Mutual centralizers in a Lie algebra

Definition

Two Lie subalgebras $\mathfrak a,\mathfrak b\subseteq\mathfrak g$ are mutual centralizers in $\mathfrak g$ if

C_{\mathfrak g}(\mathfrak a)=\mathfrak b \qquad\text{and}\qquad C_{\mathfrak g}(\mathfrak b)=\mathfrak a,

where $C_{\mathfrak g}(-)$ denotes the Lie-algebra centralizer.

Consequences

Mutual centralizers commute elementwise:

[\mathfrak a,\mathfrak b]=0.

They also satisfy the double-centralizer property. Consequently, each is determined inside $\mathfrak g$ by the other.

The converse fails: the condition $[\mathfrak a,\mathfrak b]=0$ gives only

\mathfrak a\subseteq C_{\mathfrak g}(\mathfrak b), \qquad \mathfrak b\subseteq C_{\mathfrak g}(\mathfrak a),

and either inclusion may be strict.

Their intersection is central in both:

\mathfrak a\cap\mathfrak b \subseteq Z(\mathfrak a)\cap Z(\mathfrak b).

If both subalgebras are semisimple, their centers vanish, so the intersection is zero and $\mathfrak a+\mathfrak b\cong\mathfrak a\oplus\mathfrak b$ .

Exceptional examples

Inside the complex exceptional Lie algebra $\mathfrak e_7$ , there are embeddings for which

\mathfrak{sl}_2\quad\text{and}\quad\mathfrak{so}_{12}

are mutual centralizers. Another pair is

\mathfrak{sl}_3\quad\text{and}\quad\mathfrak{sl}_6.

Their direct sums are maximal-rank subalgebras of $\mathfrak e_7$ ; the second is also a maximal proper subalgebra. Such pairs are often called reductive dual pairs when both members are reductive, though usage of that term can impose further hypotheses in representation theory.

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, §§6–7. arXiv record.