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.
Read more about Indefinite Orthogonal Group: Examples, Matrix Definition, Topology, Split Orthogonal Group, Split Orthogonal Group in Odd Dimension
Famous quotes containing the words indefinite and/or group:
“There are times when they seem so small! And then again, although they never seem large, there is a vastness behind them, a past of indefinite complexity and marvel, an amazing power of absorbing and assimilating, which forces one to suspect some power in the race so different from our own that one cannot understand that power. And ... whatever doubts or vexations one has in Japan, it is only necessary to ask oneself: Well, who are the best people to live with?”
—Lafcadio Hearn (18501904)
“The poet who speaks out of the deepest instincts of man will be heard. The poet who creates a myth beyond the power of man to realize is gagged at the peril of the group that binds him. He is the true revolutionary: he builds a new world.”
—Babette Deutsch (18951982)