orthogonal Lie algebra
Formal Lie groupoids
The orthogonal Lie algebra is the Lie algebra of the orthogonal group .
The special orthogonal Lie algebra is the Lie algebra of the special orthogonal group .
Since the two Lie groups differ by an discrete group , these two Lie algebras coincide; we traditionally write instead of .
In the classification of Lie algebras
For , is a simple Lie algebra, either when is even or when is odd. For , 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 , is the trivial Lie algebra (which is too simple to be simple by any standard).
Relation to Clifford algebra
For an inner product space, the special orthogonal Lie algebra on is naturally isomorphism to the algebra of bivectors in the Clifford algebra under the Clifford commutator bracket.
Revised on October 28, 2011 00:46:57
by Urs Schreiber