nLab
orthogonal Lie algebra

Idea

The orthogonal Lie algebra $𝔬$ is the Lie algebra of the orthogonal group $O$.

The special orthogonal Lie algebra is the Lie algebra $\mathrm{𝔰𝔬}$ of the special orthogonal group $\mathrm{SO}$.

Since the two Lie groups differ by an discrete group ${\mathbb{Z}}_2$, these two Lie algebras coincide; we traditionally write $\mathrm{𝔰𝔬}$ instead of $𝔬$.

Properties

In the classification of Lie algebras

For $n>2$, $\mathrm{𝔰𝔬}(n)$ is a simple Lie algebra, either ${𝔡}_{n/2}$ when $n$ is even or ${𝔟}_{(n-1)/2}$ when $n$ is odd. For $n=2$, $\mathrm{𝔰𝔬}(n)$ is the line, an abelian Lie algebra, which is also a simple object in LieAlg but is not counted as a simple Lie algebra. For $n<2$, $\mathrm{𝔰𝔬}(n)$ is the trivial Lie algebra (which is too simple to be simple by any standard).

Relation to Clifford algebra

For $V$ an inner product space, the special orthogonal Lie algebra on $V$ is naturally isomorphism to the algebra of bivectors in the Clifford algebra $\mathrm{Cl}(V)$ under the Clifford commutator bracket.

Related Lie concepts

• special orthogonal Lie algebra , unitary Lie algebra, Poincare Lie algebra

• string Lie 2-algebra

• fivebrane Lie 6-algebra