Projective orthogonal group
Encyclopedia
In projective geometry
and linear algebra
, the projective orthogonal group PO is the induced action
of the orthogonal group
of a quadratic space V = (V,Q)A quadratic space is a vector space
V together with a quadratic form
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
where O(V) is the orthogonal group of (V) and ZO(V)={±I} is the subgroup of all orthogonal scalar transformations of V – these consist of the identity and reflection through the origin. These scalars are quotiented out because they act trivially on the projective space and they form the kernel
of the action, and the notation "Z" is because the scalar transformations are the center 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:
where SO(V) is the special orthogonal group over V and SZO(V) is the subgroup of orthogonal scalar transformations with unit determinant
. Here SZO is the center of SO, and is trivial in odd dimension, while it equals {±1} 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 groups and pin group
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
s, and in representation theory
.
More intrinsically, the (real positive definite) projective orthogonal group PO can be defined as the isometries of real projective space
, while PSO can be defined as the orientation-preserving isometries of real projective space (when the space is orientable; otherwise PSO = PO).
but 
). This is reflected in odd-dimensional real projective space being orientable, while even-dimensional real projective space is nonorientable, and at a more abstract level, the Lie algebra
s of odd and even dimensional projective orthogonal groups are in two different families:
Thus,
subgroups – not just an abstract external direct sum.
while
and is instead a non-trivial central extension of PO(2k).
Beware that PO(2k+1) is isometries of
while PO(2k) is isometries of 
– 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
– PSO(2k+1) and PO(2k+1) are equal subsets of the target (namely, the whole space), hence the equality, while the induced map 
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(2k) → PSO(2k) and O(2k) → PO(2k) are both 2-to-1 covers, and PSO(2k) < PO(2k) is an index
2 subgroup.
PSO is the maximal compact subgroup
in the projective special linear group 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.
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.
(that have central symmetry). 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
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 groups.
For example, in 3 dimensions, 4 of the 5 Platonic solid
s have central symmetry (cube/octahedron, dodecahedron/icosahedron), while the tetrahedron does not – however, the compound of two tetrahedra has central symmetry, though the resulting symmetry group is the same as that of the cube/octahedron.
s or Spin group, respectively:
These groups are all compact real forms of the same Lie algebra.
These are all 2-to-1 covers, except for SO(2k+1) → PSO(2k+1) which is 1-to-1 (an isomorphism).
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 
s, the projective orthogonal group is the structure group of projective bundles, and the corresponding classifying space
is denoted BPO.
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 – PU is the symmetries of complex projective space (preserving the Fubini–Study metric).
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
.
s of Lie type
, namely the Chevalley groups of type Dn. 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 ({±I}) (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). The quasideterminant can be defined as
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.
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
and linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, the projective orthogonal group PO is the induced action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
of a quadratic space V = (V,Q)A quadratic space is a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V together with a quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
Q; the Q is dropped from notation when it is clear. on the associated projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
P(V). Explicitly, the projective orthogonal group is the quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
- PO(V) = O(V)/ZO(V) = O(V)/{±I}
where O(V) is the orthogonal group of (V) and ZO(V)={±I} is the subgroup of all orthogonal scalar transformations of V – these consist of the identity and reflection through the origin. These scalars are quotiented out because they act trivially on the projective space and they form the kernel
Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
of the action, and the notation "Z" is because the scalar transformations are the center 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)/SZO(V)
where SO(V) is the special orthogonal group over V and SZO(V) is the subgroup of orthogonal scalar transformations with unit determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
. Here SZO is the center of SO, and is trivial in odd dimension, while it equals {±1} 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 groups and pin group
Pin group
In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group....
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
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, and in representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
.
More intrinsically, the (real positive definite) projective orthogonal group PO can be defined as the isometries of real projective space
Real projective space
In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
, while PSO can be defined as the orientation-preserving isometries of real projective space (when the space is orientable; otherwise PSO = PO).
Odd and even dimensions
The structure of PO differs significantly between odd and even dimension, fundamentally because in even dimension, reflection through the origin is orientation-preserving, while it odd dimension it is orientation-reversing ( but ). This is reflected in odd-dimensional real projective space being orientable, while even-dimensional real projective space is nonorientable, and at a more abstract level, the Lie algebraLie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s of odd and even dimensional projective orthogonal groups are in two different families:
Thus,
subgroups – not just an abstract external direct sum.
while
and is instead a non-trivial central extension of PO(2k).Beware that PO(2k+1) is isometries of
while PO(2k) is isometries of – 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 isomorphismIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
/equality distinction in this equation is because the context is the 2-to-1 quotient map
– PSO(2k+1) and PO(2k+1) are equal subsets of the target (namely, the whole space), hence the equality, while the induced map 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(2k) → PSO(2k) and O(2k) → PO(2k) are both 2-to-1 covers, and PSO(2k) < PO(2k) is an index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
2 subgroup.
General properties
PSO and PO are centerless, as with PSL and PGL; this is because scalar matrices are not only the center of SO and O, but also the hypercenter (quotient by the center does not always yield a centerless group).PSO is the maximal compact subgroup
Maximal compact subgroup
In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups....
in the projective special linear group 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 representationProjective representation
In the mathematical field of representation theory, 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 groupwhere GL is the general linear group of invertible linear transformations of V over F and F* here is the...
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 (that have central symmetry). As always with a quotient map (by the lattice theoremLattice theorem
In mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem or the correspondence theorem, states that if N is a normal subgroup of a group G, then there exists a bijection from the set of all subgroups A of G such that A contains N, onto the set of all subgroups...
), there is a Galois connection
Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...
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
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 groups.
For example, in 3 dimensions, 4 of the 5 Platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s have central symmetry (cube/octahedron, dodecahedron/icosahedron), while the tetrahedron does not – however, the compound of two tetrahedra 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 groups, whose top are the (simply connected) Pin groupPin group
In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group....
s or Spin group, respectively:
- Pin±(n) → O(n) → PO(n).
- Spin(n) → SO(n) → PSO(n).
These groups are all compact real forms of the same Lie algebra.
These are all 2-to-1 covers, except for SO(2k+1) → PSO(2k+1) which is 1-to-1 (an isomorphism).
Homotopy groups
Homotopy groupHomotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
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 Bundles
Just as the orthogonal group is the structure group of vector bundleVector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s, the projective orthogonal group is the structure group of projective bundles, and the corresponding classifying space
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 by a free action of G...
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
Projective unitary group
In mathematics, the projective unitary group PU is the quotient of the unitary group U by the right multiplication of its center, U, embedded as scalars....
, PU(n): PO preserves a symmetric form, while PU preserves a hermitian form – PU is the symmetries of complex projective space (preserving the Fubini–Study metric).
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 invariantOrthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
.
Finite fields
The projective orthogonal group over a finite field is used in the construction of a family of finite simple groupSimple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
s of Lie type
Group of Lie type
In mathematics, a group of Lie type G is a group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups...
, namely the Chevalley groups of type Dn. 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 ({±I}) (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). The quasideterminant can be defined as
where D is the Dickson invariantOrthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
(it is the determinant defined by the Dickson invariant), or in terms of the dimension of the fixed space.