Orthogonal group

In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension 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 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 has two connected components. The one that contains the identity element is a subgroup, called the special orthogonal group, and denoted . It consists of all orthogonal matrices of determinant . 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 , and . In the other connected component all orthogonal matrices have as a determinant.
By extension, for any field , a matrix with entries in such that its inverse equals its transpose is called an "orthogonal matrix over" . The orthogonal
matrices form a subgroup, denoted , of the general linear group ; that is 
More generally, given a non-degenerate symmetric bilinear form or quadratic form 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 of dimension , the elements of the orthogonal group are, up to a uniform scaling (homothecy), the linear maps from to that map orthogonal vectors to orthogonal vectors.
In Euclidean geometry.
The orthogonal group is the subgroup of the general linear group , consisting of all endomorphisms that preserve the Euclidean norm, that is endomorphisms such that formula_2
Let be the group of the Euclidean isometries of a Euclidean space of dimension . 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 is the subgroup of the elements such that . This stabilizer is (or, more exactly, is isomorphic to) , 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 from to , 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 (for details, see ).
The kernel of is the vector space of the translations. So, the translation form a normal subgroup of , the stabilizers of two points are conjugate under the action of the translations, and all stabilizers are isomorphic to .
Moreover, the Euclidean group is a semidirect product of and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of .
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 equals , and thus the determinant of is either or . The orthogonal matrices with determinant form a subgroup called the "special orthogonal group", denoted , consisting of all direct isometries of , which are those that preserve the orientation of the space.
 is a normal subgroup of , as being the kernel of the determinant, which is a group homomorphism whose image is the multiplicative group Moreover, the orthogonal group is a semidirect product of and the group with two elements, since, given any reflection , one has .
The group with two elements } (where is the identity matrix) is a normal subgroup and even a characteristic subgroup of , and, if is even, also of . If is odd, is the internal direct product of and }. For every positive integer the cyclic group of -fold rotations is a normal subgroup of and .
Canonical form.
For any element of there is an orthogonal basis, where its matrix has the form
where the matrices are 2-by-2 rotation matrices, that is matrices of the form 
with formula_7
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 if and only if there are an even number of on the diagonal.
The special case of is known as Euler's rotation theorem, which asserts that every (non-identity) element of is a rotation about a unique axis-angle pair.
Reflections.
Reflections are the elements of whose canonical form is 
where is the 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 is the product of two reflections whose axes have an angle of .
Every element of is the product of at most 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 ) is an example of an element of that is not the product of fewer than reflections.
Symmetry group of spheres.
The orthogonal group is the symmetry group of the -sphere (for , 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 . The orientation-preserving subgroup is isomorphic (as a "real" Lie group) to the circle group, also known as , the multiplicative group of the complex numbers of absolute value equal to one. This isomorphism sends the complex number of absolute value  to the special orthogonal matrix
In higher dimension, has a more complicated structure (in particular, it is no longer commutative). The topological structures of the -sphere and are strongly correlated, and this correlation is widely used for studying both topological spaces.
Group structure.
The groups and are real compact Lie groups of dimension . The group has two connected components, with being the identity component, that is, the connected component containing the identity matrix.
As algebraic groups.
The orthogonal group can be identified with the group of the matrices such that formula_10 
Since both members of this equation are symmetric matrices, this provides formula_11 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 is an algebraic set. Moreover, it can be proved that its dimension is
which implies that is a complete intersection. This implies that all its irreducible components have the same dimension, and that it has no embedded component.
In fact, has two irreducible components, that are distinguished by the sign of the determinant (that is or ). Both are nonsingular algebraic varieties of the same dimension . The component with is .
Maximal tori and Weyl groups.
A maximal torus in a compact Lie group "G" is a maximal subgroup among those that are isomorphic to for some , where is the standard one-dimensional torus.
In and , for every maximal torus, there is a basis on which the torus consists of the block-diagonal matrices of the form 
where each belongs to . 
In and , the maximal tori have the same form, bordered by a row and a column of zeros, and 1 on the diagonal.
The Weyl group of is the semidirect product formula_14 of a normal elementary abelian 2-subgroup and a symmetric group, where the nontrivial element of each } factor of acts on the corresponding circle factor of } by inversion, and the symmetric group acts on both and } by permuting factors. The elements of the Weyl group are represented by matrices in }.
The factor is represented by block permutation matrices with 2-by-2 blocks, and a final 1 on the diagonal. The component is represented by block-diagonal matrices with 2-by-2 blocks either
with the last component chosen to make the determinant 1.
The Weyl group of is the subgroup formula_16 of that of , where is the kernel of the product homomorphism } given by formula_17; that is, is the subgroup with an even number of minus signs. The Weyl group of is represented in by the preimages under the standard injection of the representatives for the Weyl group of . Those matrices with an odd number of formula_18 blocks have no remaining final coordinate to make their determinants positive, and hence cannot be represented in .
Topology.
Low-dimensional topology.
The low-dimensional (real) orthogonal groups are familiar spaces:
Fundamental group.
In terms of algebraic topology, for the fundamental group of is cyclic of order 2, and the spin group is its universal cover. For the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the group is the unique connected 2-fold cover).
Homotopy groups.
Generally, the homotopy groups 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, is a homogeneous space for , and one has the following fiber bundle:
which can be understood as "The orthogonal group acts transitively on the unit sphere , 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 is -connected, so the homotopy groups stabilize, and for : thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.
From Bott periodicity we obtain , therefore the homotopy groups of are 8-fold periodic, meaning , and one need only to list the lower 8 homotopy groups:
Relation to KO-theory.
Via the clutching construction, homotopy groups of the stable space are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: . Setting (to make fit into the periodicity), one obtains:
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.
Lie groups.
From general facts about Lie groups, always vanishes, and is free (free abelian).
Vector bundles.
From the vector bundle point of view, is vector bundles over , 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 is dimension.
Loop spaces.
Using concrete descriptions of the loop spaces in Bott periodicity, one can interpret the higher homotopies of in terms of simpler-to-analyze homotopies of lower order. Using π0, and have two components, and have countably many components, and the rest are connected.
Interpretation of homotopy groups.
In a nutshell:
Let be any of the four division algebras , , , , and let be the tautological line bundle over the projective line , and its class in K-theory. Noting that , , , , these yield vector bundles over the corresponding spheres, and
From the point of view of symplectic geometry, can be interpreted as the Maslov index, thinking of it as the fundamental group of the stable Lagrangian Grassmannian as , so .
Whitehead tower.
The orthogonal group anchors a Whitehead tower:
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 0("O") to obtain "SO" from "O", 1("O") to obtain "Spin" from "SO", 3("O") to obtain "String" from "Spin", and then 7("O") and so on to obtain the higher order branes.
Of indefinite quadratic form over the reals.
Over the real numbers, nondegenerate quadratic forms are classified by Sylvester's law of inertia, which asserts that, on a vector space of dimension , such a form can be written as the difference of a sum of squares and a sum of squares, with . In other words, there is a basis on which the matrix of the quadratic form is a diagonal matrix, with entries equal to , and entries equal to . The pair 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 . Moreover, as a quadratic form and its opposite have the same orthogonal group, one has .
The standard orthogonal group is . So, in the remainder of this section, it is supposed that neither nor is zero.
The subgroup of the matrices of determinant 1 in is denoted . The group 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 .
The group is the Lorentz group that is fundamental in relativity theory. Here the corresponds to space coordinates, and corresponds to the time coordinate.
Of complex quadratic forms.
Over the field of complex numbers, every non-degenerate quadratic form in variables is equivalent to formula_24. Thus, up to isomorphism, there is only one non-degenerate complex quadratic space of dimension , and one associated orthogonal group, usually denoted . It is the group of "complex orthogonal matrices", complex matrices whose product with their transpose is the identity matrix.
As in the real case, has two connected components. The component of the identity consists of all matrices of determinant in ; it is denoted .
The groups and are complex Lie groups of dimension over (the dimension over is twice that). For , these groups are noncompact.
As in the real case, is not simply connected: For , the fundamental group of is cyclic of order 2, whereas the fundamental group of is .
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 can be decomposed as a direct sum of pairwise orthogonal subspaces
where each is a hyperbolic plane (that is there is a basis such that the matrix of the restriction of to has the form formula_26), and the restriction of to is anisotropic (that is, for every nonzero in ).
Chevalley–Warning theorem asserts that over a finite field the dimension of is at most two.
If the dimension of is odd, the dimension of is thus equal to one, and its matrix is congruent either to formula_27 or to formula_28 where is a non-square scalar. It results that there is only one orthogonal group that is denoted , where is the number of elements of the finite field (a power of an odd prime).
If the dimension of is two and is not a square in the ground field (that is, if its number of elements is congruent to 3 modulo 4), the matrix of the restriction of to is congruent to either or , where is the 2×2 identity matrix. If the dimension of is two and is a square in the ground field (that is, if is congruent to 1, modulo 4) the matrix of the restriction of to is congruent to formula_29 is any non-square scalar.
This implies that if the dimension of is even, there are only two orthogonal groups, depending whether the dimension of zero or two. They are denoted respectively and .
The orthogonal group is a dihedral group of order , where .
2</math> and formula_30
In the real case, the corresponding isomorphisms are:
where is the circle of the complex numbers of norm one;
with formula_33 and formula_34
When the characteristic is not two, the order of the orthogonal groups are
In characteristic two, the formulas are the same, except that the factor of formula_38 must be removed.
The Dickson invariant.
For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to the quotient group (integers modulo 2), taking the value in case the element is the product of an even number of reflections, and the value of 1 otherwise.
Algebraically, the Dickson invariant can be defined as , where is the identity . 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 and usually has index 2 in . When the characteristic of is not 2, the Dickson Invariant is whenever the determinant is . Thus when the characteristic is not 2, is commonly defined to be the elements of with determinant . Each element in has determinant . Thus in characteristic 2, the determinant is always .
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.)
The spinor norm.
The spinor norm is a homomorphism from an orthogonal group over a field to the quotient group (the multiplicative group of the field up to multiplication by square elements), that takes reflection in a vector of norm to the image of in .
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 , 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.
Here 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 , which is simply the group of -valued points, to is essentially the spinor norm, because is isomorphic to the multiplicative group of the field modulo squares.
There is also the connecting homomorphism from of the orthogonal group, to the 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 and consists of the skew-symmetric matrices, with the Lie bracket given by the commutator. One Lie algebra corresponds to both groups. It is often denoted by formula_40 or formula_41, and called the orthogonal Lie algebra or special orthogonal Lie algebra. Over real numbers, these Lie algebras for different are the compact real forms of two of the four families of semisimple Lie algebras: in odd dimension , where , while in even dimension , where .
Since the group 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 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 formula_42 (over a field with characteristic not equal to 2) with a nondegenerate symmetric bilinear form formula_43, the special orthogonal Lie algebra consists of tracefree endomorphisms formula_44 which are skew-symmetric for this form (formula_45). Over a field of characteristic 2 we consider instead the alternating endomorphisms. Concretely we can equate these with the alternating tensors formula_46. The correspondence is given by:
This description applies equally for the indefinite special orthogonal Lie algebras formula_48 for symmetric bilinear forms with signature formula_49.
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 and 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, 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:
Lie supergroups.
The orthogonal group is also an important subgroup of various Lie groups:
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 is denoted for the conformal orthogonal group, and consists of the product of the orthogonal group with the group of dilations. If is odd, these two subgroups do not intersect, and they are a direct product: , where } is the real multiplicative group, while if is even, these subgroups intersect in , so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: .
Similarly one can define ; note that this is always: .
Discrete subgroups.
As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups. 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:
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:
These are all 2-to-1 covers.
For the special orthogonal group, the corresponding groups are:
Spin is a 2-to-1 cover, while in even dimension, is a 2-to-1 cover, and in odd dimension is a 1-to-1 cover; i.e., isomorphic to . These groups, , , and are Lie group forms of the compact special orthogonal Lie algebra, formula_55 – Spin is the simply connected form, while PSO is the centerless form, and SO is in general neither.
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 is the Stiefel manifold of orthonormal bases (orthonormal -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 for of "incomplete" orthonormal bases (orthonormal -frames) are still homogeneous spaces for the orthogonal group, but not "principal" homogeneous spaces: any -frame can be taken to any other -frame by an orthogonal map, but this map is not uniquely determined.