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 complex, number and/or field:
“We must open our eyes and see that modern civilization has become so complex and the lives of civilized men so interwoven with the lives of other men in other countries as to make it impossible to be in this world and out of it.”
—Franklin D. Roosevelt (1882–1945)
“A considerable percentage of the people we meet on the street are people who are empty inside, that is, they are actually already dead. It is fortunate for us that we do not see and do not know it. If we knew what a number of people are actually dead and what a number of these dead people govern our lives, we should go mad with horror.”
—George Gurdjieff (c. 1877–1949)
“The little toy dog is covered with dust,
But sturdy and stanch he stands;
And the little toy soldier is red with rust,
And the musket moulds in his hands.
Time was when the little toy dog was new,
And the soldier was passing fair;
And that was the time when our Little Boy Blue
Kissed them and put them there.”
—Eugene Field (1850–1895)