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
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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:
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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
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while when it is equal to
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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
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( 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
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when , and the form
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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) Zbl 0221.20056 |
[2] | E. Artin, "Geometric algebra" , Interscience (1957) MR1529733 MR0082463 Zbl 0077.02101 |
[3] | , Automorphisms of the classical groups , Moscow (1976) (In Russian; translated from English and French) (Collection of translations) Zbl 0042.25603 |
[4] | H. Weyl, "The classical groups, their invariants and representations" , Princeton Univ. Press (1946) MR0000255 Zbl 1024.20502 |
[5] | D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013 |
[6] | N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) MR2333539 MR2327161 MR2325344 MR2284892 MR2272929 MR0928386 MR0896478 MR0782297 MR0782296 MR0722608 MR0682756 MR0643362 MR0647314 MR0610795 MR0583191 MR0354207 MR0360549 MR0237342 MR0205211 MR0205210 |
[7] | O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973) Zbl 0259.10018 |
[8] | D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) MR0229247 Zbl 0144.44804 |
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) MR0045125 Zbl 0042.25603 |
Orthogonal group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_group&oldid=21903