Special Orthogonal Group in N dimensions

GPTKB entity

Predicate	Object
gptkbp:instanceOf	gptkb:group_of_people
gptkbp:actsOn	N-dimensional Euclidean space
gptkbp:alsoKnownAs	gptkb:SO(N)
gptkbp:application	computer graphics crystallography mechanics physics robotics
gptkbp:centralTo	{I, -I} for even N {I} for odd N
gptkbp:compact	true
gptkbp:compactLieGroup	true
gptkbp:connectedComponentOfIdentity	true
gptkbp:containsElement	rotation matrix
gptkbp:defines	group of N×N orthogonal matrices with determinant 1
gptkbp:dimensions	N(N-1)/2
gptkbp:discreteSubgroup	rotation group of regular polyhedra
gptkbp:example	SO(2) is the circle group SO(3) is the rotation group in 3D
gptkbp:field	gptkb:geometry gptkb:mathematics group theory linear algebra
gptkbp:generation	skew-symmetric matrices
gptkbp:hasSubgroup	Orthogonal Group in N dimensions
gptkbp:homotopyGroup	π1(SO(2)) = Z π1(SO(N)) = Z2 for N ≥ 3
https://www.w3.org/2000/01/rdf-schema#label	Special Orthogonal Group in N dimensions
gptkbp:identityElement	identity matrix
gptkbp:irreducibleRepresentation	yes
gptkbp:isA	gptkb:Lie_group
gptkbp:isSimple	true for N ≥ 5
gptkbp:Lie_algebra	so(N)
gptkbp:maximalTorus	SO(2)^floor(N/2)
gptkbp:notation	gptkb:SO(N)
gptkbp:order	infinite for N > 1
gptkbp:property	preserves angles preserves orientation connected for N ≥ 2 preserves lengths
gptkbp:rank	floor(N/2)
gptkbp:realForm	yes
gptkbp:realization	gptkb:group_of_people
gptkbp:relatedTo	gptkb:rotation_group orthogonal group
gptkbp:universalCover	Spin(N)
gptkbp:bfsParent	gptkb:SO(N)
gptkbp:bfsLayer	6