Classification of Simple Lie Algebras - Core Definitions

The classification of simple Lie algebras is a monumental achievement reducing all semisimple structures to a finite list. The key insight is that simple Lie algebras correspond bijectively to irreducible root systems, which are classified by Dynkin diagrams.

Definition

A Lie algebra $\mathfrak{g}$ over an algebraically closed field of characteristic zero is simple if:

$\dim \mathfrak{g} > 1$
The only ideals of $\mathfrak{g}$ are $\{0\}$ and $\mathfrak{g}$ itself

A Lie algebra is semisimple if it is a direct sum of simple Lie algebras.

Every semisimple Lie algebra decomposes uniquely (up to order) into simple factors, so classification reduces to classifying simple algebras.

Theorem

Bijection Theorem

There is a bijection between:

Isomorphism classes of simple Lie algebras over $\mathbb{C}$
Isomorphism classes of irreducible root systems
Connected Dynkin diagrams

This establishes a purely combinatorial classification of simple Lie algebras.

Definition

The Cartan matrix of a simple Lie algebra with simple roots $\{\alpha_1, \ldots, \alpha_n\}$ is: $A = (a_{ij}) \quad \text{where} \quad a_{ij} = \frac{2\langle \alpha_j, \alpha_i\rangle}{\langle \alpha_i, \alpha_i\rangle}$

The Cartan matrix is:

Integer-valued
$a_{ii} = 2$
$a_{ij} \leq 0$ for $i \neq j$
Positive definite (for finite-dimensional algebras)

Example

For $\mathfrak{sl}_3(\mathbb{C})$ with simple roots $\alpha_1, \alpha_2$ at 120° angle: $A = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix}$ This is the Cartan matrix of type $A_2$ .

Definition

The Dynkin diagram is a graph encoding the Cartan matrix:

One node per simple root
Nodes $i, j$ connected by $a_{ij}a_{ji}$ edges
Arrow points from longer to shorter root if lengths differ

Dynkin diagrams provide visual classification: connected diagrams correspond to simple algebras.

Remark

The classification list consists of four infinite families ( $A_n, B_n, C_n, D_n$ ) and five exceptional types ( $G_2, F_4, E_6, E_7, E_8$ ). This finite list contains all possible simple Lie algebras over $\mathbb{C}$ , a remarkable finiteness result.

Isogeny classes: Over $\mathbb{C}$ , simple Lie algebras are completely determined by their root system. However, the corresponding Lie groups may differ by coverings (central extensions). For instance, $\mathfrak{sl}_n$ arises from both $SL_n$ and $PSL_n = SL_n/\mu_n$ .

Definition

The rank of a semisimple Lie algebra is the dimension of a Cartan subalgebra, equivalently the number of simple roots. For simple algebras, the rank determines the subscript in the classification (e.g., $E_8$ has rank 8).

Example

Classical simple Lie algebras and their ranks:

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

The classification provides explicit models for all simple Lie algebras, enabling concrete computations in representation theory and applications.