Orthogonal Group - Over The Complex Number Field

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.