Lie Algebras and Their Representations

Lie algebra representations provide infinitesimal descriptions of Lie group representations, connecting continuous symmetry to algebraic structures.

DefinitionLie Algebra

A Lie algebra over a field $\mathbb{F}$ is a vector space $\mathfrak{g}$ equipped with a bilinear bracket operation $[\cdot, \cdot]: \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$ satisfying:

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

ExampleClassical Lie Algebras

General linear: $\mathfrak{gl}_n(\mathbb{F}) =$ all $n \times n$ matrices with $[A,B] = AB - BA$

Special linear: $\mathfrak{sl}_n(\mathbb{F}) = \{A \in \mathfrak{gl}_n : \text{tr}(A) = 0\}$

Orthogonal: $\mathfrak{so}_n(\mathbb{F}) = \{A \in \mathfrak{gl}_n : A^T = -A\}$ (skew-symmetric matrices)

Symplectic: $\mathfrak{sp}_{2n}(\mathbb{F})$ preserving a symplectic form

DefinitionRepresentation of a Lie Algebra

A representation of a Lie algebra $\mathfrak{g}$ on a vector space $V$ is a Lie algebra homomorphism: $\rho: \mathfrak{g} \to \mathfrak{gl}(V)$ satisfying $\rho([X, Y]) = [\rho(X), \rho(Y)] = \rho(X)\rho(Y) - \rho(Y)\rho(X)$ for all $X, Y \in \mathfrak{g}$ .

Equivalently, it's a linear map making $V$ into a $\mathfrak{g}$ -module: $X \cdot (Y \cdot v) - Y \cdot (X \cdot v) = [X,Y] \cdot v$ .

DefinitionAdjoint Representation

Every Lie algebra acts on itself via the adjoint representation: $\text{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}), \quad \text{ad}(X)(Y) = [X, Y]$

The Jacobi identity ensures $\text{ad}$ is a Lie algebra homomorphism.

ExampleRepresentations of $\mathfrak{sl}_2$

The Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ has basis: $H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$

with relations $[H,E] = 2E$ , $[H,F] = -2F$ , $[E,F] = H$ .

Irreducible representations are classified by dimension: for each $n \geq 1$ , there exists a unique $(n+1)$ -dimensional irreducible representation $V_n$ .

Remark

Lie algebra representations are the infinitesimal counterparts of Lie group representations. For a Lie group $G$ with Lie algebra $\mathfrak{g} = T_e G$ , representations of $G$ differentiate to representations of $\mathfrak{g}$ . Conversely, for simply-connected $G$ , representations of $\mathfrak{g}$ integrate to representations of $G$ (Lie's third theorem).

This correspondence is fundamental in physics (generators of symmetry groups), differential geometry (connections and curvature), and harmonic analysis (orbit method).