Rank of a semisimple Lie algebra

Definition

Let $\mathfrak g$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero. Its rank is

\operatorname{rank}\mathfrak g=\dim\mathfrak h,

where $\mathfrak h\subseteq\mathfrak g$ is any Cartan subalgebra. All Cartan subalgebras of $\mathfrak g$ are conjugate, so this dimension is independent of the choice of $\mathfrak h$ . It is also the rank of the associated root system.

Examples

The classical complex Lie algebras satisfy

\operatorname{rank}\mathfrak{sl}_n(\mathbb C)=n-1, \qquad \operatorname{rank}\mathfrak{so}_{2n}(\mathbb C)=n.

For $\mathfrak{sl}_n$ , the diagonal trace-zero matrices form a Cartan subalgebra. The exceptional complex Lie algebras of types $E_6,E_7,E_8,F_4$ , and $G_2$ have ranks $6,7,8,4$ , and $2$ , respectively.

Relation to rank in other settings

For a compact connected Lie group $G$ , the rank of $G$ is the dimension of a maximal torus. Its Lie algebra has the same rank after complexification. For a general real reductive Lie algebra, complex rank and real rank are different invariants: complex rank is the rank of its complexification, whereas real rank is the dimension of a maximal abelian subspace in the noncompact part of a Cartan decomposition. The unqualified definition above is the complex semisimple one used in root-system arguments.

Why rank matters for subalgebras

A subalgebra has maximal rank when it contains a Cartan subalgebra of the ambient semisimple algebra. This is stronger than merely having large dimension. It permits both algebras to be described using roots in one common Cartan subalgebra and underlies root-removal constructions.

References

James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972, §§8–10. Publisher record.
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002, Chapters IV–V. Publisher record.