Lie Algebras - Examples and Constructions

Understanding Lie algebras requires familiarity with key examples and construction techniques. These examples illustrate the diversity of Lie algebra structures and provide concrete models for the abstract theory.

Example

The Heisenberg algebra $\mathfrak{h}_n$ is the $(2n+1)$ -dimensional Lie algebra with basis $\{p_1, \ldots, p_n, q_1, \ldots, q_n, z\}$ and bracket relations: $[q_i, p_j] = \delta_{ij} z, \quad [q_i, q_j] = [p_i, p_j] = 0, \quad [z, \cdot] = 0$ This algebra is nilpotent and fundamental in quantum mechanics, encoding the canonical commutation relations.

The classical Lie algebras form four infinite families:

$\mathfrak{sl}_n = \{X \in \mathfrak{gl}_n : \text{tr}(X) = 0\}$ - special linear algebra
$\mathfrak{so}_n = \{X \in \mathfrak{gl}_n : X^T + X = 0\}$ - special orthogonal algebra
$\mathfrak{sp}_{2n} = \{X \in \mathfrak{gl}_{2n} : X^T J + JX = 0\}$ where $J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$ - symplectic algebra
$\mathfrak{su}_n = \{X \in \mathfrak{gl}_n(\mathbb{C}) : X^* + X = 0, \text{tr}(X) = 0\}$ - special unitary algebra

Definition

A Cartan subalgebra $\mathfrak{h}$ of a Lie algebra $\mathfrak{g}$ is a maximal abelian subalgebra such that $\text{ad}_H$ is diagonalizable for all $H \in \mathfrak{h}$ . For semisimple Lie algebras, Cartan subalgebras exist, are conjugate under automorphisms, and have dimension equal to the rank of $\mathfrak{g}$ .

Example

For $\mathfrak{sl}_n(\mathbb{C})$ , a Cartan subalgebra consists of diagonal traceless matrices: $\mathfrak{h} = \left\{\text{diag}(h_1, \ldots, h_n) : h_i \in \mathbb{C}, \sum_{i=1}^n h_i = 0\right\}$ This has dimension $n-1$ , so $\mathfrak{sl}_n(\mathbb{C})$ has rank $n-1$ .

Semidirect products construct new Lie algebras from existing ones. If $\mathfrak{g}$ and $\mathfrak{h}$ are Lie algebras and $\phi: \mathfrak{h} \to \text{Der}(\mathfrak{g})$ is a homomorphism (where $\text{Der}(\mathfrak{g})$ denotes derivations), then $\mathfrak{g} \ltimes_\phi \mathfrak{h}$ has bracket: $[(X_1, Y_1), (X_2, Y_2)] = ([X_1, X_2] + \phi(Y_1)(X_2) - \phi(Y_2)(X_1), [Y_1, Y_2])$

Remark

The Lie algebra cohomology $H^*(\mathfrak{g}, V)$ for a $\mathfrak{g}$ -module $V$ provides powerful invariants. For instance, $H^2(\mathfrak{g}, \mathbb{C})$ classifies central extensions, while Whitehead's lemmas state that $H^1(\mathfrak{g}, V) = H^2(\mathfrak{g}, V) = 0$ for semisimple $\mathfrak{g}$ and finite-dimensional $V$ .

Exceptional Lie algebras complete the classification of simple Lie algebras over $\mathbb{C}$ . These are $\mathfrak{g}_2$ (dimension 14, rank 2), $\mathfrak{f}_4$ (dimension 52, rank 4), $\mathfrak{e}_6$ (dimension 78, rank 6), $\mathfrak{e}_7$ (dimension 133, rank 7), and $\mathfrak{e}_8$ (dimension 248, rank 8).

Example

The Lie algebra $\mathfrak{g}_2$ can be realized as the derivation algebra of the octonions $\mathbb{O}$ , or as the automorphism algebra of a certain 7-dimensional cross product. Its Cartan subalgebra has dimension 2, and its root system consists of 12 roots forming a hexagonal pattern in the plane.

The universal enveloping algebra $U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ is the associative algebra generated by $\mathfrak{g}$ subject to $XY - YX = [X,Y]$ . The Poincaré-Birkhoff-Witt theorem provides an explicit basis for $U(\mathfrak{g})$ , and representations of $\mathfrak{g}$ correspond bijectively to $U(\mathfrak{g})$ -modules.

These constructions and examples demonstrate the interplay between algebraic, geometric, and combinatorial aspects of Lie theory.