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Concept
Indefinite orthogonal group
Summary
In mathematics, the indefinite orthogonal group, O(p, q) 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 (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 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 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. This should not be confused with the indefinite unitary group U(p, q) which preserves a sesquilinear form of signature (p, q).
In even dimension n = 2p, O(p, p) is known as the split orthogonal group.
The basic example is the squeeze mappings, which is the group SO+(1, 1) of (the identity component of) linear transforms preserving the unit hyperbola. Concretely, these are the matrices and can be interpreted as hyperbolic rotations, just as the group SO(2) can be interpreted as circular rotations.
In physics, the Lorentz group O(1,3) is of central importance, being the setting for electromagnetism and special relativity. (Some texts use O(3,1) for the Lorentz group; however, O(1,3) is prevalent in quantum field theory because the geometric properties of the Dirac equation are more natural in O(1,3).
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