Regular Lie subalgebra

Definition

Let $\mathfrak g$ be a finite-dimensional complex semisimple Lie algebra and let $\mathfrak h\subset\mathfrak g$ be a Cartan subalgebra. A Lie subalgebra $\mathfrak k\subseteq\mathfrak g$ is regular relative to $\mathfrak h$ if

[\mathfrak h,\mathfrak k]\subseteq\mathfrak k.

A subalgebra is called regular if it is regular relative to some Cartan subalgebra of $\mathfrak g$ .

Root-space description

Write

\mathfrak g=\mathfrak h\oplus\bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha

for the root-space decomposition. Since the root spaces are the simultaneous weight spaces for $\operatorname{ad}(\mathfrak h)$ , regularity is equivalent to a decomposition

\mathfrak k=(\mathfrak k\cap\mathfrak h) \oplus\bigoplus_{\alpha\in\Psi}\mathfrak g_\alpha

for some subset $\Psi\subseteq\Phi$ , subject to the closure conditions required for the displayed vector space to be a Lie subalgebra.

When $\mathfrak k$ is reductive in $\mathfrak g$ —so that the restricted adjoint representation of $\mathfrak k$ on $\mathfrak g$ is completely reducible— $\Psi$ is symmetric under $\alpha\mapsto-\alpha$ and forms a root subsystem. Its Cartan part contains the corresponding coroots and may be larger than their span; those extra toral directions account for part or all of the center of $\mathfrak k$ . This embedded notion rules out, for example, a one-dimensional subalgebra spanned by a nilpotent root vector, even though that subalgebra is abstractly abelian.

Distinctions

“Regular” means normalized by an ambient Cartan; it does not mean that the subalgebra contains that Cartan.
A reductive regular subalgebra that contains an ambient Cartan has maximal rank, but regular subalgebras can have smaller rank.
Regularity depends on an embedding $\mathfrak k\hookrightarrow\mathfrak g$ , not merely on the abstract isomorphism type of $\mathfrak k$ .

Regular embeddings let root-system calculations replace matrix calculations. This is the mechanism behind deleting simple roots, constructing Levi subalgebras, and computing many centralizers in exceptional Lie algebras.

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, §§2–3. arXiv record.