Special Orthogonal Group in 3 dimensions

URI: https://gptkb.org/entity/Special_Orthogonal_Group_in_3_dimensions

GPTKB entity

Statements (65)

Predicate	Object
gptkbp:instanceOf	gptkb:group_of_people gptkb:topology gptkb:Lie_group
gptkbp:actsOn	gptkb:3-dimensional_Euclidean_space
gptkbp:alsoKnownAs	gptkb:SO(3)
gptkbp:application	computer graphics crystallography physics robotics rigid body rotations
gptkbp:centralTo	{identity}
gptkbp:compactSubgroupOf	GL(3, R)
gptkbp:containsElement	3x3 orthogonal matrices with determinant 1
gptkbp:determinantOfElements	1
gptkbp:dimensions	3
gptkbp:doubleCover	gptkb:Spin(3)
gptkbp:field	real numbers
gptkbp:firstHomologyGroup	gptkb:Z/2Z
gptkbp:fundamentalGroup	gptkb:Z/2Z
gptkbp:generation	skew-symmetric matrices
gptkbp:groupOrder	infinite
gptkbp:hasElementOrder	continuous
gptkbp:homogeneousSpace	sphere S^2
gptkbp:homotopyGroupPi1	gptkb:Z/2Z
gptkbp:homotopyGroupPi2	0
gptkbp:homotopyGroupPi3	Z
https://www.w3.org/2000/01/rdf-schema#label	Special Orthogonal Group in 3 dimensions
gptkbp:identityElement	3x3 identity matrix
gptkbp:isomorphicTo	gptkb:projective_special_unitary_group_PSU(2) real projective 3-space as a manifold
gptkbp:Lie_algebra	so(3)
gptkbp:LieAlgebraDimension	3
gptkbp:manifoldDimension	3
gptkbp:maximalTorus	gptkb:SO(2)
gptkbp:notation	SO(3, R)
gptkbp:parameter	gptkb:Euler_angles quaternions axis-angle rotation vector
gptkbp:quotientBySO2	2-sphere S^2
gptkbp:rank	1
gptkbp:realForm	SO(3, R)
gptkbp:relatedGroup	gptkb:Lie_group gptkb:rotation_group simple compact non-abelian connected
gptkbp:relatedTo	gptkb:Euler_angles gptkb:orthogonal_group_O(3) gptkb:Spin_group gptkb:axis-angle_representation gptkb:special_unitary_group_SU(2) quaternions rotation matrices Lie algebra so(3) orthogonal group O(3, R) projective space RP^3 rotation group in 3D special linear group SL(3, R)
gptkbp:secondHomologyGroup	0
gptkbp:thirdHomologyGroup	Z
gptkbp:universalCover	gptkb:SU(2)
gptkbp:bfsParent	gptkb:SO(3)
gptkbp:bfsLayer	7