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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**^{[1]} or **generalized orthogonal group**.^{[2]} 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. 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* = 2*p*, 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).)

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:^{[3]}

- .

More explicitly, consists of matrices such that^{[4]}

- ,

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*).

The group SO^{+}(*p*, *q*) and related subgroups of O(*p*, *q*) can be described algebraically. Partition a matrix *L* in O(*p*, *q*) as a block matrix:

where *A*, *B*, *C*, and *D* are *p*×*p*, *p*×*q*, *q*×*p*, and *q*×*q* blocks, respectively. 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. The subgroup SO^{+}(*p*, *q*) consists of matrices *L* such that det *A* and det *D* are both positive.^{[5]}^{[6]}

For all matrices *L* in O(*p*, *q*), the determinants of *A* and *D* have the property that and that ^{[7]} In particular, the subgroup SO(*p*, *q*) consists of matrices *L* such that det *A* and det *D* have the same sign.^{[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.
*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)}, each of which either preserves both orientations or reverses both orientations, in either case preserving the overall orientation.^{[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.

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

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}

In even dimensions, 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 group SO(1, 1) may be identified with the group of unit 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.

**^**Popov 2001**^**Hall 2015, p. 8, Section 1.2**^**Hall 2015 Section 1.2.3**^**Hall 2015 Chapter 1, Exercise 1- ^
^{a}^{b}Lester, J. A. (1993). "Orthochronous subgroups of O(p,q)".*Linear and Multilinear Algebra*.**36**(2): 111–113. doi:10.1080/03081089308818280. Zbl 0799.20041. **^**Shirokov 2012, pp. 88–96, Section 7.1**^**Shirokov 2012, pp. 89–91, Lemmas 7.1 and 7.2

- Hall, Brian C. (2015),
*Lie Groups, Lie Algebras, and Representations: An Elementary Introduction*, Graduate Texts in Mathematics, vol. 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 - Popov, V. L. (2001) [1994], "Orthogonal group",
*Encyclopedia of Mathematics*, EMS Press - Shirokov, D. S. (2012).
*Lectures on Clifford algebras and spinors*Лекции по алгебрам клиффорда и спинорам (PDF) (in Russian). doi:10.4213/book1373. Zbl 1291.15063. - Joseph A. Wolf,
*Spaces of constant curvature*, (1967) page. 335.