# Indefinite orthogonal group

Last updated

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  or generalized orthogonal group.  The dimension of the group is n(n − 1)/2.

## Contents

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 $z_{j}\mapsto iz_{j}$ 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 = 2p, 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 $\left[{\begin{smallmatrix}\cosh(\alpha )&\sinh(\alpha )\\\sinh(\alpha )&\cosh(\alpha )\end{smallmatrix}}\right],$ 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).)

## Matrix definition

One can define O(p, q) as a group of matrices, just as for the classical orthogonal group O(n). Consider the $(p+q)\times (p+q)$ diagonal matrix $g$ given by

$g=\mathrm {diag} (\underbrace {1,\ldots ,1} _{p},\underbrace {-1,\ldots ,-1} _{q}).$ Then we may define a symmetric bilinear form $[\cdot ,\cdot ]_{p,q}$ on $\mathbb {R} ^{p+q}$ by the formula

$[x,y]_{p,q}=\langle x,gy\rangle =x_{1}y_{1}+\cdots +x_{p}y_{p}-x_{p+1}y_{p+1}-\cdots -x_{p+q}y_{p+q}$ ,

where $\langle \cdot ,\cdot \rangle$ is the standard inner product on $\mathbb {R} ^{p+q}$ .

We then define $\mathrm {O} (p,q)$ to be the group of $(p+q)\times (p+q)$ matrices that preserve this bilinear form: 

$\mathrm {O} (p,q)=\{A\in M_{p+q}(\mathbb {R} )|[Ax,Ay]_{p,q}=[x,y]_{p,q}\,\forall x,y\in \mathbb {R} ^{p+q}\}$ .

More explicitly, $\mathrm {O} (p,q)$ consists of matrices $A$ such that 

$gA^{\mathrm {tr} }g=A^{-1}$ ,

where $A^{\mathrm {tr} }$ is the transpose of $A$ .

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).

### Subgroups

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:

$L={\begin{pmatrix}A&B\\C&D\end{pmatrix}}$ 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

$L={\begin{pmatrix}A&B\\C&D\end{pmatrix}}\;\mathrm {and} \;M={\begin{pmatrix}W&X\\Y&Z\end{pmatrix}}$ are in O(p, q), then

$(\operatorname {sgn} \det A)(\operatorname {sgn} \det W)=\operatorname {sgn} \det(AW+BY).$ 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.  

For all matrices L in O(p, q), the determinants of A and D have the property that ${\textstyle {\frac {\det A}{\det D}}=\det L}$ and that $|\det A|=|\det D|\geq 1.$ In particular, the subgroup SO(p, q) consists of matrices L such that det A and det D have the same sign. 

## 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. 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. The special orthogonal group has components π0(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, π1(SO+(p, q)) = π1(SO(p)) × π1(SO(q)), and is given by:

π1(SO+(p, q))p = 1p = 2p 3
q = 1C1ZC2
q = 2ZZ × ZZ × C2
q 3C2C2 × ZC2 × C2

## Split orthogonal group

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 so2n (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.

## Related Research Articles In mathematics, a Lie algebra is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. The vector space together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. In mathematics, a Lie group is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract, generic concept of multiplication and the taking of inverses (division). Combining these two ideas, one obtains a continuous group where points can be multiplied together, and their inverse can be taken. If, in addition, the multiplication and taking of inverses are defined to be smooth (differentiable), one obtains a Lie group. In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication. The orthogonal group is an algebraic group and a Lie group. It is compact.

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3. In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n) for positive integer n and field F. The latter is called the compact symplectic group and is also denoted by . Many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn, and Sp(n) is the compact real form of Sp(2n, C). Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension n. In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.

In mathematics, a quadratic form is a polynomial with terms all of degree two. For example, Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. In mathematics the spin group Spin(n) is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups

In mathematics, the Heisenberg group, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form

In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.

In mathematics, a maximal compact subgroupK of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups. In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.

In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature. More precisely, they are representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields. In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.

In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was initiated by Ernest Vinberg and Bertram Kostant.

In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional real Euclidean Jordan algebras, originally studied and classified by Jordan, von Neumann & Wigner (1934). The tube domain associated with a symmetric cone is a noncompact Hermitian symmetric space of tube type. All the algebraic and geometric structures associated with the symmetric space can be expressed naturally in terms of the Jordan algebra. The other irreducible Hermitian symmetric spaces of noncompact type correspond to Siegel domains of the second kind. These can be described in terms of more complicated structures called Jordan triple systems, which generalize Jordan algebras without identity.

In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first introduced by Max Koecher in his Jordan algebraic approach to Hermitian symmetric spaces and bounded symmetric domains of tube type. Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compact subgroup. Both groups act transitively on the compactification. The theory has been extended to cover all Hermitian symmetric spaces using the theory of Jordan pairs or Jordan triple systems. Koecher obtained the results in the more general case directly from the Jordan algebra case using the fact that only Jordan pairs associated with period two automorphisms of Jordan algebras are required.

• 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
• Popov, V. L. (2001) , "Orthogonal group", Encyclopedia of Mathematics , EMS Press
• Shirokov, D. S. (2012). Lectures on Clifford algebras and spinorsЛекции 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.
1. Popov 2001
2. Hall 2015 , p. 8, Section 1.2
3. Hall 2015 Section 1.2.3
4. Hall 2015 Chapter 1, Exercise 1
5. 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.
6. Shirokov 2012 , pp. 88–96, Section 7.1
7. Shirokov 2012 , pp. 89–91, Lemmas 7.1 and 7.2