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Inmathematics, theorthogonal group in dimensionn, denotedO(n), is thegroup ofdistance-preserving transformations of aEuclidean space of dimensionn that preserve a fixed point, where the group operation is given bycomposing transformations. The orthogonal group is sometimes called thegeneral orthogonal group, by analogy with thegeneral linear group. Equivalently, it is the group ofn ×northogonal matrices, where the group operation is given bymatrix multiplication (an orthogonal matrix is areal matrix whoseinverse equals itstranspose). The orthogonal group is analgebraic group and aLie group. It iscompact.

The orthogonal group in dimensionn has twoconnected components. The one that contains theidentity element is anormal subgroup, called thespecial orthogonal group, and denotedSO(n). It consists of all orthogonal matrices ofdeterminant 1. This group is also called therotation group, generalizing the fact that in dimensions 2 and 3, its elements are the usualrotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, seeSO(2),SO(3) andSO(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 fieldF, ann ×n matrix with entries inF such that its inverse equals its transpose is called anorthogonal matrix overF. Then ×n orthogonalmatrices form a subgroup, denotedO(n,F), of thegeneral linear groupGL(n,F); that is$${\displaystyle \mathrm{O}(n,F)=\left\{Q\in \mathrm{GL}(n,F)\mid Q^{\mathsf{T}}Q=Q{Q}^{\mathsf{T}}=I\right\}.}$$

More generally, given a non-degeneratesymmetric bilinear form orquadratic form^[1] on avector space over afield, theorthogonal group of the form is the group of invertiblelinear maps that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is thedot product, or, equivalently, the quadratic form is the sum of the square of the coordinates.

All orthogonal groups arealgebraic 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 aEuclidean vector spaceE of dimensionn, the elements of the orthogonal groupO(n) are,up to auniform scaling (homothecy), thelinear maps fromE toE that maporthogonal vectors to orthogonal vectors.

The orthogonalO(n) is the subgroup of thegeneral linear groupGL(n,R), consisting of allendomorphisms that preserve theEuclidean norm; that is, endomorphismsg such that${\displaystyle \Vert g(x)\Vert =\Vert x\Vert .}$

LetE(n) be the group of theEuclidean isometries of aEuclidean spaceS of dimensionn. This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension areisomorphic. Thestabilizer subgroup of a pointx ∈S is the subgroup of the elementsg ∈ E(n) such thatg(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 naturalgroup homomorphismp fromE(n) toO(n), which is defined by

${\displaystyle p(g)(y-x)=g(y)-g(x),}$

where, as usual, the subtraction of two points denotes thetranslation 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 byg (for details, seeAffine space § Subtraction and Weyl's axioms).

Thekernel ofp is the vector space of the translations. So, the translations form anormal subgroup ofE(n), the stabilizers of two points areconjugate under the action of the translations, and all stabilizers are isomorphic toO(n).

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

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

${\displaystyle Q{Q}^{\mathsf{T}}=I.}$

It follows from this equation that the square of thedeterminant ofQ equals1, and thus the determinant ofQ is either1 or−1. The orthogonal matrices with determinant1 form a subgroup called thespecial orthogonal group, denotedSO(n), consisting of alldirect isometries ofO(n), which are those that preserve theorientation of the space.

SO(n) is a normal subgroup ofO(n), as being thekernel of the determinant, which is a group homomorphism whose image is the multiplicative group{−1, +1}. This implies that the orthogonal group is an internalsemidirect product ofSO(n) and any subgroup formed with the identity and areflection.

The group with two elements{±I} (whereI is the identity matrix) is anormal subgroup and even acharacteristic subgroup ofO(n), and, ifn is even, also ofSO(n). Ifn is odd,O(n) is the internaldirect product ofSO(n) and{±I}.

The groupSO(2) isabelian (whereasSO(n) is not abelian whenn > 2). Its finite subgroups are thecyclic groupC_k ofk-fold rotations, for every positive integerk. All these groups are normal subgroups ofO(2) andSO(2).

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

where there may be any number, including zero, of ±1's; and where the matricesR₁, ...,R_k are 2-by-2 rotation matrices, that is matrices of the form

witha² +b² = 1.

This results from thespectral theorem by regroupingeigenvalues that arecomplex conjugate, and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to1.

The element belongs toSO(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 byπ and a pair of eigenvalues+1 can be identified with a rotation by0.

The special case ofn = 3 is known asEuler's rotation theorem, which asserts that every (non-identity) element ofSO(3) is arotation about a unique axis–angle pair.

Reflections are the elements ofO(n) whose canonical form is

whereI 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 itsmirror image with respect to ahyperplane.

In dimension two,every rotation can be decomposed into a product of two reflections. More precisely, a rotation of angleθ is the product of two reflections whose axes form an angle ofθ / 2.

A product of up ton elementary reflections always suffices to generate any element ofO(n). This results immediately from the above canonical form and the case of dimension two.

TheCartan–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.

Thereflection through the origin (the mapv ↦ −v) is an example of an element ofO(n) that is not a product of fewer thann reflections.

The orthogonal groupO(n) is thesymmetry group of the(n − 1)-sphere (forn = 3, this is just thesphere) and all objects with spherical symmetry, if the origin is chosen at the center.

Thesymmetry group of acircle isO(2). The orientation-preserving subgroupSO(2) is isomorphic (as areal Lie group) to thecircle group, also known asU(1), the multiplicative group of thecomplex numbers of absolute value equal to one. This isomorphism sends the complex numberexp(φi) = cos(φ) +i sin(φ) ofabsolute value 1 to the special orthogonal matrix

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

The groupsO(n) andSO(n) are realcompactLie groups ofdimensionn(n − 1) / 2. The groupO(n) has twoconnected components, withSO(n) being theidentity component, that is, the connected component containing theidentity matrix.

The orthogonal groupO(n) can be identified with the group of the matricesA such thatA^TA =I. Since both members of this equation aresymmetric matrices, this providesn(n + 1) / 2 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 thatO(n) is analgebraic set. Moreover, it can be proved^{[citation needed]} that its dimension is

${\displaystyle \frac{n(n-1)}{2}=n^2-\frac{n(n+1)}{2},}$

which implies thatO(n) is acomplete intersection. This implies that all itsirreducible components have the same dimension, and that it has noembedded component.In fact,O(n) has two irreducible components, that are distinguished by the sign of the determinant (that isdet(A) = 1 ordet(A) = −1). Both arenonsingular algebraic varieties of the same dimensionn(n − 1) / 2. The component withdet(A) = 1 isSO(n).

Amaximal torus in a compactLie groupG is a maximal subgroup among those that are isomorphic toT^k for somek, whereT = SO(2) is the standard one-dimensional torus.^[2]

InO(2n) andSO(2n), for every maximal torus, there is a basis on which the torus consists of theblock-diagonal matrices of the form

where eachR_j belongs toSO(2). InO(2n + 1) andSO(2n + 1), the maximal tori have the same form, bordered by a row and a column of zeros, and1 on the diagonal.

TheWeyl group ofSO(2n + 1) is thesemidirect product${\displaystyle \{\pm 1{\}}^n⋊S_n}$ of a normalelementary abelian2-subgroup and asymmetric group, where the nontrivial element of each{±1} factor of{±1}ⁿ acts on the corresponding circle factor ofT × {1} byinversion, and the symmetric groupS_n acts on both{±1}ⁿ andT × {1} by permuting factors. The elements of the Weyl group are represented by matrices inO(2n) × {±1}.TheS_n factor is represented by block permutation matrices with 2-by-2 blocks, and a final1 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 determinant1.

The Weyl group ofSO(2n) is the subgroup of that ofSO(2n + 1), whereH_n−1 < {±1}ⁿ is thekernel of the product homomorphism{±1}ⁿ → {±1} given by${\displaystyle \left({\epsilon }_1,\dots ,{\epsilon }_n\right)\mapsto {\epsilon }_1\cdots {\epsilon }_n}$; that is,H_n−1 < {±1}ⁿ is the subgroup with an even number of minus signs. The Weyl group ofSO(2n) is represented inSO(2n) by the preimages under the standard injectionSO(2n) → SO(2n + 1) of the representatives for the Weyl group ofSO(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 inSO(2n).

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The low-dimensional (real) orthogonal groups are familiarspaces:

• O(1) =S⁰, atwo-pointdiscrete space
• SO(1) = {1}
• SO(2) isS¹
• SO(3) isRP^3[3]
• SO(4) isdoubly covered bySU(2) × SU(2) =S³ ×S³.

In terms ofalgebraic topology, forn > 2 thefundamental group ofSO(n,R) iscyclic of order 2,^[4] and thespin groupSpin(n) is itsuniversal cover. Forn = 2 the fundamental group isinfinite cyclic and the universal cover corresponds to thereal line (the groupSpin(2) is the unique connected2-fold cover).

Generally, thehomotopy groupsπ_k(O) of the real orthogonal group are related tohomotopy 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 thedirect limit of the sequence of inclusions:

${\displaystyle \mathrm{O}(0)\subset \mathrm{O}(1)\subset \mathrm{O}(2)\subset \cdots \subset O=\bigcup _{k=0}^{\mathrm{\infty }}\mathrm{O}(k)}$

Since the inclusions are all closed, hencecofibrations, this can also be interpreted as a union. On the other hand,Sⁿ is ahomogeneous space forO(n + 1), and one has the followingfiber bundle:

${\displaystyle \mathrm{O}(n)\to \mathrm{O}(n+1)\to S^n,}$

which can be understood as "The orthogonal groupO(n + 1) actstransitively on the unit sphereSⁿ, and thestabilizer of a point (thought of as aunit vector) is the orthogonal group of theperpendicular complement, which is an orthogonal group one dimension lower." Thus the natural inclusionO(n) → O(n + 1) is(n − 1)-connected, so the homotopy groups stabilize, andπ_k(O(n + 1)) = π_k(O(n)) forn >k + 1: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.

FromBott periodicity we obtainΩ⁸O ≃O, therefore the homotopy groups ofO are 8-fold periodic, meaningπ_{k + 8}(O) = π_k(O), and so one need list only the first 8 homotopy groups:

Via theclutching construction, homotopy groups of the stable spaceO are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1:π_k(O) = π_{k + 1}(BO). SettingKO =BO ×Z = Ω⁻¹O ×Z (to makeπ₀ fit into the periodicity), one obtains:

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

• π₀(O) = π₀(O(1)) =Z / 2Z, fromorientation-preserving/reversing (this class survives toO(2) and hence stably)
• π₁(O) = π₁(SO(3)) =Z / 2Z, which isspin comes fromSO(3) =RP³ =S³ / (Z / 2Z).
• π₂(O) = π₂(SO(3)) = 0, which surjects ontoπ₂(SO(4)); this latter thus vanishes.

From general facts aboutLie groups,π₂(G) always vanishes, andπ₃(G) is free (free abelian).

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π₀(KO) is avector bundle overS⁰, 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, soπ₀(KO) =Z is thedimension.

Using concrete descriptions of the loop spaces inBott periodicity, one can interpret the higher homotopies ofO in terms of simpler-to-analyze homotopies of lower order. Using π₀,O andO/U have two components,KO =BO ×Z andKSp =BSp ×Z havecountably many components, and the rest are connected.

In a nutshell:^[5]

• π₀(KO) =Z is aboutdimension
• π₁(KO) =Z / 2Z is aboutorientation
• π₂(KO) =Z / 2Z is aboutspin
• π₄(KO) =Z is abouttopological quantum field theory.

LetR be any of the fourdivision algebrasR,C,H,O, and letL_R be thetautological line bundle over theprojective lineRP¹, and[L_R] its class in K-theory. Noting thatRP¹ =S¹,CP¹ =S²,HP¹ =S⁴,OP¹ =S⁸, these yield vector bundles over the corresponding spheres, and

• π₁(KO) is generated by[L_R]
• π₂(KO) is generated by[L_C]
• π₄(KO) is generated by[L_H]
• π₈(KO) is generated by[L_O]

From the point of view ofsymplectic geometry,π₀(KO) ≅ π₈(KO) =Z can be interpreted as theMaslov index, thinking of it as the fundamental groupπ₁(U/O) of the stableLagrangian Grassmannian asU/O ≅ Ω⁷(KO), soπ₁(U/O) = π₁₊₇(KO).

The orthogonal group anchors aWhitehead tower:

${\displaystyle \cdots \to \mathrm{Fivebrane}(n)\to \mathrm{String}(n)\to \mathrm{Spin}(n)\to \mathrm{SO}(n)\to \mathrm{O}(n)}$

which is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructingshort exact sequences starting with anEilenberg–MacLane space for the homotopy group to be removed. The first few entries in the tower are thespin group and thestring group, and are preceded by thefivebrane group. The homotopy groups that are killed are in turnπ₀(O) to obtainSO fromO,π₁(O) to obtainSpin fromSO,π₃(O) to obtainString fromSpin, and thenπ₇(O) and so on to obtain the higher orderbranes.

Over the real numbers,nondegenerate quadratic forms are classified bySylvester's law of inertia, which asserts that, on a vector space of dimensionn, such a form can be written as the difference of a sum ofp squares and a sum ofq squares, withp +q =n. In other words, there is a basis on which the matrix of the quadratic form is adiagonal matrix, withp entries equal to1, andq entries equal to−1. The pair(p,q) called theinertia, 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 denotedO(p,q). Moreover, as a quadratic form and its opposite have the same orthogonal group, one hasO(p,q) = O(q,p).

The standard orthogonal group isO(n) = O(n, 0) = O(0,n). So, in the remainder of this section, it is supposed that neitherp norq is zero.

The subgroup of the matrices of determinant 1 inO(p,q) is denotedSO(p,q). The groupO(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 denotedSO⁺(p,q).

The groupO(3, 1) is theLorentz group that is fundamental inrelativity theory. Here the3 corresponds to space coordinates, and1 corresponds to the time coordinate.

Over the fieldC ofcomplex numbers, every non-degeneratequadratic form inn variables is equivalent tox₁² + ... +x_n². Thus, up to isomorphism, there is only one non-degenerate complexquadratic space of dimensionn, and one associated orthogonal group, usually denotedO(n,C). It is the group ofcomplex orthogonal matrices, complex matrices whose product with their transpose is the identity matrix.

As in the real case,O(n,C) has two connected components. The component of the identity consists of all matrices of determinant1 inO(n,C); it is denotedSO(n,C).

The groupsO(n,C) andSO(n,C) are complex Lie groups of dimensionn(n − 1) / 2 overC (the dimension overR is twice that). Forn ≥ 2, these groups are noncompact.As in the real case,SO(n,C) is not simply connected: Forn > 2, thefundamental group ofSO(n,C) iscyclic of order 2, whereas the fundamental group ofSO(2,C) isZ.

Over a field of characteristic different from two, twoquadratic forms areequivalent if their matrices arecongruent, 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 formQ can be decomposed as a direct sum of pairwise orthogonal subspaces

${\displaystyle V=L_1\oplus L_2\oplus \cdots \oplus L_m\oplus W,}$

where eachL_i is ahyperbolic plane (that is there is a basis such that the matrix of the restriction ofQ toL_i has the form), and the restriction ofQ toW isanisotropic (that is,Q(w) ≠ 0 for every nonzerow inW).

TheChevalley–Warning theorem asserts that, over afinite field, the dimension ofW is at most two.

If the dimension ofV is odd, the dimension ofW is thus equal to one, and its matrix is congruent either to${\displaystyle \left[\begin{array}{c}1\end{array}\right]}$ or to${\displaystyle \left[\begin{array}{c}\phi \end{array}\right],}$ where𝜑 is a non-square scalar. It results that there is only one orthogonal group that is denotedO(2n + 1,q), whereq is the number of elements of the finite field (a power of an odd prime).^[6]

If the dimension ofW is two and−1 is not a square in the ground field (that is, if its number of elementsq is congruent to 3 modulo 4), the matrix of the restriction ofQ toW is congruent to eitherI or−I, whereI is the 2×2 identity matrix. If the dimension ofW is two and−1 is a square in the ground field (that is, ifq is congruent to 1, modulo 4) the matrix of the restriction ofQ toW is congruent toφ is any non-square scalar.

This implies that if the dimension ofV is even, there are only two orthogonal groups, depending whether the dimension ofW zero or two. They are denoted respectivelyO⁺(2n,q) andO⁻(2n,q).^[6]

The orthogonal groupO^ε(2,q) is adihedral group of order2(q −ε), whereε = ±.

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

${\displaystyle \left|\mathrm{O}(2n+1,q)\right|=2q^{n^2}\prod _{i=1}^n\left(q^{2i}-1\right),}$

${\displaystyle \left|{\mathrm{O}}^{+}(2n,q)\right|=2q^{n(n-1)}\left(q^n-1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right),}$

${\displaystyle \left|{\mathrm{O}}^{-}(2n,q)\right|=2q^{n(n-1)}\left(q^n+1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right).}$

In characteristic two, the formulas are the same, except that the factor2 of|O(2n + 1,q)| must be removed.

For orthogonal groups, theDickson invariant is a homomorphism from the orthogonal group to the quotient groupZ / 2Z (integers modulo 2), taking the value0 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 asD(f) = rank(I −f) modulo 2, whereI is the identity (Taylor 1992, Theorem 11.43). Over fields that are not ofcharacteristic 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 thekernel of the Dickson invariant^[8] and usually has index 2 inO(n,F ).^[9] When the characteristic ofF is not 2, the Dickson Invariant is0 whenever the determinant is1. Thus when the characteristic is not 2,SO(n,F ) is commonly defined to be the elements ofO(n,F ) with determinant1. Each element inO(n,F ) has determinant±1. Thus in characteristic 2, the determinant is always1.

The Dickson invariant can also be defined forClifford groups andpin 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 thehypoabelian 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 theWitt index is 2.^[10] A reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vectoru takes a vectorv tov +B(v,u)/Q(u) ·u whereB is the bilinear form^{[clarification needed]} andQ is the quadratic form associated to the orthogonal geometry. Compare this to theHouseholder reflection of odd characteristic or characteristic zero, which takesv tov − 2·B(v,u)/Q(u) ·u.
• Thecenter of the orthogonal group usually has order 1 in characteristic 2, rather than 2, sinceI = −I.
• In odd dimensions2n + 1 in characteristic 2, orthogonal groups overperfect fields are the same assymplectic groups in dimension2n. 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 dimension2n, 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.

Thespinor norm is a homomorphism from an orthogonal group over a fieldF to thequotient groupF^× / (F^×)² (themultiplicative group of the fieldFup to multiplication bysquare elements), that takes reflection in a vector of normn to the image ofn inF^× / (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 ofGalois cohomology ofalgebraic 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 partpost hoc, as far as the discovery of the phenomenon is concerned. The first point is thatquadratic forms over a field can be identified as a GaloisH¹, 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 thedeterminant.

The 'spin' name of the spinor norm can be explained by a connection to thespin group (more accurately apin group). This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use ofClifford algebras). The spin covering of the orthogonal group provides ashort exact sequence ofalgebraic groups.

${\displaystyle 1\to {\mu }_2\to {\mathrm{P}\mathrm{i}\mathrm{n}}_V\to {\mathrm{O}}_{\mathrm{V}}\to 1}$

Hereμ₂ is thealgebraic 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. Theconnecting homomorphism fromH⁰(O_V), which is simply the groupO_V(F) ofF-valued points, toH¹(μ₂) is essentially the spinor norm, becauseH¹(μ₂) is isomorphic to the multiplicative group of the field modulo squares.

There is also the connecting homomorphism fromH¹ of the orthogonal group, to theH² 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.

TheLie algebra corresponding to Lie groupsO(n,F ) andSO(n,F ) consists of theskew-symmetricn ×n matrices, with the Lie bracket[ , ] given by thecommutator. One Lie algebra corresponds to both groups. It is often denoted by${\displaystyle \mathfrak{o}(n,F)}$ or${\displaystyle \mathfrak{s}\mathfrak{o}(n,F)}$, and called theorthogonal Lie algebra orspecial orthogonal Lie algebra. Over real numbers, these Lie algebras for differentn are thecompact real forms of two of the four families ofsemisimple Lie algebras: in odd dimensionB_k, wheren = 2k + 1, while in even dimensionD_r, wheren = 2r.

Since the groupSO(n) is not simply connected, the representation theory of the orthogonal Lie algebras includes both representations corresponding toordinary representations of the orthogonal groups, and representations corresponding toprojective representations of the orthogonal groups. (The projective representations ofSO(n) are just linear representations of the universal cover, thespin group Spin(n).) The latter are the so-calledspin representation, which are important in physics.

More generally, given a vector spaceV (over a field with characteristic not equal to 2) with a nondegenerate symmetric bilinear form${\displaystyle ⟨u,v⟩}$, the special orthogonal Lie algebra consists of tracefree endomorphisms${\displaystyle \phi }$ which are skew-symmetric for this form (${\displaystyle ⟨\phi A,B⟩=-⟨A,\phi B⟩}$). Over a field of characteristic 2 we consider instead the alternating endomorphisms. Concretely we can equate these with thebivectors of theexterior algebra, theantisymmetric tensors of${\displaystyle {\wedge }^{2}V}$. The correspondence is given by:

${\displaystyle v\wedge w\mapsto ⟨v,\cdot ⟩w-⟨w,\cdot ⟩v}$

This description applies equally for the indefinite special orthogonal Lie algebras${\displaystyle \mathfrak{s}\mathfrak{o}(p,q)}$ for symmetric bilinear forms with signature(p,q).

Over real numbers, this characterization is used in interpreting thecurl 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 inclusionsO(n) ⊂ U(n) ⊂ USp(2n) andUSp(n) ⊂ U(n) ⊂ O(2n) are part of a sequence of 8 inclusions used in ageometric proof of the Bott periodicity theorem, and the corresponding quotient spaces aresymmetric spaces of independent interest – for example,U(n)/O(n) is theLagrangian Grassmannian.

In physics, particularly in the areas ofKaluza–Klein compactification, it is important to find out the subgroups of the orthogonal group. The main ones are:

${\displaystyle \mathrm{O}(n)\supset \mathrm{O}(n-1)}$ – preserve an axis

${\displaystyle \mathrm{O}(2n)\supset \mathrm{U}(n)\supset \mathrm{S}\mathrm{U}(n)}$ –U(n) are those that preserve a compatible complex structureor a compatible symplectic structure – see2-out-of-3 property;SU(n) also preserves a complex orientation.

${\displaystyle \mathrm{O}(2n)\supset \mathrm{U}\mathrm{S}\mathrm{p}(n)}$

${\displaystyle \mathrm{O}(7)\supset {\mathrm{G}}_2}$

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

Beingisometries, real orthogonal transforms preserveangles, and are thusconformal maps, though not all conformal linear transforms are orthogonal. In classical terms this is the difference betweencongruence andsimilarity, as exemplified by SSS (side-side-side)congruence of triangles and AAA (angle-angle-angle)similarity of triangles. The group of conformal linear maps ofRⁿ is denotedCO(n) for theconformal orthogonal group, and consists of the product of the orthogonal group with the group ofdilations. Ifn is odd, these two subgroups do not intersect, and they are adirect product:CO(2k + 1) = O(2k + 1) ×R^∗, whereR^∗ =R∖{0} is the realmultiplicative group, while ifn is even, these subgroups intersect in±1, so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar:CO(2k) = O(2k) ×R⁺.

Similarly one can defineCSO(n); 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 aspoint groups and can be realized as the symmetry groups ofpolytopes. A very important class of examples are thefinite Coxeter groups, which include the symmetry groups ofregular polytopes.

Dimension 3 is particularly studied – seepoint groups in three dimensions,polyhedral groups, andlist of spherical symmetry groups. In 2 dimensions, the finite groups are either cyclic or dihedral – seepoint groups in two dimensions.

Other finite subgroups include:

• Permutation matrices (theCoxeter groupA_n)
• Signed permutation matrices (theCoxeter groupB_n); also equals the intersection of the orthogonal group with theinteger matrices.^{[note 2]}

The orthogonal group is neithersimply connected norcenterless, and thus has both acovering group and aquotient group, respectively:

• Two coveringPin groups,Pin₊(n) → O(n) andPin₋(n) → O(n),
• The quotientprojective 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 dimensionPSO(2k + 1) is a 1-to-1 cover; i.e., isomorphic toSO(2k + 1). These groups,Spin(n),SO(n), andPSO(n) are Lie group forms of the compactspecial orthogonal Lie algebra,${\displaystyle \mathfrak{s}\mathfrak{o}(n,\mathbf{R})}$ –Spin is the simply connected form, whilePSO is the centerless form, andSO is in general neither.^{[note 3]}

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

Theprincipal homogeneous space for the orthogonal groupO(n) is theStiefel manifoldV_n(Rⁿ) oforthonormal bases (orthonormaln-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 anyorthogonal basis to any otherorthogonal basis.

The other Stiefel manifoldsV_k(Rⁿ) fork <n ofincomplete orthonormal bases (orthonormalk-frames) are still homogeneous spaces for the orthogonal group, but notprincipal homogeneous spaces: anyk-frame can be taken to any otherk-frame by an orthogonal map, but this map is not uniquely determined.

• Coordinate rotations and reflections
• Reflection through the origin

• rotation group,SO(3,R)
• SO(8)

• indefinite orthogonal group
• unitary group
• symplectic group

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

• Representations of classical Lie groups
• Brauer algebra

• "Orthogonal group",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• John Baez "This Week's Finds in Mathematical Physics" week 105
• John Baez on Octonions
• (in Italian)n-dimensional Special Orthogonal Group parametrization

• Lie groups
• Quadratic forms
• Euclidean symmetries
• Linear algebraic groups

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