|
gptkbp:instanceOf
|
gptkb:group_of_people
gptkb:Lie_group
|
|
gptkbp:actsOn
|
gptkb:n-dimensional_Euclidean_space
|
|
gptkbp:centralTo
|
{I} for odd n, {I, -I} for even n
|
|
gptkbp:compact
|
true
|
|
gptkbp:connectedTo
|
true
|
|
gptkbp:consistsOf
|
n×n orthogonal matrices with determinant 1
|
|
gptkbp:dimensions
|
n(n-1)/2
|
|
gptkbp:fundamental_group_for_n_=_1
|
trivial group
|
|
gptkbp:fundamental_group_for_n_=_2
|
Z
|
|
gptkbp:fundamental_group_for_n_≥_3
|
gptkb:Z/2Z
|
|
gptkbp:hasSubgroup
|
gptkb:orthogonal_group_O(n)
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
orthogonal group SO(n)
|
|
gptkbp:identityElement
|
identity matrix
|
|
gptkbp:is_a_classical_group
|
true
|
|
gptkbp:is_a_matrix_Lie_group
|
true
|
|
gptkbp:is_a_maximal_compact_subgroup_of
|
gptkb:general_linear_group_GL(n,_R)
|
|
gptkbp:is_a_real_algebraic_group
|
true
|
|
gptkbp:is_a_real_Lie_group
|
true
|
|
gptkbp:is_simple_for_n_≥_5
|
true
|
|
gptkbp:Lie_algebra
|
so(n)
|
|
gptkbp:notation
|
gptkb:SO(n)
|
|
gptkbp:order_for_n=1
|
2
|
|
gptkbp:order_for_n=2
|
infinite
|
|
gptkbp:order_for_n=3
|
infinite
|
|
gptkbp:preserves
|
gptkb:Euclidean_inner_product
|
|
gptkbp:realForm
|
gptkb:special_orthogonal_group_SO(n,_C)
|
|
gptkbp:relatedTo
|
gptkb:rotation_group
|
|
gptkbp:special_case_for_n=2
|
gptkb:circle_group_SO(2)
|
|
gptkbp:special_case_for_n=3
|
gptkb:rotation_group_SO(3)
|
|
gptkbp:bfsParent
|
gptkb:special_unitary_group_SU(n)
gptkb:connected_Lie_groups
|
|
gptkbp:bfsLayer
|
6
|