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 won’t be back till its over over there.”
—George M. Cohan (1878–1942)
“All of life and human relations have become so incomprehensibly complex that, when you think about it, it becomes terrifying and your heart stands still.”
—Anton Pavlovich Chekhov (1860–1904)
“A good marriage ... is a sweet association in life: full of constancy, trust, and an infinite number of useful and solid services and mutual obligations.”
—Michel de Montaigne (1533–1592)
“The planter, who is Man sent out into the field to gather food, is seldom cheered by any idea of the true dignity of his ministry. He sees his bushel and his cart, and nothing beyond, and sinks into the farmer, instead of Man on the farm.”
—Ralph Waldo Emerson (1803–1882)