Indefinite Orthogonal Group

In mathematics, the indefinite orthogonal group, O(p,q) is the Lie group of all linear transformations of a n = p + q 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 indefinite special orthogonal group, SO(p,q) is the subgroup of O(p,q) consisting of all elements with determinant 1. Unlike in the definite case, SO(p,q) is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected SO+(p,q) and O+(p,q), which has 2 components – see the topology section for definition and discussion.

The signature of the form determines the group up to isomorphism; interchanging p with q amounts to replacing the metric by its negative, and so gives the same group. If either p or q equals zero, then the group is isomorphic to the ordinary orthogonal group O(n). We assume in what follows that both p and q are positive.

The group O(p,q) is defined for vector spaces over the reals. For complex spaces, all groups O(p,q; C) are isomorphic to the usual orthogonal group O(p + q; C), since the transform changes the signature of a form.

In even dimension, the middle group O(n,n) is known as the split orthogonal group, and is of particular interest. In odd dimension, the corresponding almost-middle group O(n,n+1) is known as the quasi-split orthogonal group, and plays a similar role.