Over The Complex Number Field
Over the field C of complex numbers, O(n, C) and SO(n, C) are complex Lie groups of dimension n(n − 1)/2 over C (which means the dimension over R is twice that). O(n, C) has two connected components, and SO(n, C) is the connected component containing the identity matrix. For n ≥ 2 these groups are noncompact.
Just as in the real case SO(n, C) is not simply connected. For n > 2 the fundamental group of SO(n, C) is cyclic of order 2 whereas the fundamental group of SO(2, C) is infinite cyclic.
The complex Lie algebra associated to O(n, C) and SO(n, C) consists of the skew-symmetric complex n × n matrices, with the Lie bracket given by the commutator.
Read more about this topic: Orthogonal Group
Famous quotes containing the words over the, complex, number and/or field:
“And we wont be back till its over over there.”
—George M. Cohan (18781942)
“The money complex is the demonic, and the demonic is Gods ape; the money complex is therefore the heir to and substitute for the religious complex, an attempt to find God in things.”
—Norman O. Brown (b. 1913)
“Civilization is maintained by a very few people in a small number of places and we need only some bombs and a few prisons to blot it out altogether.”
—Cyril Connolly (19031974)
“She is as in a field a silken tent
At midday when a sunny summer breeze
Has dried the dew and all its ropes relent,”
—Robert Frost (18741963)