# Indefinite orthogonal group

In mathematics, the **indefinite orthogonal group**, O(*p*,*q*) is the Lie group of all linear transformations of a *n*-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature (*p*,*q*), where *n* = *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, split form is the almost-middle group O(*n*,*n* + 1).

## Examples

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.

## Matrix definition

One can define O(*p*,*q*) as a group of matrices, just as for the classical orthogonal group O(*n*). The standard inner product on **R**^{p,q} is given in coordinates by the diagonal matrix:

The group O(*p*,*q*) is then the group of a *n*×*n* matrices *M* (where *n* = *p*+*q*) such that ; as a bilinear form,

Here *M*^{T} denotes the transpose of the matrix *M*. One can easily verify that the set of all such matrices forms a group. The inverse of *M* is given by

One obtains an isomorphic group (indeed, a conjugate subgroup of GL(V)) by replacing η with any symmetric matrix with *p* positive eigenvalues and *q* negative ones (such a matrix is necessarily nonsingular); equivalently, any quadratic form with signature (*p*,*q*). Diagonalizing this matrix gives a conjugation of this group with the standard group O(*p*,*q*).

## Topology

Assuming both *p* and *q* are nonzero, neither of the groups O(*p*,*q*) or SO(*p*,*q*) are connected, having four and two components respectively.
*π*_{0}(O(*p*,*q*)) ≅ C_{2} × C_{2} 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. The special orthogonal group has components *π*_{0}(SO(*p*,*q*)) = {(1,1),(−1,−1)} which either preserves both orientations or reverses both orientations, in either case preserving the overall orientation.

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*) which preserves 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.

The group O(*p*,*q*) is also not compact, but contains the compact subgroups O(*p*) and O(*q*) acting on the subspaces on which the form is definite. In fact, O(*p*) × O(*q*) is a maximal compact subgroup of O(*p*,*q*), while S(O(*p*) × O(*q*)) is a maximal compact subgroup of SO(*p*,*q*).
Likewise, SO(*p*) × SO(*q*) is a maximal compact subgroup of SO^{+}(*p*, *q*).
Thus up to homotopy, the spaces are products of (special) orthogonal groups, from which algebro-topological invariants can be computed.

In particular, the fundamental group of SO^{+}(*p*,*q*) is the product of the fundamental groups of the components, *π*_{1}(SO^{+}(*p*,*q*)) = *π*_{1}(SO(*p*)) × *π*_{1}(SO(*q*)), and is given by:

*π*_{1}(SO^{+}(*p*,*q*))*p*= 1*p*= 2*p*≥ 3*q*= 1{1} **Z****Z**_{2}*q*= 2**Z****Z**×**Z****Z**×**Z**_{2}*q*≥ 3**Z**_{2}**Z**_{2}×**Z****Z**_{2}×**Z**_{2}

## Split orthogonal group

In even dimension, the middle group O(*n*,*n*) is known as the **split orthogonal group**, and is of particular interest. It is the split Lie group corresponding to the complex Lie algebra so_{2n} (the Lie group of the split real form of the Lie algebra); more precisely, the identity component is the split Lie group, as non-identity components cannot be reconstructed from the Lie algebra. 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.

The case (1,1) corresponds to the split-complex numbers.

In terms of being a group of Lie type – i.e., construction of an algebraic group from a Lie algebra – split orthogonal groups are Chevalley groups, while the non-split orthogonal groups require a slightly more complicated construction, and are Steinberg groups.

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

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In odd dimension, the split form is the almost-middle group O(*n*,*n*+1).

## See also

## References

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- Anthony Knapp,
*Lie Groups Beyond an Introduction*, Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. ISBN 0-8176-4259-5 – see page 372 for a description of the indefinite orthogonal group - Joseph A. Wolf,
*Spaces of constant curvature*, (1967) page. 335.