In even dimension, the middle group O(n,n) is known as the split orthogonal group, and is of particular interest. It is the split Lie group corresponding to the complex Lie algebra so2n (the Lie group of the split real form of the Lie algebra); more precisely, the identity component is the split Lie group, as non-identity components cannot be reconstructed from the Lie algebra. In this sense it is opposite to the definite orthogonal group O(n) := O(n,0) = O(0,n), which is the compact real form of the complex Lie algebra.
The case (1,1) corresponds to the split-complex numbers.
In terms of being a group of Lie type – i.e., construction of an algebraic group from a Lie algebra – split orthogonal groups are Chevalley groups, while the non-split orthogonal groups require a slightly more complicated construction, and are Steinberg groups.
Split orthogonal groups are used to construct the generalized flag variety over non-algebraically closed fields.
Read more about this topic: Indefinite Orthogonal Group
Famous quotes containing the words split and/or group:
“Would you convey my compliments to the purist who reads your proofs and tell him or her that I write in a sort of broken-down patois which is something like the way a Swiss waiter talks, and that when I split an infinitive, God damn it, I split it so it will stay split, and when I interrupt the velvety smoothness of my more or less literate syntax with a few sudden words of bar- room vernacular, that is done with the eyes wide open and the mind relaxed but attentive.”
—Raymond Chandler (1888–1959)
“Remember that the peer group is important to young adolescents, and there’s nothing wrong with that. Parents are often just as important, however. Don’t give up on the idea that you can make a difference.”
—The Lions Clubs International and the Quest Nation. The Surprising Years, I, ch.5 (1985)