|
gptkbp:instanceOf
|
gptkb:group_of_people
gptkb:topology
gptkb:Lie_group
|
|
gptkbp:actsOn
|
R^2
rotations
|
|
gptkbp:application
|
gptkb:geometry
gptkb:signal_processing
computer vision
physics
robotics
|
|
gptkbp:automorphismGroup
|
gptkb:Z/2Z
|
|
gptkbp:centralTo
|
gptkb:SO(2)
|
|
gptkbp:commutatorSubgroup
|
trivial
|
|
gptkbp:compact
|
compact
|
|
gptkbp:compactSubgroupOf
|
gptkb:GL(2,_R)
|
|
gptkbp:connectedness
|
connected
|
|
gptkbp:containsElement
|
2x2 real orthogonal matrices with determinant 1
|
|
gptkbp:determinant
|
1
|
|
gptkbp:dimensions
|
1
|
|
gptkbp:eigenvalues
|
complex numbers of modulus 1
|
|
gptkbp:firstHomologyGroup
|
Z
|
|
gptkbp:fullName
|
gptkb:Special_Orthogonal_Group_in_2_Dimensions
|
|
gptkbp:fundamentalGroup
|
Z
|
|
gptkbp:generation
|
[[0, -1], [1, 0]]
|
|
gptkbp:hasSubgroup
|
gptkb:O(2)
|
|
gptkbp:homologyGroupH0
|
Z
|
|
gptkbp:homologyGroupH1
|
Z
|
|
gptkbp:homologyGroupH2
|
0
|
|
gptkbp:homotopyType
|
gptkb:butter
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
SO(2)
|
|
gptkbp:identityElement
|
2x2 identity matrix
|
|
gptkbp:isomorphicTo
|
gptkb:U(1)
gptkb:circle_group
|
|
gptkbp:isQuotientOf
|
O(2)/SO(2) ≅ Z/2Z
|
|
gptkbp:Lie_algebra
|
so(2)
|
|
gptkbp:matrixRepresentation
|
[[cosθ, -sinθ], [sinθ, cosθ]]
|
|
gptkbp:maximalTorus
|
gptkb:SO(2)
|
|
gptkbp:order
|
infinite
|
|
gptkbp:parameter
|
rotation matrices
angle θ in [0, 2π)
|
|
gptkbp:rank
|
1
|
|
gptkbp:realization
|
symmetries of the circle
|
|
gptkbp:relatedGroup
|
gptkb:Lie_group
simple
compact
connected
abelian
|
|
gptkbp:represents
|
irreducible representations are 1-dimensional
|
|
gptkbp:trace
|
2cosθ
|
|
gptkbp:universalCover
|
R
|
|
gptkbp:bfsParent
|
gptkb:U(1)
gptkb:SL(2,ℝ)
gptkb:旋转群
gptkb:特殊正交群
gptkb:U(1)_symmetry
gptkb:rotation_group
|
|
gptkbp:bfsLayer
|
6
|