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gptkbp:instanceOf
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gptkb:Lie_group
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gptkbp:actsOn
|
gptkb:Euclidean_space
|
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gptkbp:application
|
gptkb:geometry
computer vision
physics
robotics
rotations in N-dimensional Euclidean space
|
|
gptkbp:centralTo
|
{I, -I} for even N
{I} for odd N
|
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gptkbp:compact
|
true
|
|
gptkbp:defines
|
Group of N×N real orthogonal matrices with determinant 1
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gptkbp:determinant
|
1
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gptkbp:dimensions
|
N(N-1)/2
|
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gptkbp:field
|
real numbers
|
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gptkbp:fullName
|
Special orthogonal group of degree N over the real numbers
|
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gptkbp:fundamentalGroup
|
Z for N=2
Z_2 for N ≥ 3
|
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gptkbp:hasConnection
|
true
|
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gptkbp:hasSubgroup
|
O(N, ℝ)
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
SO(N, ℝ)
|
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gptkbp:identityElement
|
identity matrix
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gptkbp:isClosedSubgroupOf
|
GL(N, ℝ)
|
|
gptkbp:isConnectedComponentOf
|
O(N, ℝ)
|
|
gptkbp:isHomogeneousSpace
|
true
|
|
gptkbp:isMatrixGroup
|
true
|
|
gptkbp:isNonAbelian
|
true for N ≥ 3
|
|
gptkbp:isPathConnected
|
true
|
|
gptkbp:isReductive
|
true
|
|
gptkbp:isSemisimple
|
true for N ≥ 3
|
|
gptkbp:isSimple
|
false for N=4
true for N ≥ 3 except N=4
|
|
gptkbp:Lie_algebra
|
so(N, ℝ)
|
|
gptkbp:matrixCondition
|
A^T A = I
det(A) = 1
|
|
gptkbp:maximalCompactSubgroupOf
|
GL(N, ℝ)
|
|
gptkbp:notation
|
gptkb:SO(N)
SO(N, R)
|
|
gptkbp:realForm
|
SO(N, ℂ)
|
|
gptkbp:relatedGroup
|
gptkb:group_of_people
gptkb:rotation_group
gptkb:SU(N)
gptkb:U(N)
orthogonal group
O(N, ℝ)
SL(N, ℝ)
Spin(N)
special group
|
|
gptkbp:universalCover
|
Spin(N)
|
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gptkbp:bfsParent
|
gptkb:SO(N)
|
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gptkbp:bfsLayer
|
6
|