Orthogonal

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. This is a subgroup of the general linear group GL(n,F) given by

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

where Q^T is the transpose of Q. The classical orthogonal group over the real numbers is usually just written O(n).

More generally the orthogonal group of a non-singular quadratic form over F is the group of matrices preserving the form. The Cartan-Dieudonné theorem describes the structure of the orthogonal group.

Every orthogonal matrix has determinant 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). If the characteristic of F is 2, then 1 = −1, hence O(n,F) and SO(n,F) coincide; otherwise the index of SO(n,F) in O(n,F) is 2. In characteristic 2 and even dimension, many authors define the SO(n,F) differently as the kernel of the Dickson invariant; then it usually has index 2 in O(n,F).

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.

Group (mathematics)

Group theory

Basic notions
Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product

Discrete groups
Classification of finite simple groups Cyclic group Z_n Alternating group A_n Sporadic groups Mathieu group M_11..12,M_22..24 Conway group Co_1..3 Janko group J_1..4 Fischer group F_22..24 Baby Monster group B Monster group M Other finite groups Symmetric group, S_n Dihedral group, D_n Infinite groups The integers, Z Modular groups, PSL(2,Z) and SL(2,Z)

Continuous groups
Lie group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) G₂ F₄ E₆ E₇ E₈ Lorentz group Poincaré group

Infinite dimensional groups
conformal group diffeomorphism group Loop group Quantum group O(∞) SU(∞) Sp(∞)

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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.

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 Rⁿ; it contains those which leave the origin fixed. It is the symmetry group of the sphere (n = 3) or hypersphere 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 which leave the origin fixed. It is the rotation group of the sphere and all objects with spherical symmetry, if the origin is chosen at the center.

{ I, −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 direct product of SO(n,R) and { I, −I }. The cyclic group of k-fold rotations C_k is for every positive integer k 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 R₁,...,R_k are 2-by-2 rotation matrices.

The symmetry group of a circle is O(2,R), also called Dih(S¹), where S¹ denotes the multiplicative group of complex numbers of absolute value 1.

SO(2,R) is isomorphic (as a Lie group) to the circle S¹ (circle group). This isomorphism sends the complex number exp(φi) = cos(φ) + i sin(φ) to the 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. See rotation group and the general formula for a 3 × 3 rotation matrix in terms of the axis and the angle.

In terms of algebraic topology, for n > 2 the fundamental group of SO(n,R) is cyclic of order 2, and the spinor 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 spinor group Spin(2) is the unique 2-fold cover).

The Lie algebra associated to the Lie groups O(n,R) and SO(n,R) consists of the skew-symmetric real n-by-n matrices, with the Lie bracket given by the commutator. This Lie algebra is often denoted by o(n,R) or by so(n,R).

3D isometries which leave the origin fixed

The isometries of R³ which leave the origin fixed, forming the group O(3,R), can be categorized as follows:

SO(3,R):
- identity
- rotation about an axis through the origin by an angle not equal to 180°
- rotation about an axis through the origin by an angle of 180°
the same with inversion in the origin (x is mapped to −x), i.e. respectively:
- inversion in the origin
- rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin which is perpendicular to the axis
- reflection in a plane through the origin

The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations.

See also the similar overview including translations.

Conformal group

Main article: Conformal group

Being isometries (preserving distances), orthogonal transforms also preserve angles, and are thus conformal maps, though not all conformal linear transforms are orthogonal. The group of conformal linear maps of Rⁿ is denoted CO(n), 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: $\operatorname{CO}(2n+1) = \operatorname{O}(2n+1) \times \mathbf{R}$ , while if n is even, these subgroups intersect in $\pm 1$ , so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: $\operatorname{CO}(2n) = \operatorname{O}(2n) \times \mathbf{R}^+$ .

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

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 (which 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 ≥ 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.

The complex Lie algebra associated to O(n,C) and SO(n,C) consists of the skew-symmetric complex n-by-n matrices, with the Lie bracket given by the commutator.

Topology

Low dimensional

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

$\begin{align} O(1) &= \left\{\pm 1\right\} = S^0\\ SO(1) &= \left\{1\right\} = *\\ SO(2) &= S^1\\ SO(3) &= \mathbf{RP}^3 \end{align}$

Homotopy groups

The homotopy groups of the 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

$O(0) \subset O(1)\subset O(2)\subset\cdots\subset O = \bigcup_{k=0}^\infty O(k)$

(as the inclusions are all closed inclusions, hence cofibrations, this can also be interpreted as a union).

$S n$ is a homogeneous space for $O (n + 1)$ , and one has the following fiber bundle:

$O(n) \to O(n+1) \to S^n,$

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". The map $O(n) \to O(n+1)$ is the natural inclusion.

Thus the inclusion $O(n) \to O(n+1)$ is (n-1)-connected, so the homotopy groups stabilize, and $π k (O) = π k (O (n))$ for $n > k + 1$ : thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.

Via Bott periodicity, $\Omega^8 O \simeq O$ , thus the homotopy groups of O are 8-fold periodic, meaning $π k + 8 O = π k O$ , and one need only compute the lower 8 homotopy groups to compute them all.

$\begin{align} \pi_0 O &= \mathbf Z/2\\ \pi_1 O &= \mathbf Z/2\\ \pi_2 O &= 0\\ \pi_3 O &= \mathbf Z\\ \pi_4 O &= 0\\ \pi_5 O &= 0\\ \pi_6 O &= 0\\ \pi_7 O &= \mathbf Z\\ \end{align}$

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: $π k O = π k + 1 B O$ .

Setting $KO = BO \times \mathbf Z = \Omega^{-1} O \times \mathbf Z$ (to make $π 0$ fit into the periodicity), one obtains:

$\begin{align} \pi_0 KO &= \mathbf Z\\ \pi_1 KO &= \mathbf Z/2\\ \pi_2 KO &= \mathbf Z/2\\ \pi_3 KO &= 0\\ \pi_4 KO &= \mathbf Z\\ \pi_5 KO &= 0\\ \pi_6 KO &= 0\\ \pi_7 KO &= 0\\ \end{align}$

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.

$\pi_0(O) = \pi_0(O(1)) = \mathbf Z/2$ from orientation-preserving/reversing (this class survives to $O (2)$ and hence stably)

$SO(3) = \mathbf{RP}^3 = S^3/(\mathbf Z/2)$ yields

$\pi_1(O) = \pi_1(SO(3)) = \mathbf Z/2$ which is spin
$π 2 (O) = π 2 (S O (3)) = 0$ , which surjects onto $π 2 (S O (4))$ ; 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

$\pi_0(KO) = \mathbf Z$ is dimension

Loop spaces

Using concrete descriptions of the loop spaces in Bott periodicity, one can interpret higher homotopy of O as lower homotopy of simple to analyze spaces. Using $π 0$ , O and O/U have two components, $KO = BO \times \mathbf Z$ and $KSp = BSp \times \mathbf Z$ have $\mathbf Z$ components, and the rest are connected.

Interpretation of homotopy groups

In a nutshell:¹

$\pi_0(KO) = \mathbf Z$ is dimension
$\pi_1(KO) = \mathbf Z/2$ is orientation
$\pi_2(KO) = \mathbf Z/2$ is spin
$\pi_4(KO) = \mathbf Z$ is topological quantum field theory

Let $F = \mathbf R, \mathbf C, \mathbf H, \mathbf O$ , and let $L F$ be the tautological line bundle over the projective line $\mathbf{FP}^1$ , and $L F$ its class in K-theory. Noting that $\mathbf{RP}^1 = S^1, \mathbf{CP}^1 = S^2, \mathbf{HP}^1 = S^4, \mathbf{OP}^1 = S^8$ , these yield vector bundles over the corresponding spheres, and

$π 1 (K O)$ is generated by $[L_{\mathbf R}]$
$π 2 (K O)$ is generated by $[L_{\mathbf C}]$
$π 4 (K O)$ is generated by $[L_{\mathbf H}]$
$π 8 (K O)$ is generated by $[L_{\mathbf O}]$

Over finite fields

Orthogonal groups can also be defined over finite fields $\mathbf{F}_q$ , where $q$ is a power of a prime $p$ . When defined over such fields, they come in two types in even dimension: $O + (2 n, q)$ and $O - (2 n, q)$ ; and one type in odd dimension: $O (2 n + 1, q)$ .

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 $W = 0$ , then $G$ is of plus type. If $W = < w >$ 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 these groups, O(n,q) = { A in GL(n,q) : A·A^t=I }, when the characteristic is greater than two

$|O(2n+1,q)|=2q^n\prod_{i=0}^{n-1}(q^{2n}-q^{2i}).$

If $- 1$ is a square in $\mathbf{F}_q$

$|O(2n,q)|=2(q^n-1)\prod_{i=1}^{n-1}(q^{2n}-q^{2i}).$

- 1

is a nonsquare in $\mathbf{F}_q$

$|O(2n,q)|=2(q^n+(-1)^{n+1})\prod_{i=1}^{n-1}(q^{2n}-q^{2i}).$

The Dickson invariant

For orthogonal groups in even dimensions, the Dickson invariant is a homomorphism from the orthogonal group to Z/2Z, and is 0 or 1 depending on whether an element is the product of an even or odd number of reflections. 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 extra information. In characteristic 2 many authors define the special orthogonal group to be the elements of Dickson invariant 0, rather than the elements of determinant 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 behave differently. This section lists some of the differences.

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 (Grove 2002, Theorem 6.6 and 14.16). Note that a reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthgonal to a vector u takes a vector v to v+B(v,u)/Q(u)·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·B(v,u)/Q(u)·u.

The center of the orthogonal group usually has order 1 in characteristic 2, rather than 2.

In odd dimensions 2n+1 in characteristic 2, orthogonal groups over perfect fields are the same as symplectic groups in dimension 2n. 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 2n, 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

F^*/F^*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^*/F^*2.

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 H¹, 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.

$1 \rightarrow \mu_2 \rightarrow Pin_V \rightarrow O_V \rightarrow 1$

Here μ₂ 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 H⁰(O_V) which is simply the group O_V(F) of F-valued points, to H¹(μ₂) is essentially the spinor norm, because H¹(μ₂) is isomorphic to the multiplicative group of the field modulo squares.

There is also the connecting homomorphism from H¹ of the orthogonal group, to the H² 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.

Important 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:

$O(n) \supset O(n-1)$

$O(2n) \supset SU(n)$

$O(2n) \supset USp(n)$

$O(7) \supset G_2$

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

$SU(n) \supset O(n)$

$USp(2n) \supset O(n)$

$G_2 \supset O(3)$

$F_4 \supset O(9)$

$E_6 \supset O(10)$

$E_7 \supset O(12)$

$E_8 \supset O(16)$

The group O(10) is of special importance in superstring theory because it is the symmetry group of 10 dimensional space-time.

Footnotes

^ John Baez "This Week's Finds in Mathematical Physics" week 105

References

Grove, Larry C. (2002), Classical groups and geometric algebra, Graduate Studies in Mathematics, 39, Providence, R.I.: American Mathematical Society, MR 1859189, ISBN 978-0-8218-2019-3

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