Solvable and Semisimple Algebras - Examples and Constructions

Concrete examples illuminate the abstract theory of solvable and semisimple Lie algebras. These examples demonstrate how classification theorems work in practice and provide computational tools.

Example

Borel subalgebras of $\mathfrak{sl}_n$ :

The upper triangular matrices in $\mathfrak{sl}_n(\mathbb{C})$ form a maximal solvable subalgebra: $\mathfrak{b}_n = \left\{\begin{pmatrix} h_1 & * & \cdots & * \\ 0 & h_2 & \cdots & * \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & h_n \end{pmatrix} : h_1 + \cdots + h_n = 0\right\}$

This is called a Borel subalgebra. For semisimple $\mathfrak{g}$ , Borel subalgebras are maximal solvable and all conjugate.

Definition

For a semisimple Lie algebra $\mathfrak{g}$ , a Borel subalgebra $\mathfrak{b}$ is a maximal solvable subalgebra. The quotient $G/B$ (for corresponding Lie group) is called a flag variety, fundamental in algebraic geometry.

Example

Classical semisimple algebras:

$\mathfrak{sl}_n(\mathbb{C})$ - type $A_{n-1}$ , dimension $n^2-1$ , rank $n-1$
$\mathfrak{so}_{2n+1}(\mathbb{C})$ - type $B_n$ , dimension $n(2n+1)$ , rank $n$
$\mathfrak{sp}_{2n}(\mathbb{C})$ - type $C_n$ , dimension $n(2n+1)$ , rank $n$
$\mathfrak{so}_{2n}(\mathbb{C})$ - type $D_n$ , dimension $n(2n-1)$ , rank $n$

These are the four infinite families of simple complex Lie algebras.

Remark

Beyond the classical series, there are exactly five exceptional simple Lie algebras: $\mathfrak{g}_2, \mathfrak{f}_4, \mathfrak{e}_6, \mathfrak{e}_7, \mathfrak{e}_8$ . Together with the classical series, these give a complete classification of simple Lie algebras over $\mathbb{C}$ .

Example

The 3-dimensional non-abelian solvable algebra: $\mathfrak{g} = \langle X, Y, Z : [X, Y] = Z, [X, Z] = [Y, Z] = 0\rangle$ This is the Lie algebra of the Heisenberg group. It has derived series: $\mathfrak{g}^{(0)} = \mathfrak{g}$ , $\mathfrak{g}^{(1)} = \langle Z\rangle$ , $\mathfrak{g}^{(2)} = 0$ , so it's solvable. It's also nilpotent with lower central series terminating at step 2.

Levi decomposition: Any Lie algebra $\mathfrak{g}$ admits a decomposition: $\mathfrak{g} = \mathfrak{s} \ltimes \text{Rad}(\mathfrak{g})$ where $\mathfrak{s}$ is semisimple (called a Levi subalgebra) and $\text{Rad}(\mathfrak{g})$ is the radical. While the radical is canonical, the Levi subalgebra is only unique up to conjugation.

Example

For $\mathfrak{gl}_n(\mathbb{C})$ : $\mathfrak{gl}_n(\mathbb{C}) = \mathfrak{sl}_n(\mathbb{C}) \oplus \mathbb{C} \cdot I$ where $\mathfrak{sl}_n(\mathbb{C})$ is the Levi factor (semisimple) and $\mathbb{C} \cdot I$ is the radical (abelian, hence solvable).

Parabolic subalgebras: For semisimple $\mathfrak{g}$ , a subalgebra $\mathfrak{p}$ containing a Borel subalgebra is called parabolic. These arise as stabilizers of partial flags and play a key role in representation theory and geometric constructions.

Definition

A reductive Lie algebra is one whose radical is the center: $\text{Rad}(\mathfrak{g}) = Z(\mathfrak{g})$ . Equivalently, $\mathfrak{g} = \mathfrak{s} \oplus Z(\mathfrak{g})$ for semisimple $\mathfrak{s}$ .

Example

The algebra $\mathfrak{gl}_n(\mathbb{C})$ is reductive but not semisimple. The algebra $\mathfrak{sl}_n(\mathbb{C}) \oplus \mathfrak{sl}_m(\mathbb{C})$ is both semisimple and reductive.

These examples show how the solvable/semisimple dichotomy pervades Lie theory, with most interesting structure concentrated in the semisimple case.