|
gptkbp:instanceOf
|
gptkb:group_of_people
gptkb:Lie_group
|
|
gptkbp:actsOn
|
gptkb:n-dimensional_real_Euclidean_space
|
|
gptkbp:application
|
gptkb:geometry
computer vision
mechanics
physics
robotics
|
|
gptkbp:centralTo
|
{I, -I} for even n
{I} for odd n
|
|
gptkbp:compact
|
compact
|
|
gptkbp:component
|
trivial
|
|
gptkbp:connectedness
|
connected
|
|
gptkbp:definedIn
|
group of n×n real orthogonal matrices with determinant 1
|
|
gptkbp:determinant
|
1
|
|
gptkbp:dimensions
|
n(n-1)/2
|
|
gptkbp:doubleCover
|
gptkb:Spin(n)
|
|
gptkbp:field
|
real numbers
|
|
gptkbp:fullName
|
gptkb:Special_orthogonal_group
|
|
gptkbp:fundamentalGroup
|
Z for n=2
Z/2Z for n>2
|
|
gptkbp:generation
|
rotations in coordinate planes
|
|
gptkbp:hasSubgroup
|
O(n)
|
|
gptkbp:homotopyGroup
|
π1(SO(2)) = Z
π1(SO(n)) = Z/2Z for n > 2
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
SO(n)
|
|
gptkbp:identityElement
|
identity matrix
|
|
gptkbp:isometryGroupOf
|
gptkb:n-dimensional_Euclidean_space_(orientation-preserving)
|
|
gptkbp:isQuotientOf
|
gptkb:Spin(n)
|
|
gptkbp:isSimple
|
n > 2
|
|
gptkbp:Lie_algebra
|
so(n)
|
|
gptkbp:matrixCondition
|
A^T A = I, det(A) = 1
|
|
gptkbp:maximalCompactSubgroupOf
|
gptkb:SL(n,_R)
|
|
gptkbp:notation
|
gptkb:SO(n)
|
|
gptkbp:notSimpleFor
|
n = 4
|
|
gptkbp:order
|
infinite
|
|
gptkbp:preserves
|
gptkb:Euclidean_inner_product
orientation
|
|
gptkbp:realForm
|
gptkb:SO(n,_C)
|
|
gptkbp:relatedGroup
|
gptkb:Lie_group
gptkb:matrix_Lie_group
orthogonal group
semisimple group
compact group
connected group
simple group (for n > 2)
|
|
gptkbp:relatedTo
|
gptkb:orthogonal_group_O(n)
gptkb:special_linear_group_SL(n)
spin group Spin(n)
|
|
gptkbp:represents
|
orthogonal transformations
|
|
gptkbp:universalCover
|
gptkb:Spin(n)
|
|
gptkbp:bfsParent
|
gptkb:orthogonal_group
|
|
gptkbp:bfsLayer
|
5
|