nLab
orthogonal Lie algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

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}.

Properties

In the classification of Lie algebras

For n>2n \gt 2, 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.

Revised on October 9, 2013 11:39:14 by Urs Schreiber (89.204.139.155)