nLab orthogonal Lie algebra

Idea

The orthogonal Lie algebra $\mathfrak{o}$ is the Lie algebra of the orthogonal group $O$.

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

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

Properties

In the classification of Lie algebras

For $n \gt 2$, except for $n = 4$, where $\mathfrak{so}(4)\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)$, $\mathfrak{so}(n)$ is a simple Lie algebra, either $\mathfrak{d}_{n/2}$ when $n$ is even or $\mathfrak{b}_{(n-1)/2}$ when $n$ is odd. For $n = 2$, $\mathfrak{so}(n)$ 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 $n \lt 2$, $\mathfrak{so}(n)$ 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 $Cl(V)$ under the Clifford commutator bracket.

Related concepts

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

• special linear Lie algebra

• symplectic Lie algebra

• string Lie 2-algebra

• fivebrane Lie 6-algebra

• orthosymplectic super Lie algebra