Maximal-rank Lie subalgebra

Definition

Let $\mathfrak g$ be a finite-dimensional complex reductive Lie algebra, and let $\mathfrak k\subseteq\mathfrak g$ be reductive in $\mathfrak g$ , meaning that its adjoint action on $\mathfrak g$ is completely reducible. The subalgebra $\mathfrak k$ has maximal rank in $\mathfrak g$ if

\operatorname{rank}\mathfrak k=\operatorname{rank}\mathfrak g,

where the rank of a reductive Lie algebra is the dimension of any of its Cartan subalgebras.

Equivalently, some Cartan subalgebra of $\mathfrak k$ is a Cartan subalgebra of $\mathfrak g$ . After conjugating the embedding, one may therefore choose a common Cartan subalgebra

\mathfrak h\subseteq\mathfrak k\subseteq\mathfrak g.

Root-system description

Relative to a common $\mathfrak h$ , a maximal-rank reductive subalgebra is regular, and its roots form a root subsystem of the roots of $\mathfrak g$ . If $\mathfrak k$ is semisimple, this subsystem has full rank. In general its roots can span a smaller space, with the remaining Cartan directions belonging to the center of $\mathfrak k$ . Every Levi subalgebra is of maximal rank because it retains the whole ambient Cartan.

There are also maximal-rank semisimple subalgebras that are not Levi subalgebras. A standard exceptional example is

\mathfrak{sl}_2\oplus\mathfrak{so}_{12}\subset\mathfrak e_7,

whose ranks add to $1+6=7$ .

Distinction from maximality

“Maximal rank” concerns the size of a Cartan subalgebra. It does not assert that $\mathfrak k$ is a maximal proper Lie subalgebra. Conversely, a maximal proper subalgebra can have smaller rank than its ambient algebra.

References

Eugene B. Dynkin, “Semisimple subalgebras of semisimple Lie algebras,” Matematicheskii Sbornik 30(72), no. 2 (1952), 349–462; English translation, AMS Translations, Series 2, vol. 6 (1957), 111–244. Journal record.
John C. Baez, “Three Generations in $E_7$ ,” 2026, §6. arXiv record.