Classification Theorem for Simple Lie Algebras

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero. Then $\mathfrak{g}$ is isomorphic to exactly one of the following:

Classical types:

• $A_n$ ($n \geq 1$): $\mathfrak{sl}_{n+1}$, dimension $(n+1)^2-1$
• $B_n$ ($n \geq 2$): $\mathfrak{so}_{2n+1}$, dimension $2n^2+n$
• $C_n$ ($n \geq 3$): $\mathfrak{sp}_{2n}$, dimension $2n^2+n$
• $D_n$ ($n \geq 4$): $\mathfrak{so}_{2n}$, dimension $2n^2-n$

Exceptional types:

• $G_2$: dimension 14, rank 2
• $F_4$: dimension 52, rank 4
• $E_6$: dimension 78, rank 6
• $E_7$: dimension 133, rank 7
• $E_8$: dimension 248, rank 8

This list is complete and exhaustive.

Existence and Uniqueness

For each Dynkin diagram in the classification list:

1. There exists a simple Lie algebra with that diagram (existence)
2. Any two simple algebras with the same diagram are isomorphic (uniqueness)
3. Algebras with different diagrams are non-isomorphic (completeness)

Serre's Theorem (Presentation)

Every simple Lie algebra can be presented via generators and relations determined solely by its Cartan matrix. Given Cartan matrix $A = (a_{ij})$, the algebra is generated by $\{e_i, f_i, h_i : i = 1, \ldots, n\}$ subject to:

1. $[h_i, h_j] = 0$
2. $[e_i, f_j] = \delta_{ij} h_i$
3. $[h_i, e_j] = a_{ij} e_j$, $[h_i, f_j] = -a_{ij} f_j$
4. $(\text{ad}_{e_i})^{1-a_{ij}}(e_j) = 0$ for $i \neq j$
5. $(\text{ad}_{f_i})^{1-a_{ij}}(f_j) = 0$ for $i \neq j$

This gives an explicit construction from combinatorial data.

Root System Correspondence

The map $\mathfrak{g} \mapsto \Phi_\mathfrak{g}$ (Lie algebra to its root system) gives a bijection between isomorphism classes of simple Lie algebras and isomorphism classes of irreducible reduced root systems.