algebra and its Killing form .
The finite Engel/trace argument proves that the Killing form of every finite-dimensional complex semisimple Lie algebra is invariant and nondegenerate (Engel, the trace criterion, and Killing nondegeneracy).
Proof
Using and cyclicity of trace, one gets .
The nondegeneracy clause of [L1] applies to . It proves that the radical is zero by an explicit finite trace-solvability argument, rather than leaving Cartan's criterion as an unproved appeal.
Hence is invariant and has zero radical, so it is nondegenerate.
Depends on
Used by
- Standard parabolic and Levi subalgebras from a subset of simple roots Definition
- The quadratic Casimir element Definition
- Normality of the nilpotent cone and Springer functions Lemma
- The Chevalley–Eilenberg Laplacian is scalar on weight components Lemma
- The extremal weight cochain of a Weyl element is closed and unique Lemma
- The nilpotent central fiber is reduced and integral Lemma
- The quadratic Casimir element is independent of the choice of dual bases Lemma
- Under Choice, the Killing form pairs only opposite root spaces Proposition
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Pavel Etingof, Lie Groups and Lie Algebras I (standard reference, not scraped)