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- Projective_orthogonal_group abstract "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 groupPO(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 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).".
- Projective_orthogonal_group thumbnail SO-O-PSO-PO-2k.svg?width=300.
- Projective_orthogonal_group wikiPageExternalLink books?id=E_HCwwxMbfMC&pg=PA34.
- Projective_orthogonal_group wikiPageID "4845690".
- Projective_orthogonal_group wikiPageRevisionID "598291106".
- Projective_orthogonal_group hasPhotoCollection Projective_orthogonal_group.
- Projective_orthogonal_group title "Projective General Orthogonal Group".
- Projective_orthogonal_group title "Projective Special Orthogonal Group".
- Projective_orthogonal_group urlname "ProjectiveGeneralOrthogonalGroup".
- Projective_orthogonal_group urlname "ProjectiveSpecialOrthogonalGroup".
- Projective_orthogonal_group subject Category:Lie_groups.
- Projective_orthogonal_group subject Category:Projective_geometry.
- Projective_orthogonal_group subject Category:Quadratic_forms.
- Projective_orthogonal_group type Abstraction100002137.
- Projective_orthogonal_group type Form106290637.
- Projective_orthogonal_group type Group100031264.
- Projective_orthogonal_group type LanguageUnit106284225.
- Projective_orthogonal_group type LieGroups.
- Projective_orthogonal_group type Part113809207.
- Projective_orthogonal_group type QuadraticForms.
- Projective_orthogonal_group type Relation100031921.
- Projective_orthogonal_group type Word106286395.
- Projective_orthogonal_group comment "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 groupPO(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.".
- Projective_orthogonal_group label "Projective orthogonal group".
- Projective_orthogonal_group sameAs m.0bwgtb2.
- Projective_orthogonal_group sameAs Q7249468.
- Projective_orthogonal_group sameAs Q7249468.
- Projective_orthogonal_group sameAs Projective_orthogonal_group.
- Projective_orthogonal_group wasDerivedFrom Projective_orthogonal_group?oldid=598291106.
- Projective_orthogonal_group depiction SO-O-PSO-PO-2k.svg.
- Projective_orthogonal_group isPrimaryTopicOf Projective_orthogonal_group.
