Lie Theory
The Killing Form: Detecting Semisimplicity with a Bilinear Trace
Feed a Lie algebra two vectors, compose their adjoint actions, take the trace — and you get a single number that decides everything. Cartan's Criterion says a finite-dimensional Lie algebra 𝔤 over a field of characteristic 0 is semisimple if and only if its Killing form B(x, y) = tr(ad x ∘ ad y) is nondegenerate. One determinant, computed from structure constants, separates the well-behaved algebras — sums of simple pieces, fully classifiable, with completely reducible representations — from the pathological ones.
The Killing form is the canonical symmetric invariant bilinear form on 𝔤: B: 𝔤 × 𝔤 → k, B(x, y) = tr(ad x ad y), where (ad x)(z) = [x, z]. It is symmetric, associative (B([x, y], z) = B(x, [y, z])), and invariant under all automorphisms. Its radical measures how far 𝔤 is from being semisimple, and its signature over ℝ pins down which real form you are holding.
- FieldLie theory / structure of Lie algebras
- Named for / yearWilhelm Killing (1888); form isolated by Élie Cartan (1894)
- Key hypothesisdim 𝔤 < ∞, char k = 0 (algebraically closed for cleanest form)
- Statement𝔤 semisimple ⇔ B(x,y)=tr(ad x ad y) is nondegenerate
- Proof techniqueTrace/Jordan decomposition + Cartan's solvability criterion
- Generalizes toAny invariant trace form on any faithful representation
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The precise statement
Let 𝔤 be a finite-dimensional Lie algebra over a field k of characteristic 0. The Killing form is the symmetric bilinear form B: 𝔤 × 𝔤 → k defined by
B(x, y) = tr(ad x ∘ ad y),
where ad x ∈ End(𝔤) is the linear map z ↦ [x, z]. It is symmetric because tr(AB) = tr(BA), and it is associative (or invariant): B([x, y], z) = B(x, [y, z]) for all x, y, z, which follows from the Jacobi identity applied to ad[x, y] = ad x ad y − ad y ad x.
Cartan's Criterion (semisimple form). 𝔤 is semisimple — meaning its radical (maximal solvable ideal) is 0, equivalently 𝔤 is a direct sum of simple non-abelian ideals — if and only if B is nondegenerate: its radical rad B = {x : B(x, y) = 0 ∀ y} equals {0}. Equivalently, the Gram matrix of B in any basis has nonzero determinant.
The picture: an intrinsic inner product from the bracket
A Lie algebra carries no a-priori geometry — just a bracket. The Killing form manufactures one for free, using only the bracket, by letting 𝔤 act on itself via ad and measuring how two such actions overlap. Think of it as the algebra's self-portrait: it looks at itself through the adjoint representation and records the trace-pairing of the resulting operators.
Nondegeneracy is a rigidity condition. It says every direction x is 'seen' by some y — no vector is invisible to the form. When a nonzero ideal 𝔞 is solvable, its elements act on 𝔤 in an increasingly triangular, trace-killing way, so they slip into rad B and the form collapses in those directions. Semisimplicity is exactly the absence of such blind spots. Geometrically, over ℝ a compact form makes B negative-definite, turning −B into a genuine bi-invariant Riemannian metric on the corresponding Lie group — the source of a great deal of symmetric-space geometry.
Key idea of the proof
The engine is the trace form on a linear action plus Jordan decomposition. The 'easy' direction: if 𝔤 has a nonzero abelian ideal 𝔞, pick 0 ≠ a ∈ 𝔞. For any x, (ad a)(ad x) maps 𝔤 → 𝔞 → 0, so it is nilpotent, hence B(a, x) = tr(ad a ad x) = 0; thus a ∈ rad B and B is degenerate. Since a solvable ideal always contains a nonzero abelian ideal (its last nontrivial derived term), rad(𝔤) ≠ 0 ⟹ rad B ≠ 0.
The 'hard' converse uses Cartan's solvability criterion: a subalgebra 𝔤 ⊆ 𝔤𝔩(V) is solvable if tr(xy) = 0 for all x ∈ [𝔤, 𝔤], y ∈ 𝔤. Its proof factors elements via the abstract Jordan decomposition, replacing an eigenvalue λ by its Galois conjugates / a ℚ-linear functional, and showing the semisimple part has all-zero eigenvalues, hence is 0. Applying this to ad(rad B) ⊆ 𝔤𝔩(𝔤) shows rad B is a solvable ideal; if B is nondegenerate then rad B = 0, forcing rad(𝔤) = 0. Characteristic 0 is where the eigenvalue-argument lives.
Worked example: 𝔰𝔩₂ and a degenerate cousin
Take 𝔰𝔩₂(ℂ) with basis e, h, f and brackets [h, e] = 2e, [h, f] = −2f, [e, f] = h. Computing ad on this 3-dimensional space gives the Killing form Gram matrix, in the ordered basis (e, h, f),
B = [[0, 0, 4], [0, 8, 0], [4, 0, 0]],
so B(h, h) = 8, B(e, f) = 4, and all other basis pairings vanish. The determinant is −128 ≠ 0: nondegenerate, confirming 𝔰𝔩₂ is simple, hence semisimple. In general on 𝔰𝔩ₙ one finds B(x, y) = 2n·tr(xy).
Contrast the 2-dimensional non-abelian algebra with [x, y] = x (the 'ax+b' algebra). Here ad x and ad y are strictly/degenerate upper-triangular; one computes B(x, x) = 0, B(x, y) = 0, B(y, y) = 1, giving Gram matrix [[0, 0], [0, 1]] with determinant 0. The form is degenerate — correctly, since this algebra is solvable, not semisimple, with x spanning a nonzero abelian ideal sitting inside rad B.
Why the hypotheses matter
Characteristic 0 is essential. In characteristic p the trace form can vanish or degenerate for spurious reasons. Classic counterexample: 𝔰𝔩₂ over a field of characteristic 2 is not semisimple in the usual sense (its scalar-trace behavior collapses), and 𝔰𝔩ₙ has degenerate Killing form whenever p | 2n; e.g. 𝔰𝔩ₚ(k) with char k = p has a central identity element and B ≡ 0 on a subspace. Cartan's criterion simply fails, which is why modular Lie theory develops entirely different tools.
Finite dimension is essential for tr(ad x ad y) even to be defined. In infinite dimensions (Kac–Moody, affine algebras) one replaces B with a formally-defined invariant form, and nondegeneracy becomes an added axiom, not a theorem.
Non-abelian simple summands. A 1-dimensional (abelian) algebra has B ≡ 0, so it is correctly excluded from 'semisimple'. Connections: rad B always contains the nilradical, and the general reductive case is exactly 'B nondegenerate on the derived algebra' — semisimple ⊕ center.
Applications and significance
The Killing form is the load-bearing wall of structure theory. Weyl's theorem on complete reducibility of representations of a semisimple 𝔤 is proved by building the Casimir element C = ∑ xᵢ xⁱ from a basis and its B-dual basis — which exists precisely because B is nondegenerate — and using C to split extensions. The root-space decomposition and Cartan matrix rely on B restricting to a nondegenerate form on a Cartan subalgebra 𝔥, inducing an inner product on the root system that makes ⟨α, β⟩ integers and drives the Dynkin-diagram classification of simple Lie algebras (Aₙ, Bₙ, Cₙ, Dₙ, and the exceptionals G₂, F₄, E₆, E₇, E₈).
Over ℝ, the signature of B classifies real forms: negative-definite ⇔ compact form; the split form maximizes positive eigenvalues. This feeds the theory of symmetric spaces and, via −B as a bi-invariant metric, differential geometry, harmonic analysis on Lie groups, and much of mathematical physics' gauge theory.
| Criterion | Form used | Condition | Conclusion |
|---|---|---|---|
| Solvability (Cartan) | tr(xy) on ρ(𝔤) ⊆ 𝔤𝔩(V) | tr(xy)=0 for all x∈[𝔤,𝔤], y∈𝔤 | 𝔤 is solvable |
| Semisimplicity (Cartan) | Killing form B(x,y)=tr(ad x ad y) | B nondegenerate (rad B = 0) | 𝔤 is semisimple |
| Failure of semisimplicity | Killing form B | rad B ≠ 0 | 𝔤 has a nonzero solvable ideal |
| Solvability from vanishing B | B | B ≡ 0 identically | 𝔤 is solvable (char 0) |
Frequently asked questions
Why is characteristic 0 needed — where exactly does the proof break in characteristic p?
The hard direction relies on Cartan's solvability criterion, whose proof analyzes eigenvalues of the semisimple part of a Jordan decomposition and concludes they are all zero via a ℚ-linear-functional argument. In characteristic p that argument fails because Frobenius identifies distinct 'eigenvalues' and traces can vanish spuriously. Concretely, 𝔰𝔩ₙ over a field with p | 2n has degenerate Killing form even when the algebra is 'morally' simple, so nondegeneracy no longer detects semisimplicity.
Is the Killing form the same as the trace form tr(xy) on a representation?
No — the Killing form is specifically tr(ad x ad y), the trace form of the adjoint representation. For a general faithful representation ρ: 𝔤 → 𝔤𝔩(V), the form ⟨x, y⟩ = tr(ρ(x)ρ(y)) is also invariant and, on a simple 𝔤, is a nonzero scalar multiple of B. On 𝔰𝔩ₙ the defining (vector) representation gives tr(xy), and B = 2n·tr(xy). Any nonzero invariant form on a simple algebra is proportional to the Killing form.
What does the radical of the Killing form tell you if 𝔤 is not semisimple?
rad B is always an ideal of 𝔤, and by Cartan's criterion applied to ad(rad B) it is a solvable ideal, so rad B ⊆ rad(𝔤). It is not always equal to the radical, though: for a reductive algebra like 𝔤𝔩ₙ = 𝔰𝔩ₙ ⊕ (center), the center lies in rad B, and B is nondegenerate exactly on the semisimple part. So rad B detects the 'abelian/solvable obstruction' to semisimplicity.
Does the criterion hold in infinite dimensions?
Not as stated — tr(ad x ad y) need not converge or even be defined when dim 𝔤 = ∞. For Kac–Moody and affine Lie algebras one postulates an invariant symmetric bilinear form axiomatically (the 'standard invariant form'), and its nondegeneracy is a chosen structural feature rather than a consequence. Finite-dimensionality is what makes the trace, and hence the whole criterion, meaningful.
How is the Casimir element related, and why does it need nondegeneracy?
Given a basis {xᵢ} of a semisimple 𝔤, nondegeneracy of B produces a dual basis {xⁱ} with B(xᵢ, xʲ) = δᵢʲ. The Casimir C = ∑ᵢ xᵢ xⁱ lives in the universal enveloping algebra, is central, and acts as a scalar on each irreducible by Schur. Without nondegeneracy there is no dual basis, no Casimir, and the standard proof of Weyl's complete-reducibility theorem collapses — which is why semisimplicity is the right setting.
What is the signature of the Killing form and why does it matter over ℝ?
For a real semisimple Lie algebra, B is a nondegenerate symmetric real form, so it has a signature (p, q). This signature is an invariant that distinguishes the real forms of a given complex semisimple algebra: the compact real form is the unique one where B is negative-definite (signature (0, dim 𝔤)), and −B then defines a bi-invariant Riemannian metric on the compact group. Split forms maximize the positive part. The signature thus classifies real forms and underlies symmetric-space geometry.