Orthogonal group

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. Equivalently, it is the group of $n \times n$ orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose.

An important subgroup of $O(n)$ is the special orthogonal group, denoted $SO(n)$ , of the orthogonal matrices of determinant $1$ . This group is also called the rotation group, because, in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see $SO(2)$ , $SO(3)$ and $SO(4)$ .

The term "orthogonal group" may also refer to a generalization of the above case: the group of invertible linear operators that preserve a non-degenerate symmetric bilinear form or quadratic form^[1] on a vector space over a field. In particular, when the bilinear form is the scalar product on the vector space $F n$ of dimension $n$ over a field $F$ , with quadratic form the sum of squares, then the corresponding orthogonal group, denoted $O(n, F)$ , is the set of $n \times n$ orthogonal matrices with entries from $F$ , with the group operation of matrix multiplication. This is a subgroup of the general linear group $GL(n, F)$ given by

\mathrm {O} (n,F)=\{Q\in \mathrm {GL} (n,F)\mid Q^{\mathsf {T}}Q=QQ^{\mathsf {T}}=I\},

where $Q$ ^T is the transpose of $Q$ and $I$ is the identity matrix.

This article mainly discusses the orthogonal groups of quadratic forms that may be expressed over some bases as the dot product; over the reals, they are the positive definite quadratic forms. Over the reals, for any non-degenerate quadratic form, there is a basis, on which the matrix of the form is a diagonal matrix such that the diagonal entries are either $1$ or $-1$ . Thus the orthogonal group depends only on the numbers of $1$ and of $-1$ , and is denoted $O(p, q)$ , where $p$ is the number of ones and $q$ the number of negative ones. For details, see indefinite orthogonal group.

The derived subgroup $Ω(n, F)$ of $O(n, F)$ is an often studied object because, when $F$ is a finite field, $Ω(n, F)$ is often a central extension of a finite simple group.

Both $O(n, F)$ and $SO(n, F)$ are algebraic groups, because the condition that a matrix be orthogonal, i.e. have its own transpose as inverse, can be expressed as a set of polynomial equations in the entries of the matrix. The Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form.

Name

The determinant of any orthogonal matrix is either $1$ or $-1$ . The orthogonal $n$ -by- $n$ matrices with determinant $1$ form a normal subgroup of $O(n, F)$ known as the special orthogonal group $SO(n, F)$ , consisting of all proper rotations. (More precisely, $SO(n, F)$ is the kernel of the Dickson invariant, discussed below.). By analogy with GL–SL (general linear group, special linear group), the orthogonal group is sometimes called the general orthogonal group and denoted $GO$ , though this term is also sometimes used for indefinite orthogonal groups $O(p, q)$ . The term rotation group can be used to describe either the special or general orthogonal group.

In even and odd dimension

The structure of the orthogonal group differs in certain respects between even and odd dimensions; for example, over ordered fields (such as $R$ ) the $- I$ element is orientation-preserving in even dimensions, but orientation-reversing in odd dimensions. When this distinction is to be emphasized, the groups may be denoted $O(2 k)$ and $O(2 k + 1)$ , reserving $n$ for the dimension of the space ( $n = 2 k$ or $n = 2 k + 1$ ). The letters $p$ or $r$ are also used, indicating the rank of the corresponding Lie algebra; in odd dimension the corresponding Lie algebra is ${\mathfrak {so}}(2r+1)$ , while in even dimension the Lie algebra is ${\mathfrak {so}}(2r)$ .

Difference between O(n) and SO(n) in even dimensions

In two dimensions, O(2) is the group of all rotations about the origin and all reflections along a line through the origin. SO(2) is the group of all rotations about the origin.

These groups are closely related: SO(2) is a subgroup of O(2) of index 2, since any two reflections gives a rotation.

More generally, in any number of dimensions an even number of reflections gives a rotation, and a rotation followed by reflection (or vice versa) produces a reflection. Therefore, the rotations define a subgroup of O(2), but the reflections do not define a subgroup.

A "reflection through the origin" may be generated as a combination of one reflection along each of the axes. The 'reflection through the origin' is not a reflection in the usual sense in even dimensions, but rather a rotation. In two dimensions it is the only nontrivial rotation that when applied twice gives the identity. It is its own inverse in any number of dimensions. In four dimensions it is isoclinic, and if that classification were generalised it would be isoclinic in every even number of dimensions.

Over the real number field

Over the field $R$ of real numbers, the orthogonal group $O(n, R)$ and the special orthogonal group $SO(n, R)$ are often simply denoted by $O(n)$ and $SO(n)$ if no confusion is possible. They form real compact Lie groups of dimension $n (n - 1)/2$ . $O(n, R)$ has two connected components, with $SO(n, R)$ being the identity component, i.e., the connected component containing the identity matrix.

Geometric interpretation

The real orthogonal and real special orthogonal groups have the following geometric interpretations:

$O(n, R)$ is a subgroup of the Euclidean group $E (n)$ , the group of isometries of $R n$ ; it contains those that leave the origin fixed – $O(n, R) = E (n) \cap GL(n, R)$ . It is the symmetry group of the sphere ( $n = 3$ ) or $(n - 1)$ -sphere and all objects with spherical symmetry, if the origin is chosen at the center.

$SO(n, R)$ is a subgroup of $E + (n)$ , which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed – $SO(n, R) = E + (n) \cap GL(n, R) = E (n) \cap GL + (n, R)$ . It is the rotation group of the sphere and all objects with spherical symmetry, if the origin is chosen at the center.

${\pm I$ } is a normal subgroup and even a characteristic subgroup of $O(n, R)$ , and, if $n$ is even, also of $SO(n, R)$ . If $n$ is odd, $O(n, R)$ is the internal direct product of $SO(n, R)$ and ${\pm I$ }. For every positive integer $k$ the cyclic group $C k$ of $k$ -fold rotations is a normal subgroup of $O(2, R)$ and $SO(2, R)$ .

Relative to suitable orthogonal bases, the isometries are of the form:

{\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}}

where the matrices $R 1, ..., R k$ are 2-by-2 rotation matrices in orthogonal planes of rotation. As a special case, known as Euler's rotation theorem, any (non-identity) element of $SO(3, R)$ is a rotation about a uniquely defined axis.

The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group,^{[note 1]} and elements have length at most $n$ (require at most $n$ reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map $v \mapsto - v$ ), though so are other maximal combinations of rotations (and a reflection, in odd dimension).

The symmetry group of a circle is $O(2, R)$ . The orientation-preserving subgroup $SO(2, R)$ is isomorphic (as a real Lie group) to the circle group, also known as $U (1)$ . This isomorphism sends the complex number $exp(φ i) = cos φ + i sin φ$ of absolute value $1$ to the special orthogonal matrix

{\begin{bmatrix}\cos(\phi )&-\sin(\phi )\\\sin(\phi )&\cos(\phi )\end{bmatrix}}.

The group $SO(3, R)$ , understood as the set of rotations of 3-dimensional space, is of major importance in the sciences and engineering, and there are numerous charts on $SO(3)$ .

Maximal tori and Weyl groups

A maximal torus $T$ for $SO(2 n)$ , of rank n, is given by the block-diagonal matrices

{\begin{bmatrix}R_{1}&&0\\&\ddots &\\0&&R_{n}\end{bmatrix}},

where the $R j$ are 2-by-2 rotation matrices. The image $T \times{1$ } of the same torus under the block-diagonal inclusion

\mathrm {SO} (2n)\cong \mathrm {SO} (2n)\times \{1\}<\mathrm {SO} (2n+1)

is a maximal torus for $SO(2 n +1)$ . The Weyl group of $SO(2 n +1)$ is the semidirect product $\{\pm 1\}^{n}\rtimes S_{n}$ of a normal elementary abelian 2-subgroup and a symmetric group, where the nontrivial element of each ${\pm1$ } factor of ${\pm1} n$ acts on the corresponding circle factor of $T \times{1$ } by inversion, and the symmetric group $S n$ acts on both ${\pm1} n$ and $T \times{1$ } by permuting factors. The elements of the Weyl group are represented by matrices in $O(2 n) \times {\pm1$ }. The $S n$ factor is represented by block permutation matrices with 2-by-2 blocks, and a final 1 on the diagonal. The ${\pm1} n$ component is represented by block-diagonal matrices with 2-by-2 blocks either

{\begin{bmatrix}1&0\\0&1\end{bmatrix}}\quad {\text{or}}\quad {\begin{bmatrix}0&1\\1&0\end{bmatrix}},

with the last component $\pm1$ chosen to make the determinant 1.

The Weyl group of $SO(2 n)$ is the subgroup $H_{n-1}\rtimes S_{n}<\{\pm 1\}^{n}\rtimes S_{n}$ of that of $SO(2 n + 1)$ , where $H n -1 < {\pm1} n$ is the kernel of the product homomorphism ${\pm1} n \to {\pm1$ } given by $(\epsilon _{1},\ldots ,\epsilon _{n})\mapsto \epsilon _{1}\cdots \epsilon _{n}$ ; that is $H n -1 < {\pm1} n$ is the subgroup with an even number of minus signs. The Weyl group of $SO(2 n)$ is represented in $SO(2 n)$ by the preimages under the standard injection $SO(2 n) \to SO(2 n + 1)$ of the representatives for the Weyl group of $SO(2 n + 1)$ . Those matrices with an odd number of ${\begin{bmatrix}0&1\\1&0\end{bmatrix}}$ blocks have no remaining final $-1$ coordinate to make their determinants positive, and hence cannot be represented in $SO(2 n)$ .

Low-dimensional topology

The low-dimensional (real) orthogonal groups are familiar spaces:

$O(1) = S 0$ , a two-point discrete space
$SO(1) = {1}$
$SO(2)$ is $S 1$
$SO(3)$ is $R P 3$ ^[2]
$SO(4)$ is doubly covered by $SU (2) \times SU(2) = S 3 \times S 3$ .

Homotopy groups

In terms of algebraic topology, for $n > 2$ the fundamental group of $SO(n, R)$ is cyclic of order 2,^[3] and the spin group $Spin(n)$ is its universal cover. For $n = 2$ the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the group $Spin(2)$ is the unique connected 2-fold cover).

Generally, the homotopy groups $π k (O)$ of the real orthogonal group are related to homotopy groups of spheres, and thus are in general hard to compute. However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the direct limit of the sequence of inclusions:

\mathrm {O} (0)\subset \mathrm {O} (1)\subset \mathrm {O} (2)\subset \cdots \subset O=\bigcup _{k=0}^{\infty }\mathrm {O} (k)

Since the inclusions are all closed, hence cofibrations, this can also be interpreted as a union. On the other hand, $S n$ is a homogeneous space for $O(n + 1)$ , and one has the following fiber bundle:

\mathrm {O} (n)\to \mathrm {O} (n+1)\to S^{n},

which can be understood as "The orthogonal group $O(n + 1)$ acts transitively on the unit sphere $S n$ , and the stabilizer of a point (thought of as a unit vector) is the orthogonal group of the perpendicular complement, which is an orthogonal group one dimension lower. Thus the natural inclusion $O(n) \to O(n + 1)$ is $(n - 1)$ -connected, so the homotopy groups stabilize, and $π k (O(n +1)) = π k (O(n))$ for $n > k + 1$ : thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.

From Bott periodicity we obtain $Ω 8 O ≅ O$ , therefore the homotopy groups of $O$ are 8-fold periodic, meaning $π k + 8 (O) = π k (O)$ , and one needs only to list the lower 8 homotopy groups:

{\begin{aligned}\pi _{0}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{1}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(O)&=0\\\pi _{3}(O)&=\mathbf {Z} \\\pi _{4}(O)&=0\\\pi _{5}(O)&=0\\\pi _{6}(O)&=0\\\pi _{7}(O)&=\mathbf {Z} \end{aligned}}

Relation to KO-theory

Via the clutching construction, homotopy groups of the stable space $O$ are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: $π k (O) = π k + 1 (BO)$ . Setting $KO = BO \times Z = Ω -1 O \times Z$ (to make $π 0$ fit into the periodicity), one obtains:

{\begin{aligned}\pi _{0}(KO)&=\mathbf {Z} \\\pi _{1}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{3}(KO)&=0\\\pi _{4}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{5}(KO)&=0\\\pi _{6}(KO)&=0\\\pi _{7}(KO)&=0\end{aligned}}

Computation and interpretation of homotopy groups

Low-dimensional groups

The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups.

$π 0 (O) = π 0 (O(1)) = Z /2 Z$ , from orientation-preserving/reversing (this class survives to $O(2)$ and hence stably)
$π 1 (O) = π 1 (SO(3)) = Z /2 Z$ , which is spin comes from $SO(3) = R P 3 = S 3 /(Z /2 Z)$ .
$π 2 (O) = π 2 (SO(3)) = 0$ , which surjects onto $π 2 (SO(4))$ ; this latter thus vanishes.

Lie groups

From general facts about Lie groups, $π 2 (G)$ always vanishes, and $π 3 (G)$ is free (free abelian).

Vector bundles

From the vector bundle point of view, $π 0 (K O)$ is vector bundles over $S 0$ , which is two points. Thus over each point, the bundle is trivial, and the non-triviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so $π 0 (K O) = Z$ is dimension.

Loop spaces

Using concrete descriptions of the loop spaces in Bott periodicity, one can interpret the higher homotopies of $O$ in terms of simpler-to-analyze homotopies of lower order. Using π₀, $O$ and $O /U$ have two components, $K O = B O \times Z$ and $K Sp = B Sp \times Z$ have countably many components, and the rest are connected.

Interpretation of homotopy groups

In a nutshell:^[4]

$π 0 (K O) = Z$ is about dimension
$π 1 (K O) = Z /2 Z$ is about orientation
$π 2 (K O) = Z /2 Z$ is about spin
$π 4 (K O) = Z$ is about topological quantum field theory.

Let $R$ be any of the four division algebras $R$ , $C$ , $H$ , $O$ , and let $L R$ be the tautological line bundle over the projective line $R P 1$ , and $[L R]$ its class in K-theory. Noting that $R P 1 = S 1$ , $C P 1 = S 2$ , $H P 1 = S 4$ , $O P 1 = S 8$ , these yield vector bundles over the corresponding spheres, and

$π 1 (K O)$ is generated by $[L R]$
$π 2 (K O)$ is generated by $[L C]$
$π 4 (K O)$ is generated by $[L H]$
$π 8 (K O)$ is generated by $[L O]$

From the point of view of symplectic geometry, $π 0 (K O) ≅ π 8 (K O) = Z$ can be interpreted as the Maslov index, thinking of it as the fundamental group $π 1 (U/O)$ of the stable Lagrangian Grassmannian as $U/O ≅ Ω 7 (K O)$ , so $π 1 (U/O) = π 1+7 (K O)$ .

Whitehead tower

The orthogonal group anchors a Whitehead tower:

\ldots \rightarrow {\text{Fivebrane}}(n)\rightarrow {\text{String}}(n)\rightarrow {\text{Spin}}(n)\rightarrow {\text{SO}}(n)\rightarrow {\text{O}}(n)

which is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing short exact sequences starting with an Eilenberg–MacLane space for the homotopy group to be removed. The first few entries in the tower are the spin group and the string group, and are preceded by the fivebrane group. The homotopy groups that are killed are in turn $π$ ₀(O) to obtain SO from O, $π$ ₁(O) to obtain Spin from SO, $π$ ₃(O) to obtain String from Spin, and then $π$ ₇(O) and so on to obtain the higher order branes.

Over the complex number field

Over the field $C$ of complex numbers, $O(n, C)$ and $SO(n, C)$ are complex Lie groups of dimension $n (n - 1)/2$ over $C$ (it means the dimension over $R$ is twice that). $O(n, C)$ has two connected components, and $SO(n, C)$ is the connected component containing the identity matrix. For $n \geq 2$ these groups are noncompact.

Just as in the real case $SO(n, C)$ is not simply connected. For $n > 2$ the fundamental group of $SO(n, C)$ is cyclic of order 2 whereas the fundamental group of $SO(2, C)$ is infinite cyclic.

Over finite fields

Orthogonal groups can also be defined over finite fields $F q$ , where $q$ is a power of a prime $p$ .

Over finite fields of characteristic not equal to 2, orthogonal groups come in two types in even dimension: $O + (2 n, q)$ and $O - (2 n, q)$ ; and one type in odd dimension: $O(2 n + 1, q)$ .^[5]

If $V$ is the vector space on which the orthogonal group $G$ acts, it can be written as a direct orthogonal sum as follows:

V=L_{1}\oplus L_{2}\oplus \cdots \oplus L_{m}\oplus W,

where $L i$ are hyperbolic lines and $W$ contains no singular vectors. If $W$ is the zero subspace, then $G$ is of plus type. If $W$ is one-dimensional then $G$ has odd dimension. If $W$ has dimension 2, $G$ is of minus type.

In the special case where $n = 1$ , $O ϵ (2, q)$ is a dihedral group of order $2(q - ϵ)$ .

We have the following formulas for the order of $O(n, q)$ , when the characteristic is not two:

|\mathrm {O} (2n+1,q)|=2q^{n}\prod _{i=0}^{n-1}(q^{2n}-q^{2i}).

If $-1$ is a square in $F q$

|\mathrm {O} (2n,q)|=2(q^{n}-1)\prod _{i=1}^{n-1}(q^{2n}-q^{2i}).

If $-1$ is a non-square in $F q$

|\mathrm {O} (2n,q)|=2(q^{n}+(-1)^{n+1})\prod _{i=1}^{n-1}(q^{2n}-q^{2i}).

The Dickson invariant

For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to the quotient group $Z /2 Z$ (integers modulo 2), taking the value $0$ in case the element is the product of an even number of reflections, and the value of 1 otherwise.^[6]

Algebraically, the Dickson invariant can be defined as $D (f) = rank(I - f) modulo 2$ , where $I$ is the identity (Taylor 1992, Theorem 11.43). Over fields that are not of characteristic 2 it is equivalent to the determinant: the determinant is −1 to the power of the Dickson invariant. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives more information than the determinant.

The special orthogonal group is the kernel of the Dickson invariant^[6] and usually has index 2 in $O(n, F)$ .^[7] When the characteristic of $F$ is not 2, the Dickson Invariant is $0$ whenever the determinant is $1$ . Thus when the characteristic is not 2, $SO(n, F)$ is commonly defined to be the elements of $O(n, F)$ with determinant $1$ . Each element in $O(n, F)$ has determinant $\pm1$ . Thus in characteristic 2, the determinant is always $1$ .

The Dickson invariant can also be defined for Clifford groups and Pin groups in a similar way (in all dimensions).

Orthogonal groups of characteristic 2

Over fields of characteristic 2 orthogonal groups often exhibit special behaviors, some of which are listed in this section. (Formerly these groups were known as the hypoabelian groups, but this term is no longer used.)

Any orthogonal group over any field is generated by reflections, except for a unique example where the vector space is 4-dimensional over the field with 2 elements and the Witt index is 2.^[8] Note that a reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector $u$ takes a vector $v$ to $v + B (v, u)/Q(u) \cdot u$ where $B$ is the bilinear form and $Q$ is the quadratic form associated to the orthogonal geometry. Compare this to the Householder reflection of odd characteristic or characteristic zero, which takes $v$ to $v - 2\cdot B (v, u)/Q(u) \cdot u$ .
The center of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since $I = - I$ .
In odd dimensions $2 n + 1$ in characteristic 2, orthogonal groups over perfect fields are the same as symplectic groups in dimension $2 n$ . In fact the symmetric form is alternating in characteristic 2, and as the dimension is odd it must have a kernel of dimension 1, and the quotient by this kernel is a symplectic space of dimension $2 n$ , acted upon by the orthogonal group.
In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.

The spinor norm

The spinor norm is a homomorphism from an orthogonal group over a field $F$ to the quotient group $F \times /(F \times) 2$ (the multiplicative group of the field $F$ up to square elements), that takes reflection in a vector of norm $n$ to the image of $n$ in $F \times /(F \times) 2$ .^[9]

For the usual orthogonal group over the reals it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.

Galois cohomology and orthogonal groups

In the theory of Galois cohomology of algebraic groups, some further points of view are introduced. They have explanatory value, in particular in relation with the theory of quadratic forms; but were for the most part post hoc, as far as the discovery of the phenomena is concerned. The first point is that quadratic forms over a field can be identified as a Galois $H 1$ , or twisted forms (torsors) of an orthogonal group. As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the discriminant.

The 'spin' name of the spinor norm can be explained by a connection to the spin group (more accurately a pin group). This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use of Clifford algebras). The spin covering of the orthogonal group provides a short exact sequence of algebraic groups.

1\rightarrow \mu _{2}\rightarrow \mathrm {Pin} _{V}\rightarrow \mathrm {O_{V}} \rightarrow 1

Here $μ 2$ is the algebraic group of square roots of 1; over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action. The connecting homomorphism from $H 0 (O V)$ , which is simply the group $O V (F)$ of $F$ -valued points, to $H 1 (μ 2)$ is essentially the spinor norm, because $H 1 (μ 2)$ is isomorphic to the multiplicative group of the field modulo squares.

There is also the connecting homomorphism from $H 1$ of the orthogonal group, to the $H 2$ of the kernel of the spin covering. The cohomology is non-abelian, so that this is as far as we can go, at least with the conventional definitions.

Lie algebra

The Lie algebra corresponding to Lie groups $O(n, F)$ and $SO(n, F)$ consists of the skew-symmetric $n \times n$ matrices, with the Lie bracket $[ , ]$ given by the commutator. One Lie algebra corresponds to both groups. It is often denoted by ${\mathfrak {o}}(n,F)$ or ${\mathfrak {so}}(n,F)$ , and called the orthogonal Lie algebra or special orthogonal Lie algebra. Over real numbers, these Lie algebras for different $n$ are the compact real forms of two of the four families of semisimple Lie algebras: in odd dimension $B k$ , where $n = 2 k + 1$ , while in even dimension $D r$ , where $n = 2 r$ .

More intrinsically, given a vector space with an inner product, the special orthogonal Lie algebra is given by the bivectors on the space, which are sums of simple bivectors (2-blades) $v \land w$ . The correspondence is given by the map $\mathbf {v} \wedge \mathbf {w} \mapsto \mathbf {v} ^{*}\otimes \mathbf {w} -\mathbf {w} ^{*}\otimes \mathbf {v} ,$ where $v *$ is the covector dual to the vector $v$ ; in coordinates these are exactly the elementary skew-symmetric matrices.

Over real numbers, this characterization is used in interpreting the curl of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name. Generalizing the inner product with a nondegenerate form yields the indefinite orthogonal Lie algebras ${\mathfrak {so}}(p,q).$

The representation theory of the orthogonal Lie algebras includes both representations corresponding to linear representations of the orthogonal groups, and representations corresponding to projective representations of the orthogonal groups (linear representations of spin groups), the so-called spin representation, which are important in physics.

Related groups

The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotient groups, and covering groups. These are listed below.

The inclusions $O(n) \subset U(n) \subset Sp(n) = USp(2 n)$ and $USp(n) \subset U(n) \subset O(2 n)$ are part of a sequence of 8 inclusions used in a geometric proof of the Bott periodicity theorem, and the corresponding quotient spaces are symmetric spaces of independent interest – for example, $U(n)/O(n)$ is the Lagrangian Grassmannian.

Lie subgroups

In physics, particularly in the areas of Kaluza–Klein compactification, it is important to find out the subgroups of the orthogonal group. The main ones are:

\mathrm {O} (n)\supset \mathrm {O} (n-1)

– preserve an axis

\mathrm {O} (2n)\supset \mathrm {U} (n)\supset \mathrm {SU} (n)

–

U(n)

are those that preserve a compatible complex structure or a compatible symplectic structure – see 2-out-of-3 property;

SU(n)

also preserves a complex orientation.

\mathrm {O} (2n)\supset \mathrm {USp} (n)

\mathrm {O} (7)\supset \mathrm {G} _{2}

Lie supergroups

The orthogonal group $O(n)$ is also an important subgroup of various Lie groups:

\mathrm {U} (n)\supset \mathrm {SU} (n)\supset \mathrm {O} (n)

\mathrm {USp} (2n)\supset \mathrm {O} (n)

\mathrm {G} _{2}\supset \mathrm {O} (3)

\mathrm {F} _{4}\supset \mathrm {O} (9)

\mathrm {E} _{6}\supset \mathrm {O} (10)

\mathrm {E} _{7}\supset \mathrm {O} (12)

\mathrm {E} _{8}\supset \mathrm {O} (16)

Conformal group

Being isometries, real orthogonal transforms preserve angles, and are thus conformal maps, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between congruence and similarity, as exemplified by SSS (Side-Side-Side) congruence of triangles and AAA (Angle-Angle-Angle) similarity of triangles. The group of conformal linear maps of $R n$ is denoted $CO(n)$ for the conformal orthogonal group, and consists of the product of the orthogonal group with the group of dilations. If $n$ is odd, these two subgroups do not intersect, and they are a direct product: $CO(2 k + 1) = O(2 k + 1) \times R *$ , where $R * = R ∖{0$ } is the real multiplicative group, while if $n$ is even, these subgroups intersect in $\pm1$ , so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: $CO(2 k) = O(2 k) \times R +$ .

Similarly one can define $CSO(n)$ ; note that this is always: $CSO(n) = CO(n) \cap GL + (n) = SO(n) \times R +$ .

Discrete subgroups

As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups.^{[note 2]} These subgroups are known as point group and can be realized as the symmetry groups of polytopes. A very important class of examples are the finite Coxeter groups, which include the symmetry groups of regular polytopes.

Dimension 3 is particularly studied – see point groups in three dimensions, polyhedral groups, and list of spherical symmetry groups. In 2 dimensions, the finite groups are either cyclic or dihedral – see point groups in two dimensions.

Other finite subgroups include:

Permutation matrices (the Coxeter group $A n$ )
Signed permutation matrices (the Coxeter group $B n$ ); also equals the intersection of the orthogonal group with the integer matrices.^{[note 3]}

Covering and quotient groups

The orthogonal group is neither simply connected nor centerless, and thus has both a covering group and a quotient group, respectively:

Two covering Pin groups, $Pin + (n) \to O(n)$ and $Pin - (n) \to O(n)$ ,
The quotient projective orthogonal group, $O(n) \to PO(n)$ .

These are all 2-to-1 covers.

For the special orthogonal group, the corresponding groups are:

Spin group, $Spin(n) \to SO(n)$ ,
Projective special orthogonal group, $SO(n) \to PSO(n)$ .

Spin is a 2-to-1 cover, while in even dimension, $PSO(2 k)$ is a 2-to-1 cover, and in odd dimension $PSO(2 k + 1)$ is a 1-to-1 cover, i.e., isomorphic to $SO(2 k + 1)$ . These groups, $Spin(n)$ , $SO(n)$ , and $PSO(n)$ are Lie group forms of the compact special orthogonal Lie algebra, ${\mathfrak {so}}(n,{\mathbb {R} })$ – Spin is the simply connected form, while PSO is the centerless form, and SO is in general neither.^{[note 4]}

In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles for details.

Principal homogeneous space: Stiefel manifold

The principal homogeneous space for the orthogonal group $O(n)$ is the Stiefel manifold $V n (R n)$ of orthonormal bases (orthonormal $n$ -frames).

In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis.

The other Stiefel manifolds $V k (R n)$ for $k < n$ of incomplete orthonormal bases (orthonormal $k$ -frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any $k$ -frame can be taken to any other $k$ -frame by an orthogonal map, but this map is not uniquely determined.

Notes

↑ The analogy is stronger: Weyl groups, a class of (representations of) Coxeter groups, can be considered as simple algebraic groups over the field with one element, and there are a number of analogies between algebraic groups and vector spaces on the one hand, and Weyl groups and sets on the other.
↑ Infinite subsets of a compact space have an accumulation point and are not discrete.
↑ $O(n) \cap GL (n, Z)$ equals the signed permutation matrices because an integer vector of norm 1 must have a single non-zero entry, which must be $\pm1$ (if it has two non-zero entries or a larger entry, the norm will be larger than 1), and in an orthogonal matrix these entries must be in different coordinates, which is exactly the signed permutation matrices.
↑ In odd dimension, $SO(2 k + 1) ≅ PSO(2 k + 1)$ is centerless (but not simply connected), while in even dimension $SO(2 k)$ is neither centerless nor simply connected.

Citations

↑ For base fields of characteristic not 2, the definition in terms of a symmetric bilinear form is equivalent to that in terms of a quadratic form, but in characteristic 2 these notions differ.
↑ Hall 2015 Section 1.3.4
↑ Hall 2015 Proposition 13.10
↑ John Baez "This Week's Finds in Mathematical Physics" week 105
↑ Wilson, Robert A. (2009). The finite simple groups. Graduate Texts in Mathematics. 251. London: Springer. pp. 69–75. ISBN 978-1-84800-987-5. Zbl 1203.20012.
1 2 Knus, Max-Albert (1991), Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften, 294, Berlin etc.: Springer-Verlag, p. 224, ISBN 3-540-52117-8, Zbl 0756.11008
↑ (Taylor 1992, page 160)
↑ (Grove 2002, Theorem 6.6 and 14.16)
↑ Cassels 1978, p. 178

References

Cassels, J.W.S. (1978), Rational Quadratic Forms, London Mathematical Society Monographs, 13, Academic Press, ISBN 0-12-163260-1, Zbl 0395.10029
Grove, Larry C. (2002), Classical groups and geometric algebra, Graduate Studies in Mathematics, 39, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2019-3, MR 1859189
Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer
Taylor, Donald E. (1992), The Geometry of the Classical Groups, Sigma Series in Pure Mathematics, 9, Berlin: Heldermann Verlag, ISBN 3-88538-009-9, MR 1189139, Zbl 0767.20001

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Orthogonal group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
John Baez "This Week's Finds in Mathematical Physics" week 105
John Baez on Octonions
(in Italian) n-dimensional Special Orthogonal Group parametrization

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Orthogonal group

Name

In even and odd dimension

Difference between O(n) and SO(n) in even dimensions

Over the real number field

Geometric interpretation

Maximal tori and Weyl groups

Low-dimensional topology

Homotopy groups

Relation to KO-theory

Computation and interpretation of homotopy groups

Low-dimensional groups

Lie groups

Vector bundles

Loop spaces

Interpretation of homotopy groups

Whitehead tower

Over the complex number field

Over finite fields

The Dickson invariant

Orthogonal groups of characteristic 2

The spinor norm

Galois cohomology and orthogonal groups

Lie algebra

Related groups

Lie subgroups

Lie supergroups

Conformal group

Discrete subgroups

Covering and quotient groups

Principal homogeneous space: Stiefel manifold

See also

Specific transforms

Specific groups

Related groups

Lists of groups

Notes

Citations

References

External links