Indefinite orthogonal group

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the indefinite orthogonal group, O(p, q) is the <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> of all <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformations</a> of an n-<a href="/facts/Dimension_(vector_space)/z8m02A3W">dimensional</a> <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> that leave invariant a <a href="/facts/Nondegenerate_form/9TLGBvSH">nondegenerate</a>, <a href="/facts/Symmetric_bilinear_form/UI9b6RK7">symmetric bilinear form</a> of <a href="/facts/Signature_of_a_quadratic_form/l2n1jXPe">signature</a> (p, q), where n = p + q. It is also called the pseudo-orthogonal group or generalized orthogonal group. The dimension of the group is n(n − 1)/2.
The indefinite special orthogonal group, SO(p, q) is the <a href="/facts/Subgroup/r91mBgpa">subgroup</a> of O(p, q) consisting of all elements with <a href="/facts/Determinant/eGYqJ2TF">determinant</a> 1. Unlike in the definite case, SO(p, q) is not <a href="/facts/Connected_space/5CUTxfoA">connected</a> – it has 2 <a href="/facts/Connected_component_(topology)/5CUTxfoA">components</a> – and there are two additional finite <a href="/facts/Index_of_a_subgroup/ewY2A71Y">index</a> subgroups, namely the connected SO+(p, q) and O+(p, q), which has 2 components – see § Topology for definition and discussion.
The signature of the form determines the group up to <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a>; 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 <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a> 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 <a href="/facts/Complex_number/2w7mqI7e">complex</a> spaces, all groups O(p, q; C) are isomorphic to the usual <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a> O(p + q; C), 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 <a href="/facts/Indefinite_unitary_group/lEvudIhU">indefinite unitary group</a> U(p, q) which preserves a <a href="/facts/Sesquilinear_form/Kp6qUA8v">sesquilinear form</a> of signature (p, q).
In even dimension n = 2p, O(p, p) is known as the split orthogonal group.

Indefinite orthogonal group open-in-new

Indefinite orthogonal group