CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Semisimple Lie algebras are centerless and perfect

Statement

If $g$ is finite-dimensional and semisimple over a characteristic-zero field, then $Z (g) = 0$ and $[g, g] = g$ .

Facts & Assumptions

Given: Such a Lie algebra $g$ .

[L1]

Its Killing form is nondegenerate (Cartan's semisimplicity criterion).

[L2]

The Killing form is the trace form of the adjoint representation (Killing form).

Proof

technique · direct

1.1L1L2

If $z \in Z (g)$ , then ${ad}_{z} = 0$ , so [L2] gives $K (z, x) = 0$ for every $x$ . Nondegeneracy in [L1] yields $z = 0$ .

2.1L1step 1.1algebra

Let $D = [g, g]$ . Its orthogonal complement consists exactly of the elements $z$ with $K ([x, y], z) = K (x, [y, z]) = 0$ for all $x, y$ , hence $[y, z] = 0$ by [L1]. Thus $D^{⊥} = Z (g) = 0$ by step 1.1, and finite-dimensional nondegeneracy gives $D = g$ .

3.1step 1.12.1∎

When $g = 0$ , both conclusions read $0 = 0$ ; no nonempty choice was used.

Depends on

Used by

Semisimple Hamiltonian actions have a unique equivariant moment map when one exists Corollary
Maximal split abelian subspace and real rank Definition
Complex simple lie algebra viewed as a real simple algebra Example
Jordan–Chevalley parts agree under the adjoint representation Lemma
The Killing length of a root is nonzero Lemma
Root and weight lattice sandwich Proposition
The center is the common kernel of the roots inside the Cartan subalgebra Proposition
Ado's theorem with nilpotent nilradical action Theorem
Analytic and root-system Weyl groups agree Theorem
Cartan subalgebras are exactly maximal toral subalgebras Theorem
Cayley transforms connect theta-stable Cartans in the classification Theorem
Compact connected Lie groups are classified by root data Theorem
Compact roots form a reduced crystallographic root system Theorem
Complexification dichotomy for a real simple lie algebra Theorem
Conjugacy of Cartan involutions Theorem
Conjugacy of Cartan subalgebras Theorem
Conjugacy of compact real forms Theorem
Derivations of semisimple Lie algebras are inner Theorem
Equivalent characterizations of reductive Lie algebras Theorem
Every real Cartan subalgebra is conjugate to a theta-stable one Theorem
Existence of a Cartan involution Theorem
Existence of Cartan subalgebras Theorem
First Whitehead lemma Theorem
Global Cartan decomposition for a connected finite center semisimple Lie group Theorem
Malcev conjugacy of Levi subalgebras Theorem
Roots of a complex semisimple Lie algebra form a reduced crystallographic root system Theorem
Vogan and Satake diagrams give equivalent real form classifications Theorem
Vogan diagram for a fixed Cartan involution is well defined up to equivalence Theorem
Weyl's complete reducibility theorem Theorem

Dependency tree · two levels

7 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

Milne, Lie Algebras, consequences of Theorem 4.13 (standard reference, not scraped)