|
gptkbp:instanceOf
|
gptkb:algebra
gptkb:group_of_people
|
|
gptkbp:abbreviation
|
gptkb:SO(n)
gptkb:SU(n)
gptkb:SL(n,_F)
|
|
gptkbp:actsOn
|
gptkb:Vector
gptkb:complex_vector_space
gptkb:n-dimensional_Euclidean_space
symplectic vector space
|
|
gptkbp:alternativeName
|
linear_group
orthogonal_matrix
special_linear_group
special_orthogonal_group
special_unitary_group
spin_group
symplectic_group
unitary_group
|
|
gptkbp:application
|
gptkb:geometry
gptkb:signal_processing
computer graphics
computer science
crystallography
differential geometry
mechanics
particle physics
physics
quantum mechanics
representation theory
robotics
statistics
|
|
gptkbp:category
|
gptkb:algebraic_geometry
gptkb:group_of_people
gptkb:Lie_group
|
|
gptkbp:center_for_n_even
|
{identity, -identity}
|
|
gptkbp:center_for_n_odd
|
{identity}
|
|
gptkbp:centralTo
|
cyclic group of order n
scalar matrices with determinant 1
group of scalar matrices with modulus 1
|
|
gptkbp:compact
|
true
yes
|
|
gptkbp:connectedTo
|
yes
|
|
gptkbp:consistsOf
|
orthogonal matrices with determinant 1
|
|
gptkbp:contains
|
orthogonal matrices
identity matrix
|
|
gptkbp:containsElement
|
identity matrix
unitary matrix
invertible matrix
|
|
gptkbp:definedIn
|
gptkb:Field
complex numbers
real numbers
field F
|
|
gptkbp:defines
|
group of invertible linear transformations of a vector space
group of n×n unitary matrices
group of n×n matrices with determinant 1 over a field F
|
|
gptkbp:determinant
|
1
±1
|
|
gptkbp:dimensions
|
n
n(n-1)/2
n^2/2
n^2
n(2n+1)
n^2-1
|
|
gptkbp:discoveredBy
|
gptkb:Ferdinand_Georg_Frobenius
|
|
gptkbp:example
|
gptkb:general_linear_group
orthogonal group
|
|
gptkbp:example_for_n=2
|
group of planar rotations
|
|
gptkbp:example_for_n=3
|
group of rotations in 3D space
|
|
gptkbp:field
|
gptkb:geometry
gptkb:mathematics
F
group theory
linear algebra
|
|
gptkbp:first_homotopy_group_for_n>2
|
gptkb:Z/2Z
|
|
gptkbp:firstAppearance
|
19th century mathematics
|
|
gptkbp:firstHomotopyGroup
|
trivial
|
|
gptkbp:generalizes
|
orthogonal group
|
|
gptkbp:hasConnection
|
true
|
|
gptkbp:hasMaximalTorus
|
U(1)^{n-1}
|
|
gptkbp:hasSubgroup
|
gptkb:general_linear_group
gptkb:compact_symplectic_group
gptkb:projective_symplectic_group
gptkb:symplectic_similitude_group
orthogonal group
|
|
gptkbp:identityElement
|
identity matrix
|
|
gptkbp:importantFor
|
gptkb:Lie_theory
gptkb:algebraic_geometry
gptkb:geometry
group theory
physics
quantum mechanics
representation theory
linear algebra
|
|
gptkbp:is_a_classical_group
|
yes
|
|
gptkbp:is_a_compact_group
|
yes
|
|
gptkbp:is_a_connected_group
|
yes
|
|
gptkbp:is_a_Lie_group
|
yes
|
|
gptkbp:is_a_linear_algebraic_group
|
yes
|
|
gptkbp:is_a_non-abelian_group_for_n>2
|
yes
|
|
gptkbp:is_a_real_algebraic_group
|
yes
|
|
gptkbp:is_a_real_Lie_group
|
yes
|
|
gptkbp:is_a_semisimple_group
|
yes
|
|
gptkbp:is_a_simple_group_for_n>2
|
yes
|
|
gptkbp:is_a_topological_group
|
yes
|
|
gptkbp:is_simple_for_n_>_2
|
yes
|
|
gptkbp:isA
|
group of invertible matrices
|
|
gptkbp:isMatrixGroup
|
true
yes
|
|
gptkbp:isNonAbelian
|
n > 2
|
|
gptkbp:isSimple
|
n > 1
n ≥ 2, except SL(2,2) and SL(2,3)
|
|
gptkbp:isSimpleLieGroupFor
|
n ≥ 2
|
|
gptkbp:isTopologically
|
simply connected
|
|
gptkbp:Lie_algebra
|
gptkb:special_linear_Lie_algebra
skew-Hermitian matrices
su(n)
|
|
gptkbp:namedFor
|
gptkb:mathematician
|
|
gptkbp:notation
|
gptkb:GL(n,_F)
gptkb:SO(n,_R)
gptkb:SU(n)
gptkb:SL(n,_F)
gptkb:Sp(2n,_F)
O(n)
U(n)
|
|
gptkbp:notation_(complex_numbers)
|
gptkb:SL(n,_ℂ)
|
|
gptkbp:notation_(integers)
|
gptkb:SL(n,_ℤ)
|
|
gptkbp:notation_(real_numbers)
|
gptkb:SL(n,_ℝ)
|
|
gptkbp:order
|
infinite
finite for finite fields
|
|
gptkbp:order_(finite_field)
|
product of (q^n-1)(q^n-q)...(q^n-q^{n-1})/(q-1)
|
|
gptkbp:orderForSU2
|
infinite
|
|
gptkbp:orderForSU3
|
infinite
|
|
gptkbp:originatedIn
|
gptkb:Hamiltonian_mechanics
gptkb:classical_mechanics
gptkb:gauge_theory
gptkb:quantum_field_theory
gptkb:signal_processing
gptkb:topology
gptkb:algebraic_K-theory
gptkb:string_theory
coding theory
control theory
cryptography
differential geometry
dynamical systems
group theory
integrable systems
mathematical physics
modular forms
number theory
representation theory
algebraic groups
linear algebra
algebraic combinatorics
automorphic forms
moduli spaces
symplectic geometry
|
|
gptkbp:preserves
|
gptkb:Euclidean_inner_product
inner product
quadratic form
orientation
symplectic form
Hermitian inner product
|
|
gptkbp:property
|
compact
connected
non-abelian for n > 1
|
|
gptkbp:relatedTo
|
gptkb:general_linear_group
gptkb:Euclidean_space
gptkb:group_of_people
gptkb:Lie_group
gptkb:rotation_group
gptkb:projective_special_unitary_group
gptkb:projective_unitary_group
reflection
representation theory
isometry
orthogonal group
orthogonal transformation
classical group
projective special linear group
compact group
|
|
gptkbp:studiedBy
|
gptkb:Hermann_Weyl
|
|
gptkbp:studiedIn
|
group theory
linear algebra
|
|
gptkbp:subclassOf
|
gptkb:group_of_people
gptkb:Lie_group
|
|
gptkbp:universalCover
|
orthogonal group
|
|
gptkbp:used_in
|
gptkb:geometry
physics
symmetry analysis
|
|
gptkbp:usedIn
|
gptkb:gauge_theory
gptkb:Standard_Model_of_particle_physics
particle physics
quantum mechanics
representation theory
|
|
gptkbp:bfsParent
|
gptkb:group_of_people
|
|
gptkbp:bfsLayer
|
4
|