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In mathematics, the indefinite orthogonal group, O is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where n = p + q. It is also called the pseudo-orthogonal group or generalized orthogonal group. The dimension of the group is n/2.
The indefinite special orthogonal group, SO is the subgroup of O consisting of all elements with determinant 1. Unlike in the definite case, SO is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected SO and O, which has 2 components – see § Topology 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. We assume in what follows that both p and q are positive.
The group O is defined for vector spaces over the reals. For complex spaces, all groups O are isomorphic to the usual orthogonal group O, since the transform z j ↦ i z j {\displaystyle z_{j}\mapsto iz_{j}} changes the signature of a form. This should not be confused with the indefinite unitary group U which preserves a sesquilinear form of signature.