Solvable and Semisimple Algebras - Key Properties

Understanding the properties of solvable and semisimple Lie algebras is essential for both classification and representation theory. These properties reveal deep connections between algebraic structure and geometric behavior.

Theorem

Lie's Theorem

Let $\mathfrak{g}$ be a solvable Lie algebra over an algebraically closed field of characteristic zero. For any finite-dimensional representation $\rho: \mathfrak{g} \to \mathfrak{gl}(V)$ , there exists a basis of $V$ in which all $\rho(X)$ are simultaneously upper triangular.

Lie's theorem shows that solvable algebras always have common eigenvector flags. This provides a concrete description of all representations of solvable Lie algebras.

Theorem

Cartan's Criterion for Solvability

A Lie algebra $\mathfrak{g}$ is solvable if and only if $\text{tr}(\text{ad}_X \circ \text{ad}_Y) = 0$ for all $X \in [\mathfrak{g}, \mathfrak{g}]$ and $Y \in \mathfrak{g}$ .

This gives a computable test for solvability using only the adjoint representation, without computing the entire derived series.

Remark

For semisimple Lie algebras, the Killing form $\kappa(X, Y) = \text{tr}(\text{ad}_X \circ \text{ad}_Y)$ is non-degenerate. This provides an invariant inner product on $\mathfrak{g}$ that is central to the theory. All structural constants can be expressed using the Killing form.

Theorem

Weyl's Complete Reducibility

Every finite-dimensional representation of a semisimple Lie algebra (over characteristic zero) is completely reducible: it decomposes as a direct sum of irreducible representations.

This is perhaps the most important property of semisimple algebras for representation theory. It means understanding representations reduces to classifying irreducibles.

Example

For $\mathfrak{sl}_2(\mathbb{C})$ , every representation decomposes uniquely as $V = \bigoplus_{n=0}^\infty V_n^{\oplus m_n}$ where $V_n$ is the irreducible of dimension $n+1$ and $m_n$ are multiplicities.

Definition

A Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ is a maximal abelian subalgebra such that $\text{ad}_H$ is diagonalizable for all $H \in \mathfrak{h}$ . For semisimple $\mathfrak{g}$ , all Cartan subalgebras have the same dimension, called the rank of $\mathfrak{g}$ .

Theorem

Properties of Cartan Subalgebras

For semisimple $\mathfrak{g}$ :

All Cartan subalgebras are conjugate under automorphisms of $\mathfrak{g}$
$\dim \mathfrak{h} = \text{rank}(\mathfrak{g})$ is independent of the choice of $\mathfrak{h}$
The restriction of the Killing form to $\mathfrak{h}$ is non-degenerate

Example

For $\mathfrak{sl}_n(\mathbb{C})$ , the diagonal matrices with zero trace form a Cartan subalgebra of dimension $n-1$ . Thus $\text{rank}(\mathfrak{sl}_n) = n-1$ .

Root space decomposition: Fix a Cartan subalgebra $\mathfrak{h}$ of semisimple $\mathfrak{g}$ . For $\alpha \in \mathfrak{h}^*$ , define: $\mathfrak{g}_\alpha = \{X \in \mathfrak{g} : [H, X] = \alpha(H) X \text{ for all } H \in \mathfrak{h}\}$

Then $\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha$ where $\Phi \subset \mathfrak{h}^*$ is the root system. This decomposition is fundamental for classification.

Remark

The properties of semisimple algebras (complete reducibility, non-degenerate Killing form, root space decomposition) make them far more tractable than general Lie algebras. The classification of simple Lie algebras via root systems (Dynkin diagrams) is one of the great achievements of 19th and 20th century mathematics.