Centralizer of a Lie subalgebra

Definition

Let $\mathfrak k\subseteq\mathfrak g$ be a Lie subalgebra. The centralizer of $\mathfrak k$ in $\mathfrak g$ is

C_{\mathfrak g}(\mathfrak k) =\{x\in\mathfrak g:[x,y]=0\text{ for every }y\in\mathfrak k\}.

It is a Lie subalgebra of $\mathfrak g$ .

Basic properties

For subalgebras $\mathfrak a\subseteq\mathfrak b\subseteq\mathfrak g$ , centralizers reverse inclusion:

C_{\mathfrak g}(\mathfrak b)\subseteq C_{\mathfrak g}(\mathfrak a).

Moreover,

Z(\mathfrak g)\subseteq C_{\mathfrak g}(\mathfrak k), \qquad Z(\mathfrak k)=\mathfrak k\cap C_{\mathfrak g}(\mathfrak k), \qquad \mathfrak k\subseteq C_{\mathfrak g}\!\left(C_{\mathfrak g}(\mathfrak k)\right).

The last containment need not be equality; equality is a double-centralizer property.

Centralizer versus normalizer

The centralizer requires $[x,\mathfrak k]=0$ . The normalizer

N_{\mathfrak g}(\mathfrak k) =\{x\in\mathfrak g:[x,\mathfrak k]\subseteq\mathfrak k\}

only requires $x$ to preserve $\mathfrak k$ under the adjoint action. Hence

C_{\mathfrak g}(\mathfrak k)\subseteq N_{\mathfrak g}(\mathfrak k).

Matrix interpretation

If $\mathfrak g$ is a matrix Lie algebra, then $C_{\mathfrak g}(\mathfrak k)$ consists of the matrices in $\mathfrak g$ commuting, in the ordinary matrix sense, with every element of $\mathfrak k$ . For regular reductive subalgebras of a semisimple algebra, the centralizer can often be read directly from roots.

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002, Chapters I–II. Publisher record.
Arkady L. Onishchik and Ernest B. Vinberg, eds., Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras, Springer, 1994, Chapter 3. Publisher record.