Orthogonal group
Algebraic structure → Group theory Group theory |
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In
The orthogonal group in dimension n has two
By extension, for any field F, an 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.
Name
The name of "orthogonal group" originates from the following characterization of its elements. Given a
In Euclidean geometry
The orthogonal O(n) is the subgroup of the
Let E(n) be the group of the
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
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).
Special orthogonal group
By choosing an
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.
SO(n) is a normal subgroup of O(n), as being the
The group with two elements {±I} (where I is the identity matrix) is a normal subgroup and even a characteristic subgroup of O(n), and, if n is even, also of SO(n). If n is odd, O(n) is the internal direct product of SO(n) and {±I}.
The group SO(2) is abelian (whereas SO(n) is not abelian when n > 2). Its finite subgroups are the cyclic group Ck of k-fold rotations, for every positive integer k. All these groups are normal subgroups of O(2) and SO(2).
Canonical form
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
with a2 + b2 = 1.
This results from the
The element belongs to SO(n) if and only if there are an even number of −1 on the diagonal. A pair of eigenvalues −1 can be identified with a rotation by π and a pair of eigenvalues +1 can be identified with a rotation by 0.
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.
Reflections
Reflections are the elements of O(n) whose canonical form is
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 can be decomposed into a product of two reflections. More precisely, a rotation of angle θ is the product of two reflections whose axes form an angle of θ / 2.
A product of up to n elementary reflections always suffices to generate any element of O(n). 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
Symmetry group of spheres
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
Group structure
The groups O(n) and SO(n) are real
As algebraic groups
The orthogonal group O(n) can be identified with the group of the matrices A such that ATA = I. Since both members of this equation are
This proves that O(n) is an
which implies that O(n) is a
Maximal tori and Weyl groups
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.
The Weyl group of SO(2n + 1) is the semidirect product of a normal elementary abelian 2-subgroup and a symmetric group, where the nontrivial element of each {±1} factor of {±1}n acts on the corresponding circle factor of T × {1} by inversion, and the symmetric group Sn acts on both {±1}n and T × {1} by permuting factors. The elements of the Weyl group are represented by matrices in O(2n) × {±1}. The Sn factor is represented by block permutation matrices with 2-by-2 blocks, and a final 1 on the diagonal. The {±1}n component is represented by block-diagonal matrices with 2-by-2 blocks either
with the last component ±1 chosen to make the determinant 1.
The Weyl group of SO(2n) is the subgroup of that of SO(2n + 1), where Hn−1 < {±1}n is the kernel of the product homomorphism {±1}n → {±1} given by ; that is, Hn−1 < {±1}n is the subgroup with an even number of minus signs. The Weyl group of SO(2n) is represented in SO(2n) by the preimages under the standard injection SO(2n) → SO(2n + 1) of the representatives for the Weyl group of SO(2n + 1). Those matrices with an odd number of blocks have no remaining final −1 coordinate to make their determinants positive, and hence cannot be represented in SO(2n).
Topology
![]() | This section may be confusing or unclear to readers. In particular, most notations are undefined; no context for explaining why these consideration belong to the article. Moreover, the section consists essentially in a list of advanced results without providing the information that is needed for a non-specialist for verifying them (no reference, no link to articles about the methods of computation that are used, no sketch of proofs. (November 2019) |
![]() | This section may be too technical for most readers to understand.(November 2019) |
Low-dimensional topology
The low-dimensional (real) orthogonal groups are familiar spaces:
- O(1) = S0, a discrete space
- SO(1) = {1}
- SO(2) is S1
- SO(4) is × S3.
Fundamental group
In terms of
Homotopy groups
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
From
Relation to KO-theory
Via the
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.
- π0(O) = π0(O(1)) = Z / 2Z, from orientation-preserving/reversing (this class survives to O(2) and hence stably)
- π1(O) = π1(SO(3)) = Z / 2Z, which is spin comes from SO(3) = RP3 = S3 / (Z / 2Z).
- π2(O) = π2(SO(3)) = 0, which surjects onto π2(SO(4)); this latter thus vanishes.
Lie groups
From general facts about Lie groups, π2(G) always vanishes, and π3(G) is free (free abelian).
Vector bundles
![]() | This section may be confusing or unclear to readers. (January 2024) |
π0(KO) is a
Loop spaces
Using concrete descriptions of the loop spaces in
Interpretation of homotopy groups
In a nutshell:[5]
- π0(KO) = Z is about dimension
- π1(KO) = Z / 2Z is about orientation
- π2(KO) = Z / 2Z is about spin
- π4(KO) = Z is about topological quantum field theory.
Let R be any of the four
- π1(KO) is generated by [LR]
- π2(KO) is generated by [LC]
- π4(KO) is generated by [LH]
- π8(KO) is generated by [LO]
From the point of view of
Whitehead tower
The orthogonal group anchors a
which is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing
Of indefinite quadratic form over the reals
Over the real numbers,
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
Of complex quadratic forms
Over the field C of
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 finite fields
Characteristic different from two
Over a field of characteristic different from two, two
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,
where each Li is a
The Chevalley–Warning theorem asserts that, over a finite field, the dimension of W is at most two.
If the dimension of V is odd, the dimension of W is thus equal to one, and its matrix is congruent either to or to where 𝜑 is a non-square scalar. It results that there is only one orthogonal group that is denoted O(2n + 1, q), where q is the number of elements of the finite field (a power of an odd prime).[6]
If the dimension of W is two and −1 is not a square in the ground field (that is, if its number of elements q is congruent to 3 modulo 4), the matrix of the restriction of Q to W is congruent to either I or –I, where I is the 2×2 identity matrix. If the dimension of W is two and −1 is a square in the ground field (that is, if q is congruent to 1, modulo 4) the matrix of the restriction of Q to W is congruent to φ is any non-square scalar.
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(q − ε), where ε = ±.
For studying the orthogonal group of Oε(2, q), one can suppose that the matrix of the quadratic form is because, given a quadratic form, there is a basis where its matrix is diagonalizable. A matrix belongs to the orthogonal group if AQAT = Q, that is, a2 – ωb2 = 1, ac – ωbd = 0, and c2 – ωd2 = –ω. As a and b cannot be both zero (because of the first equation), the second equation implies the existence of ε in Fq, such that c = εωb and d = εa. Reporting these values in the third equation, and using the first equation, one gets that ε2 = 1, and thus the orthogonal group consists of the matrices
where a2 – ωb2 = 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 = a – αb, one has
If and are two matrices of determinant one in the orthogonal group then
This is an orthogonal matrix with a = a1a2 + ωb1b2, and b = a1b2 + b1a2. Thus
It follows that the map (a, b) ↦ a + αb is a homomorphism of the group of orthogonal matrices of determinant one into the multiplicative group of Fq2.
In the case of O+(2n, q), the image is the multiplicative group of Fq, which is a cyclic group of order q.
In the case of O–(2n, q), the above x and y are
For finishing the proof, it suffices to verify that the group all orthogonal matrices is not abelian, and is the semidirect product of the group {1, −1} and the group of orthogonal matrices of determinant one.
The comparison of this proof with the real case may be illuminating.
Here two group isomorphisms are involved:
where g is a primitive element of Fq2 and T is the multiplicative group of the element of norm one in Fq2 ;
with and
In the real case, the corresponding isomorphisms are:
where C is the circle of the complex numbers of norm one;
with and
When the characteristic is not two, the order of the orthogonal groups are[7]
In characteristic two, the formulas are the same, except that the factor 2 of |O(2n + 1, q)| must be removed.
Dickson invariant
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(I − f) 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
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).
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 Householder reflectionof odd characteristic or characteristic zero, which takes v to v − 2·B(v, u)/Q(u) · u.
- The centerof the orthogonal group usually has order 1 in characteristic 2, rather than 2, since I = −I.
- 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 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.
Galois cohomology and orthogonal groups
In the theory of
The 'spin' name of the spinor norm can be explained by a connection to the
Here μ2 is the
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.
Lie algebra
The Lie algebra corresponding to Lie groups O(n, F ) and SO(n, F ) consists of the skew-symmetric n × n matrices, with the Lie bracket [ , ] given by the commutator. One Lie algebra corresponds to both groups. It is often denoted by or , and called the orthogonal Lie algebra or special orthogonal Lie algebra. Over real numbers, these Lie algebras for different n are the
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.
More generally, given a vector space V (over a field with characteristic not equal to 2) with a nondegenerate symmetric bilinear form (⋅, ⋅), the special orthogonal Lie algebra consists of tracefree endomorphisms which are skew-symmetric for this form (). Over a field of characteristic 2 we consider instead the alternating endomorphisms. Concretely we can equate these with the alternating tensors Λ2V. The correspondence is given by:
This description applies equally for the indefinite special orthogonal Lie algebras for symmetric bilinear forms with signature (p, q).
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 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.
Lie subgroups
In physics, particularly in the areas of
- – preserve an axis
- – U(n) are those that preserve a compatible complex structure or a compatible symplectic structure – see 2-out-of-3 property; SU(n) also preserves a complex orientation.
Lie supergroups
The orthogonal group O(n) is also an important subgroup of various Lie groups:
Conformal group
Being
Similarly one can define CSO(n); this is always: CSO(n) = CO(n) ∩ GL+(n) = SO(n) × R+.
Discrete subgroups
As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups.
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 groupAn)
- Signed permutation matrices (the Coxeter group Bn); also equals the intersection of the orthogonal group with the integer matrices.[note 2]
Covering and quotient groups
The orthogonal group is neither
- Two covering Pin groups, Pin+(n) → O(n) and Pin−(n) → O(n),
- The quotient projective orthogonal group, O(n) → PO(n).
These are all 2-to-1 covers.
For the special orthogonal group, the corresponding groups are:
- Spin group, Spin(n) → SO(n),
- Projective special orthogonal group, SO(n) → PSO(n).
Spin is a 2-to-1 cover, while in even dimension, PSO(2k) is a 2-to-1 cover, and in odd dimension PSO(2k + 1) is a 1-to-1 cover; i.e., isomorphic to SO(2k + 1). These groups, Spin(n), SO(n), and PSO(n) are Lie group forms of the compact
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 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.
See also
Specific transforms
- Coordinate rotations and reflections
- Reflection through the origin
Specific groups
- rotation group, SO(3, R)
- SO(8)
Related groups
Lists of groups
- list of finite simple groups
- list of simple Lie groups
Representation theory
Notes
- ^ Infinite subsets of a compact space have an accumulation point and are not discrete.
- ^ O(n) ∩ GL(n, Z) equals the signed permutation matrices because an integer vector of norm 1 must have a single non-zero entry, which must be ±1 (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.
- ^ In odd dimension, SO(2k + 1) ≅ PSO(2k + 1) is centerless (but not simply connected), while in even dimension SO(2k) is neither centerless nor simply connected.
Citations
- ^ 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.
- ^ Hall 2015 Theorem 11.2
- ^ Hall 2015 Section 1.3.4
- ^ Hall 2015 Proposition 13.10
- ^ Baez, John. "Week 105". This Week's Finds in Mathematical Physics. Retrieved 2023-02-01.
- ^ Zbl 1203.20012.
- ^ (Taylor 1992, p. 141)
- ^ Zbl 0756.11008
- ^ (Taylor 1992, page 160)
- ^ (Grove 2002, Theorem 6.6 and 14.16)
- ^ Cassels 1978, p. 178
References
- Zbl 0395.10029
- Grove, Larry C. (2002), Classical groups and geometric algebra, MR 1859189
- Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666
- Taylor, Donald E. (1992), The Geometry of the Classical Groups, Sigma Series in Pure Mathematics, vol. 9, Berlin: Heldermann Verlag, Zbl 0767.20001