Orthogonal Lie algebra

The Lie algebra of the orthogonal group: skew-symmetric endomorphisms (or their indefinite analogues).

Orthogonal Lie algebra

Definition (Euclidean signature)

The orthogonal Lie algebra $\mathfrak{so}(n)$ is the Lie algebra of the orthogonal group $O(n)$ . Concretely,

\mathfrak{so}(n)=\{X\in \mathfrak{gl}(n,\Bbb R)\mid X^T+X=0\},

with Lie bracket given by the commutator $[X,Y]=XY-YX$ (the standard bracket on matrix Lie algebras). Here $\mathfrak{gl}(n,\Bbb R)$ is the general linear Lie algebra .

A standard basis is given by $E_{ij}-E_{ji}$ for $1\le i<j\le n$ , so

\dim \mathfrak{so}(n)=\frac{n(n-1)}{2}.

Indefinite signature

If $\eta$ is a symmetric, invertible matrix of signature $(p,q)$ , define

\mathfrak{so}(p,q)=\{X\in \mathfrak{gl}(n,\Bbb R)\mid X^T\eta+\eta X=0\}, \quad n=p+q.

This is the Lie algebra of $O(p,q)$ . In particular, the Lie algebra of the Lorentz group $O(1,n-1)$ is $\mathfrak{so}(1,n-1)$ .

Context

Orthogonal Lie algebras are basic examples of classical semisimple Lie algebras and play a central role in the classification via Dynkin diagrams and root data.