Lie Algebras - Core Definitions

A Lie algebra is a vector space equipped with a bilinear bracket operation satisfying specific algebraic identities. Lie algebras arise naturally as the tangent space at the identity of a Lie group, encoding the infinitesimal structure of the group.

Definition

A Lie algebra over a field $\mathbb{F}$ (typically $\mathbb{R}$ or $\mathbb{C}$ ) is a vector space $\mathfrak{g}$ together with a bilinear map $[\cdot, \cdot]: \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$ , called the Lie bracket, satisfying:

Antisymmetry: $[X, Y] = -[Y, X]$ for all $X, Y \in \mathfrak{g}$
Jacobi identity: $[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$ for all $X, Y, Z \in \mathfrak{g}$

The antisymmetry property implies $[X, X] = 0$ for all $X$ . In characteristic zero, the converse also holds: if $[X, X] = 0$ for all $X$ , then the bracket is antisymmetric.

Example

For any associative algebra $A$ over $\mathbb{F}$ , the commutator bracket $[a, b] = ab - ba$ defines a Lie algebra structure on $A$ . In particular, the space $M_n(\mathbb{F})$ of $n \times n$ matrices becomes a Lie algebra under the commutator, denoted $\mathfrak{gl}_n(\mathbb{F})$ .

The connection between Lie groups and Lie algebras comes from considering vector fields on the group that are invariant under group operations.

Definition

Let $G$ be a Lie group. A vector field $X$ on $G$ is left-invariant if $(L_g)_* X = X$ for all $g \in G$ , where $L_g$ denotes left translation by $g$ . The space of all left-invariant vector fields on $G$ forms a Lie algebra called the Lie algebra of $G$ , denoted $\mathfrak{g} = \text{Lie}(G)$ .

A left-invariant vector field is uniquely determined by its value at the identity $e \in G$ , so we have a natural isomorphism $\text{Lie}(G) \cong T_e G$ as vector spaces. The Lie bracket on $\mathfrak{g}$ is defined as the commutator of vector fields: $[X, Y](f) = X(Y(f)) - Y(X(f))$ for functions $f$ on $G$ .

Example

Matrix Lie algebras:

$\mathfrak{gl}_n(\mathbb{R})$ = all $n \times n$ real matrices (dimension $n^2$ )
$\mathfrak{sl}_n(\mathbb{R})$ = traceless matrices (dimension $n^2 - 1$ )
$\mathfrak{so}(n)$ = skew-symmetric matrices: $A^T = -A$ (dimension $\frac{n(n-1)}{2}$ )
$\mathfrak{u}(n)$ = skew-Hermitian matrices: $A^* = -A$ (dimension $n^2$ )
$\mathfrak{su}(n)$ = traceless skew-Hermitian matrices (dimension $n^2 - 1$ )

Remark

The Lie algebra captures the "infinitesimal" structure of the Lie group. Many properties of the group (connectedness, compactness, simplicity) have algebraic counterparts in the Lie algebra. However, the Lie algebra does not determine the global structure: non-isomorphic groups can have isomorphic Lie algebras (e.g., $SO(3)$ and $SU(2)$ ).

The adjoint representation is a fundamental construction. For $X \in \mathfrak{g}$ , the linear map $\text{ad}_X: \mathfrak{g} \to \mathfrak{g}$ defined by $\text{ad}_X(Y) = [X, Y]$ is a derivation of the Lie algebra. The Jacobi identity is equivalent to $\text{ad}_{[X,Y]} = [\text{ad}_X, \text{ad}_Y]$ , showing that $\text{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})$ is a Lie algebra homomorphism.