Classification of complex simple Lie algebras

Theorem (classification). Every finite-dimensional complex simple Lie algebra is isomorphic to exactly one of the Lie algebras in the following families:

Type $A_n$ ( $n\ge 1$ ): $\mathfrak{sl}_{n+1}(\mathbb{C})$ ,
Type $B_n$ ( $n\ge 2$ ): $\mathfrak{so}_{2n+1}(\mathbb{C})$ ,
Type $C_n$ ( $n\ge 3$ ): $\mathfrak{sp}_{2n}(\mathbb{C})$ ,
Type $D_n$ ( $n\ge 4$ ): $\mathfrak{so}_{2n}(\mathbb{C})$ ,
Exceptional types: $E_6,E_7,E_8,F_4,G_2$ .

These ranges avoid duplication: $\mathfrak{sp}_4(\mathbb C)\cong\mathfrak{so}_5(\mathbb C)$ is counted as $B_2$ . The type $C_2$ remains a valid name for that same isomorphism class.

Equivalent characterizations

Equivalently, complex simple Lie algebras are classified by connected Dynkin diagrams, or by indecomposable Cartan matrices satisfying the Cartan axioms.

Semisimple corollary. Every complex semisimple Lie algebra is a direct sum of simple ideals, so its isomorphism type is determined by a (finite) multiset of Dynkin diagram types.

Remarks

Context. The classification proceeds by choosing a Cartan subalgebra $\mathfrak{h}$ , analyzing the associated root system in $\mathfrak{h}^*$ , and encoding the relative geometry of simple roots in the Dynkin diagram. The root-system combinatorics precisely controls the Lie bracket via the root space decomposition.

For selected global Lie groups with these types, see the Lie-group table and its guide to forms and coverings.

References

Pavel Etingof, Lie Groups and Lie Algebras, §23.2, Theorem 23.7; §23.8, Remark 23.18; §24.2. The low-rank identification $B_2\cong C_2$ is stated in Remark 23.18, printed p. 128.

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Classification of complex simple Lie algebras

Types and forms

Global forms and maps

Small ranks and the wider catalogue

References