Generalized orthogonal group

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In mathematics, the generalized orthogonal group, O(p, q) is the Lie group of all linear transformations of a p + q = n dimensional real vector space which leave invariant a nondegenerate, symmetric, bilinear form of signature (p, q). The dimension of the group is

n(n + 1)/2.

The generalized special orthogonal group, SO(p, q) is the subgroup of O(p, q) having determinant 1.

The signature of the metric—(p positive and q negative eigenvalues)—determines the group up to isomorphism; interchanging p with q amounts to replacing the metric by its negative, and so gives the same group. In all cases where pq ≠ 0 the group is not compact, but contains the compact subgroups O(p) and O(q) acting on the subspaces on which the form is definite. In fact if p ≥ q, the maximal compact subgroup of O(p,q) is O(p), showing that there are no further isomorphisms in the family of groups.

Note that O(p, q) is defined for vector spaces over the realss. For complex spaces, all groups O(p, q; C) are isomorphic with the usual orthogonal group O(p + q; C).

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