nLab orthogonal Lie algebra



Lie theory

∞-Lie theory (higher geometry)


Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids



Related topics


\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras



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

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

Since the two Lie groups differ by an discrete group 2\mathbb{Z}_2, these two Lie algebras coincide; we traditionally write 𝔰𝔬\mathfrak{so} instead of 𝔬\mathfrak{o}.


In the classification of Lie algebras

For n>2n \gt 2, except for n=4n = 4, where 𝔰𝔬(4)𝔰𝔬(3)𝔰𝔬(3)\mathfrak{so}(4)\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3), 𝔰𝔬(n)\mathfrak{so}(n) is a simple Lie algebra, either 𝔡 n/2\mathfrak{d}_{n/2} when nn is even or 𝔟 (n1)/2\mathfrak{b}_{(n-1)/2} when nn is odd. For n=2n = 2, 𝔰𝔬(n)\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<2n \lt 2, 𝔰𝔬(n)\mathfrak{so}(n) is the trivial Lie algebra (which is too simple to be simple by any standard).

Relation to Clifford algebra

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

Last revised on December 1, 2023 at 12:04:18. See the history of this page for a list of all contributions to it.