In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the indefinite orthogonal group, is the
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
of all
linear transformations of an ''n''-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
real
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
that leave invariant a
nondegenerate
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.
T ...
,
symmetric bilinear form of
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
, where . It is also called the pseudo-orthogonal group or generalized orthogonal group. The dimension of the group is .
The indefinite special orthogonal group, is the
subgroup of consisting of all elements with
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
1. Unlike in the definite case, is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected and , which has 2 components – see ' for definition and discussion.
The signature of the form determines the group up to
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
; 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
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
O(''n''). We assume in what follows that both ''p'' and ''q'' are positive.
The group is defined for vector spaces over the
reals. For
complex spaces, all groups are isomorphic to the usual
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
, since the transform
changes the signature of a form. This should not be confused with the
indefinite unitary group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
which preserves a
sesquilinear form of signature .
In even dimension , is known as the
split orthogonal group
In mathematics, the indefinite orthogonal group, 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 , where . It is also called the ...
.
Examples

The basic example is the
squeeze mappings, which is the group of (the identity component of) linear transforms preserving the
unit hyperbola
In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative radi ...
. 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 is of central importance, being the setting for
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
and
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
. (Some texts use for the Lorentz group; however, is prevalent in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
because the geometric properties of the
Dirac equation are more natural in .)
Matrix definition
One can define as a group of
matrices, just as for the classical
orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
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
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
on
.
We then define
to be the group of
matrices that preserve this bilinear form:
:
.
More explicitly,
consists of matrices
such that
:
,
where
is the
transpose of
.
One obtains an isomorphic group (indeed, a conjugate subgroup of ) 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 .
Subgroups
The group and related subgroups of can be described algebraically. Partition a matrix ''L'' in as a
block matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original mat ...
:
:
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 whose upper-left ''p''×''p'' block ''A'' has positive determinant is a subgroup. Or, to put it another way, if
:
are in , then
:
The analogous result for the bottom-right ''q''×''q'' block also holds. The subgroup consists of matrices ''L'' such that and are both positive.
For all matrices ''L'' in , the determinants of ''A'' and ''D'' have the property that
and that
In particular, the subgroup consists of matrices ''L'' such that and have the same sign.
[
]
Topology
Assuming both ''p'' and ''q'' are positive, neither of the groups nor are connected, having four and two components respectively.
is the Klein four-group
In mathematics, the Klein four-group is a Group (mathematics), group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity)
and in which composing any two of the three ...
, 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 , each of which either preserves both orientations or reverses both orientations, in either case preserving the overall orientation.
The identity component of is often denoted and can be identified with the set of elements in that preserve both orientations. This notation is related to the notation for the orthochronous Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physic ...
, where the + refers to preserving the orientation on the first (temporal) dimension.
The group is also not compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, but contains the compact subgroups O(''p'') and O(''q'') acting on the subspaces on which the form is definite. In fact, is a maximal compact subgroup of , while is a maximal compact subgroup of .
Likewise, is a maximal compact subgroup of .
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
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of is the product of the fundamental groups of the components, , and is given by:
:
Split orthogonal group
In even dimensions, the middle group 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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
so2''n'' (the Lie group of the split real form
In mathematics, the notion of a real form relates objects defined over the Field (algebra), field of Real number, real and Complex number, complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is ...
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 , which is the ''compact'' real form of the complex Lie algebra.
The case corresponds to the multiplicative group of the split-complex number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
s.
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 group
In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phras ...
s, 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 In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smoot ...
over non-algebraically closed fields.
See also
*Orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
* Lorentz group
* Poincaré group
* Symmetric bilinear form
References
*
*Anthony Knapp
Anthony W. Knapp (born 2 December 1941, Morristown, New Jersey) is an American mathematician at the State University of New York, Stony Brook working on representation theory, who classified the tempered representations of a semisimple Lie group ...
, ''Lie Groups Beyond an Introduction'', Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. – see page 372 for a description of the indefinite orthogonal group
*
*
*Joseph A. Wolf
Joseph Albert Wolf (born October 18, 1936 in Chicago) is an American mathematician. He is now professor emeritus at the University of California, Berkeley.
Wolf graduated from at the University of Chicago with a bachelor's degree in 1956 and with ...
, ''Spaces of constant curvature'', (1967) page. 335.
{{refend
Lie groups