4.2 The Killing form and Cartan's criterion

The Killing form is a powerful tool in Lie algebra theory. It's a special bilinear form that helps us understand the structure of Lie algebras. By looking at how the Killing form behaves, we can figure out if a Lie algebra is semisimple or not.

Cartan's criterion uses the Killing form to give us a simple way to check for semisimplicity. This is super useful because semisimple Lie algebras are some of the most important ones we study. They're like the building blocks of more complex Lie algebras.

The Killing Form of a Lie Algebra

Definition and Notation

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• The Killing form is a symmetric bilinear form defined on a Lie algebra, commonly denoted as $B(X, Y)$ or $\kappa(X, Y)$
• For a finite-dimensional Lie algebra $\mathfrak{g}$ over a field $F$, the Killing form is defined as $B(X, Y) = \text{tr}(\text{ad}(X) \circ \text{ad}(Y))$, where $X, Y \in \mathfrak{g}$, and $\text{ad}$ represents the adjoint representation of $\mathfrak{g}$
• The trace of the composition of adjoint representations $\text{ad}(X)$ and $\text{ad}(Y)$ quantifies the extent to which the adjoint representation preserves the Lie algebra structure

Properties and Invariance

• The Killing form gauges how closely the adjoint representation resembles a Lie algebra homomorphism
• Lie algebra automorphisms leave the Killing form unchanged, meaning $B(\phi(X), \phi(Y)) = B(X, Y)$ for any automorphism $\phi$ of $\mathfrak{g}$ ($\phi$ is a bijective linear map preserving the Lie bracket)
• This invariance property highlights the intrinsic nature of the Killing form, as it remains unaffected by Lie algebra isomorphisms

Properties of the Killing Form

Bilinearity, Symmetry, and Associativity

• The Killing form is bilinear, satisfying $B(aX + bY, Z) = aB(X, Z) + bB(Y, Z)$ and $B(X, aY + bZ) = aB(X, Y) + bB(X, Z)$ for any $X, Y, Z \in \mathfrak{g}$ and $a, b \in F$ (scalar multiplication and vector addition)
• Symmetry of the Killing form implies $B(X, Y) = B(Y, X)$ for any $X, Y \in \mathfrak{g}$, a property stemming from the trace of the composition of adjoint representations
• The associativity property, $B([X, Y], Z) = B(X, [Y, Z])$ for any $X, Y, Z \in \mathfrak{g}$, relates the Killing form to the Lie bracket (fundamental operation in Lie algebras)

Invariance and Abelian Lie Algebras

• The adjoint representation leaves the Killing form invariant, expressed as $B(\text{ad}(X)(Y), Z) + B(Y, \text{ad}(X)(Z)) = 0$ for any $X, Y, Z \in \mathfrak{g}$
• For abelian Lie algebras, where the Lie bracket is identically zero ($[X, Y] = 0$ for all $X, Y$), the Killing form vanishes entirely ($B(X, Y) = 0$ for all $X, Y$)
• Examples of abelian Lie algebras include the space of diagonal matrices and the Lie algebra of a torus

Cartan's Criterion for Semisimplicity

Statement and Equivalence

• Cartan's criterion asserts that a finite-dimensional Lie algebra $\mathfrak{g}$ over a field of characteristic zero is semisimple if and only if its Killing form is non-degenerate
• A non-degenerate bilinear form has a trivial kernel, meaning $B(X, Y) = 0$ for all $Y \in \mathfrak{g}$ implies $X = 0$
• Semisimplicity is equivalent to the Lie algebra being a direct sum of simple ideals (non-abelian Lie algebras with no non-trivial ideals)

Proof Sketch

• To prove the forward direction, assume $\mathfrak{g}$ is semisimple and use the decomposition into simple ideals, along with the non-degeneracy of the Killing form on simple Lie algebras
• For the converse, assume the Killing form is non-degenerate and show that $\mathfrak{g}$ has no non-zero solvable ideals by leveraging the invariance of the Killing form under the adjoint representation and the vanishing property for solvable Lie algebras
• The absence of non-zero solvable ideals, combined with the finite-dimensionality of $\mathfrak{g}$, implies semisimplicity

Semisimplicity of Lie Algebras

Determining Semisimplicity

• To check if a given Lie algebra is semisimple, calculate its Killing form and verify its non-degeneracy
• For a Lie algebra with a basis $\{X_1, X_2, \ldots, X_n\}$, the matrix of the Killing form is given by $(B(X_i, X_j))_{i,j=1}^n$, and the Lie algebra is semisimple if and only if this matrix is invertible
• The invertibility of the Killing form matrix provides a computational criterion for semisimplicity

Examples

• Semisimple Lie algebras:
  • The special linear algebra $\mathfrak{sl}(n, F)$ ($n \times n$ matrices with trace zero)
  • The orthogonal algebra $\mathfrak{so}(n, F)$ for $n \geq 3$ (skew-symmetric matrices)
  • The symplectic algebra $\mathfrak{sp}(n, F)$ for $n \geq 2$ (matrices preserving a symplectic form)
• Non-semisimple Lie algebras:
  • The Heisenberg algebra (upper triangular matrices with constant diagonal)
  • The Euclidean algebra (isometries of Euclidean space)
  • The Poincaré algebra (isometries of Minkowski spacetime)