Orthogonal group

The group of all linear transformations of an -dimensional vector space over a field which preserve a fixed non-singular quadratic form on (i.e. linear transformations such that for all ). An orthogonal group is a classical group. The elements of an orthogonal group are called orthogonal transformations of (with respect to ), or also automorphisms of the form . Furthermore, let (for orthogonal groups over fields with characteristic 2 see [1], [7]) and let be the non-singular symmetric bilinear form on related to by the formula

The orthogonal group then consists of those linear transformations of that preserve , and is denoted by , or (when one is talking of a specific field and a specific form ) simply by . If is the matrix of with respect to some basis of , then the orthogonal group can be identified with the group of all -matrices with coefficients in such that ( is transposition).

The description of the algebraic structure of an orthogonal group is a classical problem. The determinant of any element from is equal to 1 or . Elements with determinant 1 are called rotations; they form a normal subgroup (or simply ) of index 2 in the orthogonal group, called the rotation group. Elements from are called inversions. Every rotation (inversion) is the product of an even (odd) number of reflections from .

Let be the group of all homotheties , , , of the space . Then is the centre of ; it consists of two elements: and . If is odd, then is the direct product of its centre and . If , the centre of is trivial if is odd, and coincides with the centre of if is even. If , the group is commutative and is isomorphic either to the multiplicative group of (when the Witt index of is equal to 1), or to the group of elements with norm 1 in , where is the discriminant of (when ). The commutator subgroup of is denoted by , or simply by ; it is generated by the squares of the elements from . When , the commutator subgroup of coincides with . The centre of is .

Other classical groups related to orthogonal groups include the canonical images of and in the projective group; they are denoted by and (or simply by and ) and are isomorphic to and , respectively.

The basic classical facts about the algebraic structure describe the successive factors of the following series of normal subgroups of an orthogonal group:

The group has order 2. Every element in has order 2, thus this group is defined completely by its cardinal number, and this number can be either infinite or finite of the form where is an integer. The description of the remaining factors depends essentially on the Witt index of the form .

First, let . Then when . This isomorphism is defined by the spinor norm, which defines an epimorphism from on with kernel . The group is non-trivial (and consists of the transformations and ) if and only if is even and . If , then the group is simple. The cases where are studied separately. Namely, is isomorphic to (see Special linear group) and is also simple if has at least 4 elements (the group is isomorphic to the projective group ). When , the group is isomorphic to the group and is simple (in this case ), while when , the group is isomorphic to and is not simple. In the particular case when and is a form of signature , the group is called the Lorentz group.

When (i.e. is an anisotropic form), these results are not generally true. For example, if and is a positive-definite form, then , although consists of two elements; when , , one can have , but . When , the structures of an orthogonal group and its related groups essentially depend on . For example, if , then , , , , is simple (and is isomorphic to the direct product of two simple groups); if is the field of -adic numbers and , there exists in (and ) an infinite normal series with Abelian quotients. Important special cases are when is a locally compact field or an algebraic number field. If is the field of -adic numbers, then is impossible when . If is an algebraic number field, then there is no such restriction and one of the basic results is that , when and , is simple. In this case, the study of orthogonal groups is closely connected with the theory of equivalence of quadratic forms, where one needs the forms obtained from by extension of coefficients to the local fields defined by valuations of (the Hasse principle).

If is the finite field of elements, then an orthogonal group is finite. The order of for odd is equal to

while when it is equal to

where if and otherwise. These formulas and general facts about orthogonal groups when also allow one to calculate the orders of and , since when , while the order of is equal to 2. The group , , is one of the classical simple finite groups (see also Chevalley group).

One of the basic results on automorphisms of orthogonal groups is the following: If , then every automorphism of has the form , , where is a fixed homomorphism of into its centre and is a fixed bijective semi-linear mapping of onto itself satisfying for all , where while is an automorphism of . If and , then every automorphism of is induced by an automorphism of (see [1], [3]).

Like the other classical groups, an orthogonal group has a geometric characterization (under certain hypotheses). Indeed, let be an anisotropic form such that for all . In this case is a Pythagorean orderable field. For a fixed order of the field , any sequence constructed from a linearly independent basis , where is the set of all linear combinations of the form , , is called an -dimensional chain of incident half-spaces in . The group has the property of free mobility, i.e. for any two -dimensional chains of half-spaces there exists a unique transformation from which transforms the first chain into the second. This property characterizes an orthogonal group: If is any ordered skew-field and is a subgroup in , , having the property of free mobility, then is a Pythagorean field, while , where is an anisotropic symmetric bilinear form such that for any vector .

Let be a fixed algebraic closure of the field . The form extends naturally to a non-singular symmetric bilinear form on , and the orthogonal group is a linear algebraic group defined over with as group of -points. The linear algebraic groups thus defined (for various ) are isomorphic over (but in general not over ); the corresponding linear algebraic group over is called the orthogonal algebraic group . Its subgroup is also a linear algebraic group over , and is called a properly orthogonal, or special orthogonal algebraic group (notation: ); it is the connected component of the identity of . The group is an almost-simple algebraic group (i.e. does not contain infinite algebraic normal subgroups) of type when , , and of type when , . The universal covering group of is a spinor group.

If or a -adic field, then has a canonical structure of a real, complex or -adic analytic group. The Lie group is defined up to isomorphism by the signature of the form ; if this signature is , , then is denoted by and is called a pseudo-orthogonal group. It can be identified with the Lie group of all real -matrices which satisfy

( denotes the unit -matrix). The Lie algebra of this group is the Lie algebra of all real -matrices that satisfy the condition . In the particular case , the group is denoted by and is called a real orthogonal group; its Lie algebra consists of all skew-symmetric real -matrices. The Lie group has four connected components when , and two connected components when . The connected component of the identity is its commutator subgroup, which, when , coincides with the subgroup in consisting of all transformations with determinant 1. The group is compact only when . The topological invariants of have been studied. One of the classical results is the calculation of the Betti numbers of the manifold : Its Poincaré polynomial has the form

when , and the form

when . The fundamental group of the manifold is . The calculation of the higher homotopy groups is directly related to the classification of locally trivial principal -fibrations over spheres. An important part in topological -theory is played by the periodicity theorem, according to which, when , there are the isomorphisms

further,

if ;

if ; and

if . The study of the topology of the group reduces in essence to the previous case, since the connected component of the identity of is diffeomorphic to the product on a Euclidean space.

References

[1]	J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955)
[2]	E. Artin, "Geometric algebra" , Interscience (1957)
[3]	, Automorphisms of the classical groups , Moscow (1976) (In Russian; translated from English and French) (Collection of translations)
[4]	H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946)
[5]	D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)
[6]	N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)
[7]	O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973)
[8]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)

Comments

A Pythagorean field is a field in which the sum of two squares is again a square.

References

[a1]	J. Dieudonné, "On the automorphisms of the classical groups" , Mem. Amer. Math. Soc. , 2 , Amer. Math. Soc. (1951)