|
gptkbp:instanceOf
|
gptkb:group_of_people
gptkb:topology
gptkb:Lie_group
abelian group
compact group
|
|
gptkbp:actsOn
|
Euclidean plane
|
|
gptkbp:application
|
gptkb:geometry
gptkb:signal_processing
quantum mechanics
robotics
|
|
gptkbp:automorphismGroup
|
gptkb:Z/2Z
|
|
gptkbp:centralTo
|
gptkb:SO(2)
|
|
gptkbp:commutative
|
true
|
|
gptkbp:compact
|
true
|
|
gptkbp:connects
|
true
|
|
gptkbp:containsElement
|
2x2 real orthogonal matrices with determinant 1
|
|
gptkbp:dimensions
|
1
|
|
gptkbp:field
|
real numbers
|
|
gptkbp:firstCohomologyGroup
|
Z
|
|
gptkbp:firstHomologyGroup
|
Z
|
|
gptkbp:form
|
[[cosθ, -sinθ], [sinθ, cosθ]]
|
|
gptkbp:fundamentalGroup
|
Z
|
|
gptkbp:generation
|
[[0, -1], [1, 0]]
|
|
gptkbp:hasCoveringGroup
|
real numbers under addition
|
|
gptkbp:hasSubgroup
|
gptkb:SO(3)
gptkb:O(2)
|
|
gptkbp:homotopyType
|
circle S^1
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
circle group SO(2)
|
|
gptkbp:identityElement
|
[[1, 0], [0, 1]]
|
|
gptkbp:isomorphicTo
|
gptkb:U(1)
|
|
gptkbp:isSimple
|
false
|
|
gptkbp:Lie_algebra
|
so(2)
|
|
gptkbp:LieAlgebraDimension
|
1
|
|
gptkbp:maximalTorusOf
|
gptkb:SU(2)
gptkb:SO(3)
gptkb:U(2)
|
|
gptkbp:notation
|
gptkb:U(1)
gptkb:SO(2)
gptkb:S^1
|
|
gptkbp:order
|
infinite
|
|
gptkbp:parameter
|
angle θ
|
|
gptkbp:realization
|
gptkb:unit_circle_in_complex_plane
|
|
gptkbp:relatedGroup
|
matrix multiplication
|
|
gptkbp:relatedTo
|
rotations in 2D
|
|
gptkbp:representationTheory
|
characters are e^{inθ}, n∈Z
|
|
gptkbp:universalCover
|
gptkb:real_line_R
|
|
gptkbp:bfsParent
|
gptkb:orthogonal_group_SO(n)
|
|
gptkbp:bfsLayer
|
7
|