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Trace forms are symmetric and invariant

Statement

For every finite-dimensional representation $ρ : g \to g l (V)$ , the trace form is bilinear and symmetric, and it is invariant in the sense that

$B_{ρ} ([z, x], y) + B_{ρ} (x, [z, y]) = 0.$

Consequently the Killing form has all three properties.

Facts & Assumptions

Given: A representation $ρ : g \to g l (V)$ with $V$ finite-dimensional and elements $x, y, z \in g$ .

[L1]

The trace form is $B_{ρ} (x, y) = tr (ρ (x) ρ (y))$ (Trace form of a representation).

[L2]

Finite-dimensional endomorphisms satisfy $tr (A B) = tr (B A)$ (For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ ).

Proof

technique · direct trace calculation

1.1L1L2algebra

Linearity of $ρ$ , composition, and trace makes $B_{ρ}$ bilinear. By [L2], $B_{ρ} (x, y) = tr (ρ (x) ρ (y)) = tr (ρ (y) ρ (x)) = B_{ρ} (y, x)$ , so it is symmetric.

1.2L1L2algebra

Put $X = ρ (x)$ , $Y = ρ (y)$ , and $Z = ρ (z)$ . Since $ρ$ preserves brackets, the left side of the invariance identity is $tr ((Z X - X Z) Y) + tr (X (Z Y - Y Z))$ . After expansion, the middle terms cancel directly and [L2] gives $tr (Z X Y) = tr (X Y Z)$ , so the remaining terms cancel as well.

2.1L1step 1.11.2∎

The Killing form is the trace form for $ρ = ad$ , so steps 1.1–1.2 apply verbatim. The zero representation and zero-dimensional space cause no exception: every displayed trace is then zero.

Depends on

Used by

Arakawa-Suzuki Casimir operators on tensor functors Definition
Cayley transform of a theta-stable Cartan subalgebra Definition
Standard parabolic and Levi subalgebras from a subset of simple roots Definition
The type-A nilpotent pyramid, good grading and Whittaker character Definition
Chevalley basis and real structure constants Lemma
Parabolic semi-regular duality and the reverse-column label map Lemma
The Casimir operator is basis-independent and intertwining Lemma
The Killing length of a root is nonzero Lemma
Trace forms of faithful representations of semisimple Lie algebras are nondegenerate Lemma
Orthogonality of root spaces and nondegeneracy on the Cartan subalgebra Proposition
The bracket of opposite root spaces is the root line Proposition
Cartan subalgebras are exactly maximal toral subalgebras Theorem
Cayley transforms connect theta-stable Cartans in the classification Theorem
Classification of real forms by Vogan diagrams 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
Every real Cartan subalgebra is conjugate to a theta-stable one Theorem
Existence and uniqueness up to isomorphism of the split real form Theorem
Existence of a Cartan involution Theorem
Existence of Cartan subalgebras Theorem
Global Cartan decomposition for a connected finite center semisimple Lie group Theorem
Maximal abelian subspaces of p are conjugate by K Theorem
Restricted root space decomposition Theorem
Semisimple compact groups up to isogeny Theorem
The Casimir and place-permutation operators satisfy the degenerate affine Hecke relations Theorem
The root sl₂ triple Theorem
Vogan and Satake diagrams give equivalent real form classifications Theorem
Weyl's complete reducibility theorem Theorem

Dependency tree · two levels

8 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

Kirillov, An Introduction to Lie Groups and Lie Algebras, Lemma 6.1 (standard reference, not scraped)