In
projective geometry and
linear algebra, the projective orthogonal group PO is the induced
action
Action may refer to:
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Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of the
orthogonal group of a
quadratic space ''V'' = (''V'',''Q'')
[A quadratic space is a vector space ''V'' together with a ]quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong t ...
''Q''; the ''Q'' is dropped from notation when it is clear. on the associated
projective space P(''V''). Explicitly, the projective orthogonal group is the
quotient group
:PO(''V'') = O(''V'')/ZO(''V'') = O(''V'')/
where O(''V'') is the orthogonal group of (''V'') and ZO(''V'')= is the subgroup of all orthogonal
scalar transformations of ''V'' – these consist of the identity and
reflection through the origin
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
. These scalars are quotiented out because they act
trivially on the projective space and they form the
kernel
Kernel may refer to:
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of the action, and the notation "Z" is because the scalar transformations are the
center
Center or centre may refer to:
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*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...
of the orthogonal group.
The projective special orthogonal group, PSO, is defined analogously, as the induced action of the
special orthogonal group on the associated projective space. Explicitly:
:PSO(''V'') = SO(''V'')/ZSO(''V'')
where SO(''V'') is the special orthogonal group over ''V'' and ZSO(''V'') is the subgroup of orthogonal scalar transformations with unit
determinant. Here ZSO is the center of SO, and is trivial in odd dimension, while it equals in even dimension – this odd/even distinction occurs throughout the structure of the orthogonal groups. By analogy with GL/SL and GO/SO, the projective orthogonal group is also sometimes called the projective ''general'' orthogonal group and denoted PGO.
Like the orthogonal group, the projective orthogonal group can be defined over any field and with varied quadratic forms, though, as with the ordinary orthogonal group, the main emphasis is on the ''real'' ''positive definite'' projective orthogonal group; other fields are elaborated in
generalizations, below. Except when mentioned otherwise, in the sequel PO and PSO will refer to the real positive definite groups.
Like the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
s and
pin group
The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies.
History and shareholding
The PIN Group originally traded under ...
s, which are covers rather than quotients of the (special) orthogonal groups, the projective (special) orthogonal groups are of interest for (projective) geometric analogs of Euclidean geometry, as related
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
s, and in
representation theory.
More intrinsically, the (real positive definite) projective orthogonal group PO can be defined as the
isometries of
elliptic space (in the sense of
elliptic geometry), while PSO can be defined as the
orientation-preserving isometries of elliptic space (when the space is orientable; otherwise PSO = PO).
Structure
Odd and even dimensions
The structure of PO differs significantly between odd and even dimension, fundamentally because in even dimension,
reflection through the origin
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
is orientation-preserving, while in odd dimension it is orientation-reversing (
but
). This is seen in the fact that each odd-dimensional real projective space is orientable, while each even-dimensional real projective space of positive dimension is non-orientable. At a more abstract level, the
Lie algebras of odd- and even-dimensional projective orthogonal groups form two different families:
Thus, O(2''k''+1) = SO(2''k''+1) × ,
[This product is an internal direct sum – a product of subgroups – not just an abstract external direct sum.]
while
and is instead a non-trivial
central extension of PO(2''k'').
Beware that PO(2''k''+1) is isometries of RP
2''k'' = P(R
2''k''+1), while PO(2''k'') is isometries of RP
2''k''−1 = P(R
2''k'') – the odd-dimensional (vector) group is isometries of even-dimensional projective space, while the even-dimensional (vector) group is isometries of odd-dimensional projective space.
In odd dimension,
[The isomorphism/equality distinction in this equation is because the context is the 2-to-1 quotient map O → PO – PSO(2''k''+1) and PO(2''k''+1) are equal subsets of the target (namely, the whole space), hence the equality, while the induced map SO → PSO is an isomorphism but the two groups are subsets of different spaces, hence the isomorphism rather than an equality. See for an example of this distinction being made.] so the group of projective isometries can be identified with the group of rotational isometries.
In even dimension, SO(2''k'') → PSO(2''k'') and O(2''k'') → PO(2''k'') are both 2-to-1 covers, and PSO(2''k'') < PO(2''k'') is an
index
Index (or its plural form indices) may refer to:
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* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
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2 subgroup.
General properties
PSO and PO are
centerless
In abstract algebra, the center of a group, , is the set of elements that commute with every element of . It is denoted , from German '' Zentrum,'' meaning ''center''. In set-builder notation,
:.
The center is a normal subgroup, . As a subgr ...
, as with PSL and PGL; this is because scalar matrices are not only the center of SO and O, but also the
hypercenter
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a centr ...
(quotient by the center does not always yield a centerless group).
PSO is the
maximal compact subgroup in the
projective special linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
PSL, while PO is maximal compact in the
projective general linear group PGL. This is analogous to SO being maximal compact in SL and O being maximal compact in GL.
Representation theory
PO is of basic interest in representation theory: a group homomorphism ''G'' → PGL is called a
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group
\mathrm(V) = \mathrm(V) / F^*,
where ...
of ''G,'' just as a map ''G'' → GL is called a linear representation of ''G'', and just as any linear representation can be reduced to a map ''G'' → O (by taking an invariant inner product), any projective representation can be reduced to a map ''G'' → PO.
See
projective linear group: representation theory for further discussion.
Subgroups
Subgroups of the projective orthogonal group correspond to subgroups of the orthogonal group that contain −''I'' (that have
central symmetry
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
). As always with a quotient map (by the
lattice theorem), there is a
Galois connection between subgroups of O and PO, where the adjunction on O (given by taking the image in PO and then the preimage in O) simply adds −''I'' if absent.
Of particular interest are discrete subgroups, which can be realized as symmetries of
projective polytopes – these correspond to the (discrete) point groups that include central symmetry. Compare with
discrete subgroups of the Spin group, particularly the 3-dimensional case of
binary polyhedral group
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries tha ...
s.
For example, in 3 dimensions, 4 of the 5
Platonic solids have central symmetry (cube/octahedron, dodecahedron/icosahedron), while the tetrahedron does not – however, the
stellated octahedron
The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted ...
has central symmetry, though the resulting symmetry group is the same as that of the cube/octahedron.
Topology
PO and PSO, as centerless topological groups, are at the bottom of a sequence of
covering group
In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. ...
s, whose top are the (
simply connected)
Pin group
The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies.
History and shareholding
The PIN Group originally traded under ...
s or
Spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
, respectively:
:Pin
±(''n'') → O(''n'') → PO(''n'').
:Spin(''n'') → SO(''n'') → PSO(''n'').
These groups are all
compact real form
In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is the complexification of ''g''0:
: \mathfrak ...
s of the same Lie algebra.
These are all 2-to-1 covers, except for SO(2''k''+1) → PSO(2''k''+1) which is 1-to-1 (an isomorphism).
Homotopy groups
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
s above
do not change under covers, so they agree with those of the orthogonal group. The lower homotopy groups are given as follows.
:
:
The fundamental group of (centerless) PSO(''n'') equals the center of (simply connected) Spin(''n''), which is always true about covering groups:
:
Using the
table of centers of Spin groups yields (for
):
:
:
:
In low dimensions:
:
as the group is trivial.
:
as it is topologically a circle, though note that the preimage of the identity in Spin(2) is
as for other
Homology groups
Bundles
Just as the orthogonal group is the
structure group
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E an ...
of
vector bundles, the projective orthogonal group is the structure group of
projective bundle
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces.
By definition, a scheme ''X'' over a Noetherian scheme ''S'' is a P''n''-bundle if it is locally a projective ''n''-space; i.e., X \times_S U \simeq \math ...
s, and the corresponding
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
is denoted BPO.
Generalizations
As with the orthogonal group, the projective orthogonal group can be generalized in two main ways: changing the field or changing the quadratic form. Other than the real numbers, primary interest is in complex numbers or finite fields, while (over the reals) quadratic forms can also be
indefinite forms, and are denoted PO(''p'',''q'') by their signature.
The complex projective orthogonal group, PO(''n'',C) should not be confused with the
projective unitary group, PU(''n''): PO preserves a symmetric form, while PU preserves a
hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
– PU is the symmetries of complex projective space (preserving the
Fubini–Study metric
In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CP''n'' endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and ...
).
In fields of characteristic 2 there are added complications: quadratic forms and symmetric bilinear forms are no longer equivalent, , and the determinant needs to be replaced by the
Dickson invariant.
Finite fields
The projective orthogonal group over a finite field is used in the construction of a family of finite
simple groups of
Lie type, namely the
Chevalley group
In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The ...
s of type D
''n''. The orthogonal group over a finite field, O(''n'',''q'') is not simple, since it has SO as a subgroup and a non-trivial center () (hence PO as quotient). These are both fixed by passing to PSO, but PSO itself is not in general simple, and instead one must use a subgroup (which may be of index 1 or 2), defined by the
spinor norm (in odd characteristic) or the quasideterminant (in even characteristic).
ATLAS
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geographi ...
p. xi
/ref> The quasideterminant can be defined as (−1)''D'', where ''D'' is the Dickson invariant (it is the determinant defined by the Dickson invariant), or in terms of the dimension of the fixed space.
Notes
See also
* Projective linear group
* Projective unitary group
* Orthogonal group
* Spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
References
*
* Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GO''n''(''q''), SO''n''(''q''), PGO''n''(''q''), and PSO''n''(''q''), and O''n''(''q'')." §2.4 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, pp. xi–xii, 1985.
External links
*
*
{{DEFAULTSORT:Projective Orthogonal Group
Lie groups
Projective geometry
Quadratic forms