Projective linear group PGL(2, ℂ)

URI: https://gptkb.org/entity/Projective_linear_group_PGL(2,_ℂ)
GPTKB entity
Statements (47)
Predicate Object
gptkbp:instanceOf gptkb:group_of_people
gptkb:Lie_group
projective linear group
gptkbp:actsOn gptkb:Riemann_sphere
complex projective line
gptkbp:appearsIn gptkb:algebraic_geometry
gptkb:geometry
complex analysis
group theory
gptkbp:automorphismGroup gptkb:Riemann_sphere
complex projective line
gptkbp:centralTo trivial group
gptkbp:compact true
gptkbp:definedIn complex numbers
gptkbp:dimensions 3
gptkbp:hasComplexDimension 3
gptkbp:hasConnection false
true
gptkbp:hasSubgroup gptkb:PSL(2,_ℂ)
Borel subgroup
diagonal matrices modulo scalars
upper triangular matrices modulo scalars
https://www.w3.org/2000/01/rdf-schema#label Projective linear group PGL(2, ℂ)
gptkbp:isAlgebraicGroup true
gptkbp:isHomogeneousSpaceFor itself
gptkbp:isMatrixGroup true
gptkbp:isNonAbelian true
gptkbp:isomorphicTo gptkb:PSL(2,_ℂ)
gptkbp:isProjectiveGroup true
gptkbp:isQuotientOf gptkb:GL(2,_ℂ)
center of GL(2, ℂ)
gptkbp:isReductive true
gptkbp:isSimple true
gptkbp:Lie_algebra gptkb:sl(2,_ℂ)
gptkbp:notation gptkb:PGL(2,_ℂ)
gptkb:PGL_2(ℂ)
gptkbp:order infinite
gptkbp:realForm gptkb:PGL(2,_ℝ)
gptkbp:relatedGroup projective transformations of the complex projective line
gptkbp:relatedTo gptkb:Möbius_transformations
gptkbp:universalCover none
gptkbp:usedIn gptkb:conformal_geometry
gptkb:hyperbolic_geometry
moduli theory
Kleinian groups
gptkbp:bfsParent gptkb:Möbius_transformation
gptkbp:bfsLayer 6