Structure and Classification of Semisimple Lie Algebras

The finite-dimensional semisimple Lie algebras over $\mathbb{C}$ are completely classified by a small combinatorial object — a Dynkin diagram — through the intermediate machinery of Cartan subalgebras, root systems, and Cartan matrices. The answer is the celebrated $A$–$B$–$C$–$D$–$E$–$F$–$G$ list. This is the continuous counterpart of the finite simple group classification, and it underlies the representation theory of reps-lie.md. It builds on lie-algebras.md.

Cartan subalgebra and root space decomposition

Let $\mathfrak{g}$ be a complex semisimple Lie algebra (Killing form non-degenerate; see lie-algebras.md). A Cartan subalgebra $\mathfrak{h}$ is a maximal abelian subalgebra of simultaneously diagonalisable (ad-semisimple) elements; its dimension is the rank $r$. Diagonalising the commuting operators $\{{\mathrm{ad}}_H:H\in \mathfrak{h}\}$ decomposes $\mathfrak{g}$ into simultaneous eigenspaces: $$\mathfrak{g}=\mathfrak{h}\oplus \underset{\alpha \in \Phi }{⨁}{\mathfrak{g}}_{\alpha },[H,E_{\alpha }]=\alpha (H)E_{\alpha }.$$ The nonzero linear functionals $\alpha \in {\mathfrak{h}}^{\ast }$ that occur are the roots; the set $\Phi$ is the root system. Each root space ${\mathfrak{g}}_{\alpha }$ is one-dimensional, and the $E_{\alpha }$ are the raising/lowering operators that move between weight spaces in reps-lie.md.

Root systems

Using the Killing form to give ${\mathfrak{h}}^{\ast }$ an inner product, the roots form a finite configuration of vectors with rigid geometry:

• (closure under reflection) $\Phi$ is invariant under the reflections $s_{\alpha }$ in the hyperplanes $\perp \alpha$ — these generate the Weyl group $W$, a finite Coxeter group.
• (integrality) for any roots $\alpha ,\beta$, the Cartan integer $⟨\beta ,{\alpha }^{\vee }⟩=2(\beta ,\alpha )/(\alpha ,\alpha )\in \mathbb{Z}$.
• (rationality of angles) the only possible angles between roots are $90°,120°,135°,150°$ (and $60°$), forcing highly constrained diagrams.

A choice of simple roots $\{{\alpha }_1,\dots ,{\alpha }_r\}$ — a basis such that every root is a non-negative or non-positive integer combination — reduces all this data to the $r\times r$ Cartan matrix $A_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$.

Dynkin diagrams and the classification

The Cartan matrix is encoded pictorially in a Dynkin diagram: one node per simple root, with $⟨{\alpha }_i,{\alpha }_j^{\vee }⟩⟨{\alpha }_j,{\alpha }_i^{\vee }⟩\in \{0,1,2,3\}$ edges between nodes $i,j$ (and an arrow toward the shorter root when lengths differ). Classifying semisimple Lie algebras reduces to classifying the connected admissible diagrams:

Cartan–Killing classification. Every finite-dimensional complex simple Lie algebra is exactly one of:

• four infinite families $A_n({\mathfrak{sl}}_{n+1})$, $B_n({\mathfrak{so}}_{2n+1})$, $C_n({\mathfrak{sp}}_{2n})$, $D_n({\mathfrak{so}}_{2n})$;
• five exceptional algebras $G_2,F_4,E_6,E_7,E_8$. Semisimple algebras are direct sums of these (disconnected diagrams).

The four families are precisely the complexified classical matrix algebras; the exceptions arise from the octonions and stop the list at rank $8$. The accidental isomorphisms of matrix-groups.md — $A_1=B_1=C_1$ ($\mathfrak{su}(2)$), $B_2=C_2$, $A_3=D_3$, $D_2=A_1\times A_1$ — are exactly the coincidences among small-rank diagrams.

Why the classification is so rigid

The whole edifice is forced by two facts about the rank-$1$ subalgebra $\mathfrak{su}(2)$ generated by each root: its representation theory is quantised (spins are half-integers, reps-lie.md), which makes the Cartan integers integers, which makes root systems crystallographic, which allows only finitely many diagrams. Thus the entire classification descends from the quantised spectrum of angular momentum — a striking unity between the smallest example and the general theorem.

From algebra to group

Each complex simple Lie algebra integrates to a simply connected complex Lie group, and has a distinguished compact real form (the compact group of compact-groups.md — $SU(n+1)$, $\mathrm{Spin}(2n+1)$, $\mathrm{Sp}(n)$, $\mathrm{Spin}(2n)$, and the compact exceptionals). Different real forms of the same complex algebra give the different real groups of physics (e.g. $\mathfrak{su}(2)$ vs. $\mathfrak{sl}(2,\mathbb{R})$ vs. $\mathfrak{su}(1,1)$). The representation theory is then uniform across all forms — the highest-weight theory of reps-lie.md.

References

• Humphreys, Introduction to Lie Algebras and Representation Theory, Ch. II–IV.
• Fulton & Harris, Representation Theory, Part III.
• Bourbaki, Lie Groups and Lie Algebras, Ch. 4–6 (root systems).