Orthogonal group

Template:Group theory sidebar In mathematics, the orthogonal group in dimension Template:Math, denoted Template:Math, is the group of distance-preserving transformations of a Euclidean space of dimension Template:Math 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 Template:Math orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact.

The orthogonal group in dimension Template:Math has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted Template:Math. It consists of all orthogonal matrices of determinant 1. This group is also called the rotation group, generalizing the fact that 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 Template:Math, Template:Math and Template:Math. The other component consists of all orthogonal matrices of determinant Template:Math. This component does not form a group, as the product of any two of its elements is of determinant 1, and therefore not an element of the component.

By extension, for any field Template:Math, an Template:Math matrix with entries in Template:Math such that its inverse equals its transpose is called an orthogonal matrix over Template:Math. The Template:Math orthogonal matrices form a subgroup, denoted Template:Math, of the general linear group Template:Math; that is <math display="block">\operatorname{O}(n, F) = \left\{Q \in \operatorname{GL}(n, F) \mid Q^\mathsf{T} Q = Q Q^\mathsf{T} = I \right\} .</math>

More generally, given a non-degenerate symmetric bilinear form or quadratic form<ref>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.</ref> on a vector space over a field, the orthogonal group of the form is the group of invertible linear maps that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot product, or, equivalently, the quadratic form is the sum of the square of the coordinates.

All orthogonal groups are algebraic groups, since the condition of preserving a form can be expressed as an equality of matrices.

Name

The name of "orthogonal group" originates from the following characterization of its elements. Given a Euclidean vector space Template:Math of dimension Template:Math, the elements of the orthogonal group Template:Math are, up to a uniform scaling (homothecy), the linear maps from Template:Math to Template:Math that map orthogonal vectors to orthogonal vectors.

In Euclidean geometry

The orthogonal Template:Math is the subgroup of the general linear group Template:Math, consisting of all endomorphisms that preserve the Euclidean norm; that is, endomorphisms Template:Math such that <math>\|g(x)\| = \|x\|.</math>

Let Template:Math be the group of the Euclidean isometries of a Euclidean space Template:Math of dimension Template:Math. This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are isomorphic. The stabilizer subgroup of a point Template:Math is the subgroup of the elements Template:Math such that Template:Math. This stabilizer is (or, more exactly, is isomorphic to) Template:Math, since the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space.

There is a natural group homomorphism Template:Math from Template:Math to Template:Math, which is defined by

<math>p(g)(y-x) = g(y)-g(x),</math>

where, as usual, the subtraction of two points denotes the translation vector that maps the second point to the first one. This is a well defined homomorphism, since a straightforward verification shows that, if two pairs of points have the same difference, the same is true for their images by Template:Math (for details, see Template:Slink).

The kernel of Template:Math is the vector space of the translations. So, the translations form a normal subgroup of Template:Math, the stabilizers of two points are conjugate under the action of the translations, and all stabilizers are isomorphic to Template:Math.

Moreover, the Euclidean group is a semidirect product of Template:Math and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of Template:Math.

Special orthogonal group

By choosing an orthonormal basis of a Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of orthogonal matrices, which are the matrices such that

<math> Q Q^\mathsf{T} = I. </math>

It follows from this equation that the square of the determinant of Template:Mvar equals Template:Math, and thus the determinant of Template:Mvar is either Template:Math or Template:Math. The orthogonal matrices with determinant Template:Math form a subgroup called the special orthogonal group, denoted Template:Math, consisting of all direct isometries of Template:Math, which are those that preserve the orientation of the space.

Template:Math is a normal subgroup of Template:Math, as being the kernel of the determinant, which is a group homomorphism whose image is the multiplicative group Template:Math. This implies that the orthogonal group is an internal semidirect product of Template:Math and any subgroup formed with the identity and a reflection.

The group with two elements Template:Math (where Template:Mvar is the identity matrix) is a normal subgroup and even a characteristic subgroup of Template:Math, and, if Template:Math is even, also of Template:Math. If Template:Math is odd, Template:Math is the internal direct product of Template:Math and Template:Math.

The group Template:Math is abelian (whereas Template:Math is not abelian when Template:Math). Its finite subgroups are the cyclic group Template:Math of [[rotational symmetry|Template:Math-fold rotations]], for every positive integer Template:Mvar. All these groups are normal subgroups of Template:Math and Template:Math.

Canonical form

For any element of Template:Math there is an orthogonal basis, where its matrix has the form

<math>\begin{bmatrix}

 \begin{matrix}
   R_1 &        & \\
       & \ddots & \\
       &        & R_k
 \end{matrix} & 0 \\
 0 & \begin{matrix}
   \pm 1 &        & \\
         & \ddots & \\
         &        & \pm 1
 \end{matrix}\\

\end{bmatrix},</math> where there may be any number, including zero, of ±1's; and where the matrices Template:Math are 2-by-2 rotation matrices, that is matrices of the form

<math>\begin{bmatrix}a&-b\\b&a\end{bmatrix},</math>

with Template:Math.

This results from the spectral theorem by regrouping eigenvalues that are complex conjugate, and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to Template:Math.

The element belongs to Template:Math if and only if there are an even number of Template:Math on the diagonal. A pair of eigenvalues Template:Math can be identified with a rotation by Template:Math and a pair of eigenvalues Template:Math can be identified with a rotation by Template:Math.

The special case of Template:Math is known as Euler's rotation theorem, which asserts that every (non-identity) element of Template:Math is a rotation about a unique axis–angle pair.

Reflections

Reflections are the elements of Template:Math whose canonical form is

<math>\begin{bmatrix}-1&0\\0&I\end{bmatrix},</math>

where Template:Mvar is the Template:Math identity matrix, and the zeros denote row or column zero matrices. In other words, a reflection is a transformation that transforms the space in its mirror image with respect to a hyperplane.

In dimension two, every rotation can be decomposed into a product of two reflections. More precisely, a rotation of angle Template:Math is the product of two reflections whose axes form an angle of Template:Math.

A product of up to Template:Math elementary reflections always suffices to generate any element of Template:Math. This results immediately from the above canonical form and the case of dimension two.

The Cartan–Dieudonné theorem is the generalization of this result to the orthogonal group of a nondegenerate quadratic form over a field of characteristic different from two.

The reflection through the origin (the map Template:Math) is an example of an element of Template:Math that is not a product of fewer than Template:Math reflections.

Symmetry group of spheres

The orthogonal group Template:Math is the symmetry group of the [[n-sphere|Template:Math-sphere]] (for Template:Math, this is just the sphere) and all objects with spherical symmetry, if the origin is chosen at the center.

The symmetry group of a circle is Template:Math. The orientation-preserving subgroup Template:Math is isomorphic (as a real Lie group) to the circle group, also known as Template:Math, the multiplicative group of the complex numbers of absolute value equal to one. This isomorphism sends the complex number Template:Math of absolute value Template:Math to the special orthogonal matrix

<math>\begin{bmatrix}

 \cos(\varphi) & -\sin(\varphi) \\
 \sin(\varphi) &  \cos(\varphi)

\end{bmatrix}.</math>

In higher dimension, Template:Math has a more complicated structure (in particular, it is no longer commutative). The topological structures of the Template:Mvar-sphere and Template:Math are strongly correlated, and this correlation is widely used for studying both topological spaces.

Group structure

The groups Template:Math and Template:Math are real compact Lie groups of dimension Template:Math. The group Template:Math has two connected components, with Template:Math being the identity component, that is, the connected component containing the identity matrix.

As algebraic groups

The orthogonal group Template:Math can be identified with the group of the matrices Template:Mvar such that Template:Math. Since both members of this equation are symmetric matrices, this provides Template:Math equations that the entries of an orthogonal matrix must satisfy, and which are not all satisfied by the entries of any non-orthogonal matrix.

This proves that Template:Math is an algebraic set. Moreover, it can be provedTemplate:Cn that its dimension is

<math>\frac{n(n - 1)}{2} = n^2 - \frac{n(n + 1)}{2},</math>

which implies that Template:Math is a complete intersection. This implies that all its irreducible components have the same dimension, and that it has no embedded component. In fact, Template:Math has two irreducible components, that are distinguished by the sign of the determinant (that is Template:Math or Template:Math). Both are nonsingular algebraic varieties of the same dimension Template:Math. The component with Template:Math is Template:Math.

Maximal tori and Weyl groups

A maximal torus in a compact Lie group G is a maximal subgroup among those that are isomorphic to Template:Math for some Template:Mvar, where Template:Math is the standard one-dimensional torus.<ref>Template:Harvnb Theorem 11.2</ref>

In Template:Math and Template:Math, for every maximal torus, there is a basis on which the torus consists of the block-diagonal matrices of the form

<math>\begin{bmatrix}

 R_1 &        & 0    \\
     & \ddots &      \\
 0   &        & R_n

\end{bmatrix},</math> where each Template:Math belongs to Template:Math. In Template:Math and Template:Math, the maximal tori have the same form, bordered by a row and a column of zeros, and Template:Math on the diagonal.

The Weyl group of Template:Math is the semidirect product <math>\{\pm 1\}^n \rtimes S_n</math> of a normal elementary abelian 2-subgroup and a symmetric group, where the nontrivial element of each Template:Math factor of Template:Math acts on the corresponding circle factor of Template:Math} by inversion, and the symmetric group Template:Math acts on both Template:Math and Template:Math} by permuting factors. The elements of the Weyl group are represented by matrices in Template:Math. The Template:Math factor is represented by block permutation matrices with 2-by-2 blocks, and a final Template:Math on the diagonal. The Template:Math component is represented by block-diagonal matrices with 2-by-2 blocks either

<math>\begin{bmatrix}

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

</math> with the last component Template:Math chosen to make the determinant Template:Math.

The Weyl group of Template:Math is the subgroup <math>H_{n-1} \rtimes S_n < \{\pm 1\}^n \rtimes S_n</math> of that of Template:Math, where Template:Math is the kernel of the product homomorphism Template:Math given by <math>\left(\varepsilon_1, \ldots, \varepsilon_n\right) \mapsto \varepsilon_1 \cdots \varepsilon_n</math>; that is, Template:Math is the subgroup with an even number of minus signs. The Weyl group of Template:Math is represented in Template:Math by the preimages under the standard injection Template:Math of the representatives for the Weyl group of Template:Math. Those matrices with an odd number of <math>\begin{bmatrix}

 0 & 1 \\
 1 & 0

\end{bmatrix}</math> blocks have no remaining final Template:Math coordinate to make their determinants positive, and hence cannot be represented in Template:Math.

Topology

Template:Confusing section {{safesubst:#invoke:Unsubst||date=__DATE__|$B= Template:Ambox }}

Low-dimensional topology

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

• Template:Math, a two-point discrete space
• Template:Math
• Template:Math is Template:Math
• Template:Math is Template:Math <ref>Template:Harvnb Section 1.3.4</ref>
• Template:Math is doubly covered by Template:Math.

Fundamental group

In terms of algebraic topology, for Template:Math the fundamental group of Template:Math is cyclic of order 2,<ref>Template:Harvnb Proposition 13.10</ref> and the spin group Template:Math is its universal cover. For Template:Math the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the group Template:Math is the unique connected 2-fold cover).

Homotopy groups

Generally, the homotopy groups Template:Math 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:

<math>\operatorname{O}(0) \subset \operatorname{O}(1)\subset \operatorname{O}(2) \subset \cdots \subset O = \bigcup_{k=0}^\infty \operatorname{O}(k)</math>

Since the inclusions are all closed, hence cofibrations, this can also be interpreted as a union. On the other hand, Template:Math is a homogeneous space for Template:Math, and one has the following fiber bundle:

<math>\operatorname{O}(n) \to \operatorname{O}(n + 1) \to S^n,</math>

which can be understood as "The orthogonal group Template:Math acts transitively on the unit sphere Template:Math, 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 Template:Math is [[n-connected|Template:Math-connected]], so the homotopy groups stabilize, and Template:Math for Template:Math: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.

From Bott periodicity we obtain Template:Math, therefore the homotopy groups of Template:Math are 8-fold periodic, meaning Template:Math, and so one need list only the first 8 homotopy groups:

<math>\begin{align}

 \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{align}</math>

Relation to KO-theory

Via the clutching construction, homotopy groups of the stable space Template:Math are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: Template:Math. Setting Template:Math (to make Template:Math fit into the periodicity), one obtains:

<math>\begin{align}

 \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}\\
 \pi_5 (KO) &= 0\\
 \pi_6 (KO) &= 0\\
 \pi_7 (KO) &= 0

\end{align}</math>

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.

• Template:Math, from orientation-preserving/reversing (this class survives to Template:Math and hence stably)
• Template:Math, which is spin comes from Template:Math.
• Template:Math, which surjects onto Template:Math; this latter thus vanishes.

Lie groups

From general facts about Lie groups, Template:Math always vanishes, and Template:Math is free (free abelian).

Vector bundles

Template:Confusing section Template:Math is a vector bundle over Template:Math, which consists of 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 Template:Math is the dimension.

Loop spaces

Using concrete descriptions of the loop spaces in Bott periodicity, one can interpret the higher homotopies of Template:Math in terms of simpler-to-analyze homotopies of lower order. Using π₀, Template:Math and Template:Math have two components, Template:Math and Template:Math have countably many components, and the rest are connected.

Interpretation of homotopy groups

In a nutshell:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

• Template:Math is about dimension
• Template:Math is about orientation
• Template:Math is about spin
• Template:Math is about topological quantum field theory.

Let Template:Math be any of the four division algebras Template:Math, Template:Math, Template:Math, Template:Math, and let Template:Math be the tautological line bundle over the projective line Template:Math, and Template:Math its class in K-theory. Noting that Template:Math, Template:Math, Template:Math, Template:Math, these yield vector bundles over the corresponding spheres, and

• Template:Math is generated by Template:Math
• Template:Math is generated by Template:Math
• Template:Math is generated by Template:Math
• Template:Math is generated by Template:Math

From the point of view of symplectic geometry, Template:Math can be interpreted as the Maslov index, thinking of it as the fundamental group Template:Math of the stable Lagrangian Grassmannian as Template:Math, so Template:Math.

Whitehead tower

The orthogonal group anchors a Whitehead tower:

<math>\cdots \rightarrow \operatorname{Fivebrane}(n) \rightarrow \operatorname{String}(n) \rightarrow \operatorname{Spin}(n) \rightarrow \operatorname{SO}(n) \rightarrow \operatorname{O}(n)</math>

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 Template:Pi₀(O) to obtain SO from O, Template:Pi₁(O) to obtain Spin from SO, Template:Pi₃(O) to obtain String from Spin, and then Template:Pi₇(O) and so on to obtain the higher order branes.

Of indefinite quadratic form over the reals

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Over the real numbers, nondegenerate quadratic forms are classified by Sylvester's law of inertia, which asserts that, on a vector space of dimension Template:Mvar, such a form can be written as the difference of a sum of Template:Mvar squares and a sum of Template:Mvar squares, with Template:Math. In other words, there is a basis on which the matrix of the quadratic form is a diagonal matrix, with Template:Mvar entries equal to Template:Math, and Template:Mvar entries equal to Template:Math. The pair Template:Math called the inertia, is an invariant of the quadratic form, in the sense that it does not depend on the way of computing the diagonal matrix.

The orthogonal group of a quadratic form depends only on the inertia, and is thus generally denoted Template:Math. Moreover, as a quadratic form and its opposite have the same orthogonal group, one has Template:Math.

The standard orthogonal group is Template:Math. So, in the remainder of this section, it is supposed that neither Template:Mvar nor Template:Mvar is zero.

The subgroup of the matrices of determinant 1 in Template:Math is denoted Template:Math. The group Template:Math has four connected components, depending on whether an element preserves orientation on either of the two maximal subspaces where the quadratic form is positive definite or negative definite. The component of the identity, whose elements preserve orientation on both subspaces, is denoted Template:Math.

The group Template:Math is the Lorentz group that is fundamental in relativity theory. Here the Template:Math corresponds to space coordinates, and Template:Math corresponds to the time coordinate.

Of complex quadratic forms

Over the field Template:Math of complex numbers, every non-degenerate quadratic form in Template:Mvar variables is equivalent to Template:Math. Thus, up to isomorphism, there is only one non-degenerate complex quadratic space of dimension Template:Mvar, and one associated orthogonal group, usually denoted Template:Math. It is the group of complex orthogonal matrices, complex matrices whose product with their transpose is the identity matrix.

As in the real case, Template:Math has two connected components. The component of the identity consists of all matrices of determinant Template:Math in Template:Math; it is denoted Template:Math.

The groups Template:Math and Template:Math are complex Lie groups of dimension Template:Math over Template:Math (the dimension over Template:Math is twice that). For Template:Math, these groups are noncompact. As in the real case, Template:Math is not simply connected: For Template:Math, the fundamental group of Template:Math is cyclic of order 2, whereas the fundamental group of Template:Math is Template:Math.

Over finite fields

Characteristic different from two

Over a field of characteristic different from two, two quadratic forms are equivalent if their matrices are congruent, that is if a change of basis transforms the matrix of the first form into the matrix of the second form. Two equivalent quadratic forms have clearly the same orthogonal group.

The non-degenerate quadratic forms over a finite field of characteristic different from two are completely classified into congruence classes, and it results from this classification that there is only one orthogonal group in odd dimension and two in even dimension.

More precisely, Witt's decomposition theorem asserts that (in characteristic different from two) every vector space equipped with a non-degenerate quadratic form Template:Mvar can be decomposed as a direct sum of pairwise orthogonal subspaces

<math>V = L_1 \oplus L_2 \oplus \cdots \oplus L_m \oplus W,</math>

where each Template:Mvar is a hyperbolic plane (that is there is a basis such that the matrix of the restriction of Template:Mvar to Template:Mvar has the form <math>\textstyle\begin{bmatrix}0&1\\1&0\end{bmatrix}</math>), and the restriction of Template:Mvar to Template:Mvar is anisotropic (that is, Template:Math for every nonzero Template:Mvar in Template:Mvar).

The Chevalley–Warning theorem asserts that, over a finite field, the dimension of Template:Mvar is at most two.

If the dimension of Template:Mvar is odd, the dimension of Template:Mvar is thus equal to one, and its matrix is congruent either to <math>\textstyle\begin{bmatrix}1\end{bmatrix}</math> or to <math>\textstyle\begin{bmatrix}\varphi\end{bmatrix},</math> where Template:Mvar is a non-square scalar. It results that there is only one orthogonal group that is denoted Template:Math, where Template:Mvar is the number of elements of the finite field (a power of an odd prime).<ref name=Wil6975>Template:Cite book</ref>

If the dimension of Template:Mvar is two and Template:Math is not a square in the ground field (that is, if its number of elements Template:Mvar is congruent to 3 modulo 4), the matrix of the restriction of Template:Mvar to Template:Mvar is congruent to either Template:Mvar or Template:Math, where Template:Mvar is the 2×2 identity matrix. If the dimension of Template:Mvar is two and Template:Math is a square in the ground field (that is, if Template:Mvar is congruent to 1, modulo 4) the matrix of the restriction of Template:Mvar to Template:Mvar is congruent to <math>\textstyle\begin{bmatrix}1&0\\0&\varphi\end{bmatrix},</math> Template:Mvar is any non-square scalar.

This implies that if the dimension of Template:Mvar is even, there are only two orthogonal groups, depending whether the dimension of Template:Mvar zero or two. They are denoted respectively Template:Math and Template:Math.<ref name=Wil6975 />

The orthogonal group Template:Math is a dihedral group of order Template:Math, where Template:Math. Template:Math proof2</math> and <math>b = \frac {x- x^{-1}}{2\alpha}.</math>

In the real case, the corresponding isomorphisms are:

<math>\begin{align}

\mathbf R/2\pi\mathbf R &\to C\\ \theta&\mapsto e^{i\theta}, \end{align}</math> where Template:Mvar is the circle of the complex numbers of norm one;

<math>\begin{align}

\mathbf C &\to \operatorname{SO}(2, \mathbf R) \\ x&\mapsto \begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}, \end{align}</math> with <math>\cos\theta = \frac {e^{i\theta}+e^{-i\theta}}2</math> and <math>\sin\theta = \frac {e^{i\theta}-e^{-i\theta}}{2i}.</math> }}

When the characteristic is not two, the order of the orthogonal groups are<ref>Template:Harv</ref>

<math>\left|\operatorname{O}(2n + 1, q)\right| = 2q^{n^2}\prod_{i=1}^{n}\left(q^{2i} - 1\right),</math>

<math>\left|\operatorname{O}^+(2n, q)\right| = 2q^{n(n-1)}\left(q^n-1\right)\prod_{i=1}^{n-1}\left(q^{2i} - 1\right),</math>

<math>\left|\operatorname{O}^-(2n, q)\right| = 2q^{n(n-1)}\left(q^n+ 1\right)\prod_{i=1}^{n-1}\left(q^{2i} - 1\right).</math>

In characteristic two, the formulas are the same, except that the factor Template:Math of Template:Math must be removed.

Dickson invariant

For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to the quotient group Template:Math (integers modulo 2), taking the value Template:Math in case the element is the product of an even number of reflections, and the value of 1 otherwise.<ref name=Knus224>Template:Citation</ref>

Algebraically, the Dickson invariant can be defined as Template:Math, where Template:Math is the identity Template:Harv. Over fields that are not of characteristic 2 it is equivalent to the determinant: the determinant is Template:Math 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<ref name=Knus224/> and usually has index 2 in Template:Math.<ref>Template:Harv</ref> When the characteristic of Template:Math is not 2, the Dickson Invariant is Template:Math whenever the determinant is Template:Math. Thus when the characteristic is not 2, Template:Math is commonly defined to be the elements of Template:Math with determinant Template:Math. Each element in Template:Math has determinant Template:Math. Thus in characteristic 2, the determinant is always Template:Math.

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.<ref>Template:Harv</ref> A reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector Template:Math takes a vector Template:Math to Template:Math where Template:Math is the bilinear formTemplate:Clarify and Template:Math is the quadratic form associated to the orthogonal geometry. Compare this to the Householder reflection of odd characteristic or characteristic zero, which takes Template:Math to Template:Math.
• The center of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since Template:Math.
• In odd dimensions Template:Math in characteristic 2, orthogonal groups over perfect fields are the same as symplectic groups in dimension Template:Math. 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 Template:Math, 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 Template:Math to the quotient group Template:Math (the multiplicative group of the field Template:Math up to multiplication by square elements), that takes reflection in a vector of norm Template:Math to the image of Template:Math in Template:Math.<ref name=C178>Template:Harvnb</ref>

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 phenomenon is concerned. The first point is that quadratic forms over a field can be identified as a Galois Template:Math, 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 determinant.

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.

<math> 1 \rightarrow \mu_2 \rightarrow \mathrm{Pin}_V \rightarrow \mathrm{O_V} \rightarrow 1 </math>

Here Template:Math 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 Template:Math, which is simply the group Template:Math of Template:Math-valued points, to Template:Math is essentially the spinor norm, because Template:Math is isomorphic to the multiplicative group of the field modulo squares.

There is also the connecting homomorphism from Template:Math of the orthogonal group, to the Template:Math 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

Template:AnchorTemplate:Anchor The Lie algebra corresponding to Lie groups Template:Math and Template:Math consists of the skew-symmetric Template:Math matrices, with the Lie bracket Template:Math given by the commutator. One Lie algebra corresponds to both groups. It is often denoted by <math>\mathfrak{o}(n, F)</math> or <math>\mathfrak{so}(n, F)</math>, and called the orthogonal Lie algebra or special orthogonal Lie algebra. Over real numbers, these Lie algebras for different Template:Math are the compact real forms of two of the four families of semisimple Lie algebras: in odd dimension Template:Math, where Template:Math, while in even dimension Template:Math, where Template:Math.

Since the group Template:Math is not simply connected, the representation theory of the orthogonal Lie algebras includes both representations corresponding to ordinary representations of the orthogonal groups, and representations corresponding to projective representations of the orthogonal groups. (The projective representations of Template:Math are just linear representations of the universal cover, the spin group Spin(n).) The latter are the so-called spin representation, which are important in physics.

More generally, given a vector space Template:Math (over a field with characteristic not equal to 2) with a nondegenerate symmetric bilinear form <math>\langle u,v \rangle</math>, the special orthogonal Lie algebra consists of tracefree endomorphisms <math>\varphi</math> which are skew-symmetric for this form (<math>\langle\varphi A, B\rangle = -\langle A, \varphi B\rangle</math>). Over a field of characteristic 2 we consider instead the alternating endomorphisms. Concretely we can equate these with the bivectors of the exterior algebra, the antisymmetric tensors of <math>\wedge^2 V</math>. The correspondence is given by:

<math>v\wedge w \mapsto \langle v,\cdot\rangle w - \langle w,\cdot\rangle v</math>

This description applies equally for the indefinite special orthogonal Lie algebras <math>\mathfrak{so}(p, q)</math> for symmetric bilinear forms with signature Template:Math.

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.

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 Template:Math and Template:Math 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, Template:Math 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:

<math>\mathrm{O}(n) \supset \mathrm{O}(n - 1)</math> – preserve an axis

<math>\mathrm{O}(2n) \supset \mathrm{U}(n) \supset \mathrm{SU}(n)</math> – Template:Math are those that preserve a compatible complex structure or a compatible symplectic structure – see 2-out-of-3 property; Template:Math also preserves a complex orientation.

<math>\mathrm{O}(2n) \supset \mathrm{USp}(n)</math>

<math>\mathrm{O}(7) \supset \mathrm{G}_2</math>

Lie supergroups

The orthogonal group Template:Math is also an important subgroup of various Lie groups:

<math>\begin{align}

    \mathrm{U}(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)

\end{align}</math>

Conformal group

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

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 Template:Math is denoted Template:Math for the conformal orthogonal group, and consists of the product of the orthogonal group with the group of dilations. If Template:Math is odd, these two subgroups do not intersect, and they are a direct product: Template:Math, where Template:Math} is the real multiplicative group, while if Template:Math is even, these subgroups intersect in Template:Math, so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: Template:Math.

Similarly one can define Template:Math; this is always: Template:Math.

Discrete subgroups

As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups.<ref group="note">Infinite subsets of a compact space have an accumulation point and are not discrete.</ref> These subgroups are known as point groups 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 Template:Math)
• Signed permutation matrices (the Coxeter group Template:Math); also equals the intersection of the orthogonal group with the integer matrices.<ref group="note">Template:Math equals the signed permutation matrices because an integer vector of norm 1 must have a single non-zero entry, which must be Template:Math (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.</ref>

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, Template:Math and Template:Math,
• The quotient projective orthogonal group, Template:Math.

These are all 2-to-1 covers.

For the special orthogonal group, the corresponding groups are:

• Spin group, Template:Math,
• Projective special orthogonal group, Template:Math.

Spin is a 2-to-1 cover, while in even dimension, Template:Math is a 2-to-1 cover, and in odd dimension Template:Math is a 1-to-1 cover; i.e., isomorphic to Template:Math. These groups, Template:Math, Template:Math, and Template:Math are Lie group forms of the compact special orthogonal Lie algebra, <math>\mathfrak{so}(n, \mathbf{R})</math> – Template:Math is the simply connected form, while Template:Math is the centerless form, and Template:Math is in general neither.<ref group="note">In odd dimension, Template:Math is centerless (but not simply connected), while in even dimension Template:Math is neither centerless nor simply connected.</ref>

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

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The principal homogeneous space for the orthogonal group Template:Math is the Stiefel manifold Template:Math of orthonormal bases (orthonormal [[k-frame|Template:Math-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 Template:Math for Template:Math of incomplete orthonormal bases (orthonormal Template:Math-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any Template:Math-frame can be taken to any other Template:Math-frame by an orthogonal map, but this map is not uniquely determined.

See also

Specific transforms

• Coordinate rotations and reflections
• Reflection through the origin

Specific groups

• rotation group, [[Rotation group SO(3)|Template:Math]]
• Template:Math

Related groups

• indefinite orthogonal group
• unitary group
• symplectic group

Lists of groups

• list of finite simple groups
• list of simple Lie groups

Representation theory

• Representations of classical Lie groups
• Brauer algebra

References

• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation

External links

• Template:Springer
• John Baez "This Week's Finds in Mathematical Physics" week 105
• John Baez on Octonions
• Template:In lang n-dimensional Special Orthogonal Group parametrization