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) on the associated projective space P(V). Explicitly, the projective orthogonal group is the quotient group
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 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)/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 {±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 groups, 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 groups, 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 is orientation-preserving, while in odd dimension it is orientation-reversing ( − I ∈ SO ( 2 k ) {\displaystyle -I\in \operatorname {SO} (2k)} but − I ∉ SO ( 2 k + 1 ) {\displaystyle -I\not \in \operatorname {SO} (2k+1)} ). 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: B k = s o 2 k + 1 , D k = s o 2 k . {\displaystyle B_{k}={\mathfrak {so}}_{2k+1},D_{k}={\mathfrak {so}}_{2k}.}
Thus, O(2k+1) = SO(2k+1) × {±I}, while O ( 2 k ) ≠ SO ( 2 k ) × { ± I } {\displaystyle \operatorname {O} (2k)\neq \operatorname {SO} (2k)\times \{\pm I\}} and is instead a non-trivial central extension of PO(2k). Beware that PO(2k+1) is isometries of RP2k = P(R2k+1), while PO(2k) is isometries of RP2k−1 = P(R2k) – 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, SO ( 2 k + 1 ) ≅ PSO ( 2 k + 1 ) = PO ( 2 k + 1 ) , {\displaystyle \operatorname {SO} (2k+1)\cong \operatorname {PSO} (2k+1)=\operatorname {PO} (2k+1),} 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.
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 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 representation 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
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