Indefinite orthogonal group

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.

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 § 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.

The group O(p, q) is defined for vector spaces over the reals.

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.

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).)

matrices that preserve this bilinear form:[3] More explicitly,

One obtains an isomorphic group (indeed, a conjugate subgroup of GL(p + q)) by replacing g with any symmetric matrix with p positive eigenvalues and q negative ones.

The group SO+(p, q) and related subgroups of O(p, q) can be described algebraically.

It can be shown that the set of matrices in O(p, q) whose upper-left p×p block A has positive determinant is a subgroup.

Or, to put it another way, if are in O(p, q), then The analogous result for the bottom-right q×q block also holds.

[5] Assuming both p and q are positive, neither of the groups O(p, q) nor SO(p, q) are connected, having four and two components respectively.

π0(O(p, q)) ≅ C2 × C2 is the Klein four-group, with each factor being whether an element preserves or reverses the respective orientations on the p and q dimensional subspaces on which the form is definite; note that reversing orientation on only one of these subspaces reverses orientation on the whole space.

[clarification needed] The identity component of O(p, q) is often denoted SO+(p, q) and can be identified with the set of elements in SO(p, q) that preserve both orientations.

This notation is related to the notation O+(1, 3) for the orthochronous Lorentz group, where the + refers to preserving the orientation on the first (temporal) dimension.

Thus, the spaces are homotopy equivalent to products of (special) orthogonal groups, from which algebro-topological invariants can be computed.

In this sense it is opposite to the definite orthogonal group O(n) := O(n, 0) = O(0, n), which is the compact real form of the complex Lie algebra.

Split orthogonal groups are used to construct the generalized flag variety over non-algebraically closed fields.