Compatible Cartan subalgebras

Definition

For an inclusion $\mathfrak k\subseteq\mathfrak g$ of finite-dimensional Lie algebras, Cartan subalgebras $\mathfrak t\subseteq\mathfrak k$ and $\mathfrak h\subseteq\mathfrak g$ are compatible with the inclusion if

\mathfrak t=\mathfrak k\cap\mathfrak h.

In the common maximal-rank case, one can take $\mathfrak t=\mathfrak h\subseteq\mathfrak k$ , so the two algebras use the same Cartan subalgebra.

What compatibility supplies

Compatible Cartans let the root-space decomposition of the subalgebra be compared directly with that of the ambient algebra. If $\mathfrak k$ is a regular subalgebra reductive in $\mathfrak g$ and normalized by $\mathfrak h$ , then $\mathfrak k$ is assembled from a subspace of $\mathfrak h$ and selected $\mathfrak g$ -root spaces. Its roots can therefore be recorded as a root subsystem of the ambient root system, subject to the relevant span and closure conditions.

For inclusions that are not of maximal rank, roots of $\mathfrak g$ restrict from $\mathfrak h^*$ to $\mathfrak t^*$ . Several ambient roots may have the same nonzero restriction, and some may restrict to zero. Compatibility alone does not make the smaller root system a literal subset of the larger one.

Scope and terminology

“Compatible” describes a choice made for a particular inclusion; it is not an additional intrinsic structure on either algebra. Authors sometimes use the term more loosely for nested Cartans $\mathfrak t\subseteq\mathfrak h$ . Stating the intersection equation removes that ambiguity. Existence and conjugacy statements depend on hypotheses such as reductivity, regularity, and the ground field, and should not be inferred from the definition alone.

References

Eugene B. Dynkin, “Semisimple subalgebras of semisimple Lie algebras,” American Mathematical Society Translations, Series 2, vol. 6, 1957, pp. 111–244.
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002, Chapters IV–VI. Publisher record.