Orthogonal group

Group of isometries of a Euclidean vector space or, more generally, of a vector space equipped with a quadratic form

In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication (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 n has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). 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 SO(2), SO(3) and SO(4). The other component consists of all orthogonal matrices of determinant –1. 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 F, a n×n matrix with entries in F such that its inverse equals its transpose is called an orthogonal matrix over F. The n×n orthogonal matrices form a subgroup, denoted O(n, F), of the general linear group GL(n, F); that is

More generally, given a non-degenerate symmetric bilinear form or quadratic form[1] 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.

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

Let E(n) be the group of the Euclidean isometries of a Euclidean space S of dimension n. 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 xS is the subgroup of the elements g ∈ E(n) such that g(x) = x. This stabilizer is (or, more exactly, is isomorphic to) O(n), 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 p from E(n) to O(n), which is defined by

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 g (for details, see Affine space § Subtraction and Weyl's axioms).

The kernel of p is the vector space of the translations. So, the translation form a normal subgroup of E(n), the stabilizers of two points are conjugate under the action of the translations, and all stabilizers are isomorphic to O(n).

Moreover, the Euclidean group is a semidirect product of O(n) and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of O(n).

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

It follows from this equation that the square of the determinant of Q equals 1, and thus the determinant of Q is either 1 or –1. The orthogonal matrices with determinant 1 form a subgroup called the special orthogonal group, denoted SO(n), consisting of all direct isometries of O(n), which are those that preserve the orientation of the space.

For any element of O(n) there is an orthogonal basis, where its matrix has the form

where the matrices R1, ..., Rk are 2-by-2 rotation matrices, that is matrices of the form

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 1.

The element belongs to SO(n) if and only if there are an even number of –1 on the diagonal.

The special case of n = 3 is known as Euler's rotation theorem, which asserts that every (non-identity) element of SO(3) is a rotation about a unique axis-angle pair.

where I is the (n–1)×(n–1) 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 is the product of two reflections. More precisely, a rotation of angle 𝜃 is the product of two reflections whose axes have an angle of 𝜃 / 2.

Every element of O(n) is the product of at most n reflections. 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 v ↦ −v) is an example of an element of O(n) that is not the product of fewer than n reflections.

The orthogonal group O(n) is the symmetry group of the (n − 1)-sphere (for n = 3, 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 O(2). The orientation-preserving subgroup SO(2) is isomorphic (as a real Lie group) to the circle group, also known as U(1), the multiplicative group of the complex numbers of absolute value equal to one. This isomorphism sends the complex number exp(φ i) = cos(φ) + i sin(φ) of absolute value 1 to the special orthogonal matrix

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

The groups O(n) and SO(n) are real compact Lie groups of dimension n(n − 1)/2. The group O(n) has two connected components, with SO(n) being the identity component, that is, the connected component containing the identity matrix.

This proves that O(n) is an algebraic set. Moreover, it can be proved that its dimension is

which implies that O(n) is a complete intersection. This implies that all its irreducible components have the same dimension, and that it has no embedded component. In fact, O(n) has two irreducible components, that are distinguished by the sign of the determinant (that is det(A) = 1 or det(A) = –1). Both are nonsingular algebraic varieties of the same dimension n(n – 1) / 2. The component with det(A) = 1 is SO(n).

A maximal torus in a compact Lie group G is a maximal subgroup among those that are isomorphic to Tk for some k, where T = SO(2) is the standard one-dimensional torus.[2]

In O(2n) and SO(2n), for every maximal torus, there is a basis on which the torus consists of the block-diagonal matrices of the form

where each Rj belongs to SO(2). In O(2n + 1) and SO(2n + 1), the maximal tori have the same form, bordered by a row and a column of zeros, and 1 on the diagonal.

In terms of algebraic topology, for n > 2 the fundamental group of SO(n, R) is cyclic of order 2,[4] 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 πk(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:

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

which can be understood as "The orthogonal group O(n + 1) acts transitively on the unit sphere Sn, 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) → O(n + 1) is (n − 1)-connected, so the homotopy groups stabilize, and πk(O(n + 1)) = πk(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 Ω8OO, therefore the homotopy groups of O are 8-fold periodic, meaning πk + 8(O) = πk(O), and one need only to list the lower 8 homotopy groups:

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: πk(O) = πk + 1(BO). Setting KO = BO × Z = Ω−1O × Z (to make π0 fit into the periodicity), one obtains:

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

From general facts about Lie groups, π2(G) always vanishes, and π3(G) is free (free abelian).

From the vector bundle point of view, π0(KO) is vector bundles over S0, 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 π0(KO) = Z is dimension.

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, KO = BO × Z and KSp = BSp × Z have countably many components, and the rest are connected.

Let R be any of the four division algebras R, C, H, O, and let LR be the tautological line bundle over the projective line RP1, and [LR] its class in K-theory. Noting that RP1 = S1, CP1 = S2, HP1 = S4, OP1 = S8, these yield vector bundles over the corresponding spheres, and

From the point of view of symplectic geometry, π0(KO) ≅ π8(KO) = Z can be interpreted as the Maslov index, thinking of it as the fundamental group π1(U/O) of the stable Lagrangian Grassmannian as U/O ≅ Ω7(KO), so π1(U/O) = π1+7(KO).

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 π0(O) to obtain SO from O, π1(O) to obtain Spin from SO, π3(O) to obtain String from Spin, and then π7(O) and so on to obtain the higher order branes.

Over the real numbers, nondegenerate quadratic forms are classified by Sylvester's law of inertia, which asserts that, on a vector space of dimension n, such a form can be written as the difference of a sum of p squares and a sum of q squares, with p + q = n. In other words, there is a basis on which the matrix of the quadratic form is a diagonal matrix, with p entries equal to 1, and q entries equal to –1. The pair (p, q) 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 O(p, q). Moreover, as a quadratic form and its opposite have the same orthogonal group, one has O(p, q) = O(q, p).

The standard orthogonal group is O(n) = O(n, 0) = O(0, n). So, in the remainder of this section, it is supposed that neither p nor q is zero.

The subgroup of the matrices of determinant 1 in O(p, q) is denoted SO(p, q). The group O(p, q) 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 SO+(p, q).

The group O(3, 1) is the Lorentz group that is fundamental in relativity theory. Here the 3 corresponds to space coordinates, and 1 corresponds to the time coordinate.

As in the real case, O(n, C) has two connected components. The component of the identity consists of all matrices of determinant 1 in O(n, C); it is denoted SO(n, C).

The groups O(n, C) and SO(n, C) are complex Lie groups of dimension n(n − 1)/2 over C (the dimension over R is twice that). For n ≥ 2, these groups are noncompact. 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 Z.

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 Q can be decomposed as a direct sum of pairwise orthogonal subspaces

Chevalley–Warning theorem asserts that over a finite field the dimension of W is at most two.

This implies that if the dimension of V is even, there are only two orthogonal groups, depending whether the dimension of W zero or two. They are denoted respectively O+(2n, q) and O(2n, q).[6]

The orthogonal group Oϵ(2, q) is a dihedral group of order 2(qϵ), where ϵ = ±.

where a2ωb2 = 1 and ϵ = ±1. Moreover, the determinant of the matrix is ϵ.

For further studying the orthogonal group, it is convenient to introduce a square root α of ω. This square root belongs to Fq if the orthogonal group is O+(2, q), and to Fq2 otherwise. Setting x = a + αb, and y = aαb, one has

In the case of O+(2n, q), the image is the multiplicative group of Fq, which is a cyclic group of order q.

where g is a primitive element of Fq2 and T is the multiplicative group of the element of norm one in Fq2 ;

When the characteristic is not two, the order of the orthogonal groups are[7]

For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to the quotient group Z/2Z (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.[8]

Algebraically, the Dickson invariant can be defined as D(f) = rank(If) 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[8] and usually has index 2 in O(n, F ).[9] 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 ±1. 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).

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.)

The spinor norm is a homomorphism from an orthogonal group over a field F to the quotient group F×/(F×)2 (the multiplicative group of the field F up to multiplication by square elements), that takes reflection in a vector of norm n to the image of n in F×/(F×)2.[11]

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.

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 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.

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(OV), which is simply the group OV(F) of F-valued points, to H12) is essentially the spinor norm, because H12) 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.

Since the group SO(n) 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 SO(n) 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.

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.

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) ⊂ U(n) ⊂ USp(2n) and USp(n) ⊂ U(n) ⊂ O(2n) 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.

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:

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

Similarly one can define CSO(n); note that this is always: CSO(n) = CO(n) ∩ GL+(n) = SO(n) × R+.

As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups.[note 1] 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.

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

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

The principal homogeneous space for the orthogonal group O(n) is the Stiefel manifold Vn(Rn) 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 Vk(Rn) 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.