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
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