# 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 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 § 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. 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 *n* = 2*p*, O(*p*, *p*) is known as the split orthogonal group.

## 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(3,1) 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*). Consider the diagonal matrix given by

Then we may define a symmetric bilinear form on by the formula

- ,

where is the standard inner product on .

We then define to be the group of matrices that preserve this bilinear form:[1]

- .

More explicitly, consists of matrices such that[2]

- ,

where is the transpose of .

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. Diagonalizing this matrix gives a conjugation of this group with the standard group O(*p*, *q*).

## Topology

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.
*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*= 1C _{1}Z C _{2}*q*= 2Z Z × Z Z × C _{2}*q*≥ 3C _{2}C _{2}× ZC _{2}× C_{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, as it occurs as the group of T-duality transformations in string theory, for example. 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 multiplicative group of 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.

## References

- Hall, Brian C. (2015),
*Lie Groups, Lie Algebras, and Representations: An Elementary Introduction*, Graduate Texts in Mathematics,**222**(2nd ed.), Springer, ISBN 978-3319134666 - 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 - V. L. Popov (2001) [1994], "Orthogonal group", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Joseph A. Wolf,
*Spaces of constant curvature*, (1967) page. 335.