Lie Algebras - Key Properties

Lie algebras possess rich structural properties that govern their classification and representation theory. Understanding these properties is essential for working with both the abstract theory and concrete applications.

Definition

A Lie subalgebra of a Lie algebra $\mathfrak{g}$ is a vector subspace $\mathfrak{h} \subseteq \mathfrak{g}$ that is closed under the Lie bracket: $[X, Y] \in \mathfrak{h}$ for all $X, Y \in \mathfrak{h}$ . An ideal is a subalgebra $\mathfrak{i}$ satisfying $[X, Y] \in \mathfrak{i}$ for all $X \in \mathfrak{g}$ and $Y \in \mathfrak{i}$ .

Ideals are precisely the kernels of Lie algebra homomorphisms, allowing quotient constructions. If $\mathfrak{i}$ is an ideal in $\mathfrak{g}$ , then $\mathfrak{g}/\mathfrak{i}$ inherits a Lie algebra structure via $[X + \mathfrak{i}, Y + \mathfrak{i}] = [X, Y] + \mathfrak{i}$ .

Definition

A Lie algebra $\mathfrak{g}$ is simple if $\dim \mathfrak{g} > 1$ and $\mathfrak{g}$ has no proper ideals. A Lie algebra is semisimple if it is a direct sum of simple Lie algebras, equivalently, if it has no nonzero abelian ideals.

Example

The Lie algebra $\mathfrak{sl}_n(\mathbb{C})$ is simple for $n \geq 2$ . The only ideals are $\{0\}$ and $\mathfrak{sl}_n(\mathbb{C})$ itself. In contrast, $\mathfrak{gl}_n(\mathbb{C})$ is not semisimple due to the ideal of scalar matrices (which is the center).

The center of a Lie algebra $\mathfrak{g}$ is defined as: $Z(\mathfrak{g}) = \{X \in \mathfrak{g} : [X, Y] = 0 \text{ for all } Y \in \mathfrak{g}\}$ A Lie algebra with $Z(\mathfrak{g}) = 0$ is called centerless. Simple Lie algebras are necessarily centerless (except for the $1$ -dimensional algebra).

Remark

The derived series of $\mathfrak{g}$ is defined recursively: $\mathfrak{g}^{(0)} = \mathfrak{g}$ , $\mathfrak{g}^{(k+1)} = [\mathfrak{g}^{(k)}, \mathfrak{g}^{(k)}]$ . If $\mathfrak{g}^{(k)} = 0$ for some $k$ , we say $\mathfrak{g}$ is solvable. The Lie algebra is nilpotent if the lower central series $\mathfrak{g}^0 = \mathfrak{g}$ , $\mathfrak{g}^{k+1} = [\mathfrak{g}, \mathfrak{g}^k]$ terminates in zero.

Cartan's criterion provides a powerful test for semisimplicity: a Lie algebra $\mathfrak{g}$ over a field of characteristic zero is semisimple if and only if the Killing form $\kappa(X, Y) = \text{tr}(\text{ad}_X \circ \text{ad}_Y)$ is non-degenerate.

Example

For $\mathfrak{sl}_n(\mathbb{C})$ , the Killing form is $\kappa(X, Y) = 2n \cdot \text{tr}(XY)$ . This is non-degenerate, confirming that $\mathfrak{sl}_n(\mathbb{C})$ is semisimple.

The Levi decomposition states that any finite-dimensional Lie algebra $\mathfrak{g}$ can be written as $\mathfrak{g} = \mathfrak{s} \ltimes \mathfrak{r}$ , where $\mathfrak{r}$ is the maximal solvable ideal (the radical) and $\mathfrak{s}$ is a semisimple subalgebra. This reduces the study of general Lie algebras to understanding semisimple algebras and solvable algebras separately.

Representations provide another viewpoint. A representation of $\mathfrak{g}$ is a Lie algebra homomorphism $\rho: \mathfrak{g} \to \mathfrak{gl}(V)$ for some vector space $V$ . The adjoint representation $\text{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})$ is always available and plays a central role in structure theory.

For semisimple Lie algebras, Weyl's theorem guarantees that every finite-dimensional representation is completely reducible: it decomposes as a direct sum of irreducible representations. This beautiful property underlies much of the representation theory of semisimple Lie algebras and their associated Lie groups.