Solvable and Semisimple Algebras - Core Definitions

The structure theory of Lie algebras centers on understanding their decomposition into simpler pieces. The concepts of solvability and semisimplicity provide the fundamental classification framework for this theory.

Definition

The derived series of a Lie algebra $\mathfrak{g}$ is the sequence: $\mathfrak{g}^{(0)} = \mathfrak{g}, \quad \mathfrak{g}^{(k+1)} = [\mathfrak{g}^{(k)}, \mathfrak{g}^{(k)}]$ A Lie algebra is solvable if $\mathfrak{g}^{(n)} = 0$ for some $n \geq 0$ .

Solvable Lie algebras generalize the notion of abelian algebras (where $[\mathfrak{g}, \mathfrak{g}] = 0$ ). The derived series measures how far the algebra is from being abelian by repeatedly taking commutators.

Definition

The lower central series of $\mathfrak{g}$ is defined by: $\mathfrak{g}^0 = \mathfrak{g}, \quad \mathfrak{g}^{k+1} = [\mathfrak{g}, \mathfrak{g}^k]$ A Lie algebra is nilpotent if $\mathfrak{g}^n = 0$ for some $n \geq 0$ .

Nilpotent Lie algebras form a subclass of solvable algebras. Every nilpotent algebra is solvable, but not conversely. The lower central series descends faster than the derived series.

Example

The algebra of strictly upper triangular $n \times n$ matrices is nilpotent. The algebra of all upper triangular matrices (with zero trace for $\mathfrak{sl}_n$ ) is solvable but not nilpotent for $n \geq 3$ .

Definition

A Lie algebra $\mathfrak{g}$ is simple if $\dim \mathfrak{g} > 1$ and $\mathfrak{g}$ has no proper ideals (nontrivial normal subalgebras). A Lie algebra is semisimple if it is a direct sum of simple Lie algebras.

Simple Lie algebras are the "atoms" of Lie theory—they cannot be broken down further into smaller pieces via ideal decomposition. Semisimple algebras are built entirely from these atoms.

Remark

An equivalent characterization: $\mathfrak{g}$ is semisimple if and only if it has no nonzero abelian ideals. This is often more convenient for checking semisimplicity than decomposing into simple factors.

Definition

The radical of a Lie algebra $\mathfrak{g}$ , denoted $\text{Rad}(\mathfrak{g})$ , is the maximal solvable ideal of $\mathfrak{g}$ . A Lie algebra is semisimple if and only if $\text{Rad}(\mathfrak{g}) = 0$ .

The radical always exists: it is the sum of all solvable ideals, which is itself solvable. This provides a systematic way to detect and remove the solvable part of any Lie algebra.

Example

For $\mathfrak{gl}_n(\mathbb{C})$ , the radical consists of scalar matrices $\mathbb{C} \cdot I$ . The quotient $\mathfrak{gl}_n(\mathbb{C})/\mathbb{C} \cdot I \cong \mathfrak{sl}_n(\mathbb{C})$ is semisimple.

The Killing form provides a powerful tool for detecting semisimplicity. For $X, Y \in \mathfrak{g}$ : $\kappa(X, Y) = \text{tr}(\text{ad}_X \circ \text{ad}_Y)$ This symmetric bilinear form encodes crucial structural information. Cartan's criterion states that $\mathfrak{g}$ is semisimple if and only if $\kappa$ is non-degenerate.

Remark

The dichotomy between solvable and semisimple algebras is fundamental. The Levi decomposition theorem states that every Lie algebra splits as a semidirect product of its radical (solvable part) and a semisimple subalgebra. This reduces the study of general Lie algebras to understanding solvable and semisimple algebras separately.