Orthogonal group

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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). More generally the orthogonal group of a non-singular quadratic form over F is the group of matrices preserving the form.

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.

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 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 Ck 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:

where the matrices R1,...,Rk 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

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 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 (x is mapped to −x), i.e. respectively:
    • inversion
    • 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.

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.

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 a rotation is the product of an even or odd number of reflections. Over fields that are not of characteristic 2 it is more or less 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.
  • 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.

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.

See also

de:Orthogonale Gruppe es:Grupo ortogonal fr:Groupe orthogonal