Statements (364)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:algebra
gptkb:group_of_people gptkb:mathematical_concept gptkb:Riemannian_manifold gptkb:Lie_group |
| gptkbp:alternativeName |
Algebraic_group
Clifford_algebra Hopf_algebra Lie_algebra affine_Lie_algebra compact_Lie_group compact_abelian_Lie_group complex_Lie_group exceptional_Lie_group groupe_de_Lie groupe_mathématique linear_algebraic_group real_Lie_group simple_Lie_algebra simple_Lie_group |
| gptkbp:application |
gptkb:Dirac_equation
gptkb:geometry gptkb:theoretical_physics computer graphics partial differential equations particle physics quantum mechanics relativity representation theory robotics geometric algebra spin geometry |
| gptkbp:automorphismGroup |
diagram automorphism
|
| gptkbp:basisFor |
Cartan basis
|
| gptkbp:category |
gptkb:Kac–Moody_category
noncommutative geometry graded algebra |
| gptkbp:characterizedBy |
gptkb:Pontryagin_duality
|
| gptkbp:class |
gptkb:Weyl_group
gptkb:compact_Lie_groups gptkb:Cartan_classification gptkb:Abelian_Lie_groups gptkb:connected_Lie_groups gptkb:semisimple_Lie_groups gptkb:simple_Lie_groups classified by maximal torus and root system classified by Cartan |
| gptkbp:containsModule |
Lie algebra module
|
| gptkbp:definedIn |
gptkb:Vector
gptkb:Field quadratic form bilinear form connected non-abelian Lie group with no nontrivial connected normal subgroups |
| gptkbp:defines |
A group that is also a smooth manifold where the group operations are smooth.
|
| gptkbp:dimensions |
infinite
finite infinite-dimensional finite-dimensional |
| gptkbp:Dynkin_diagram |
gptkb:affine_Dynkin_diagram
|
| gptkbp:example |
gptkb:general_linear_group
gptkb:Sweedler's_4-dimensional_Hopf_algebra gptkb:torus_group_T^n gptkb:Lie_group gptkb:rotation_group gptkb:special_linear_group_SL(n,_R) gptkb:special_orthogonal_group_SO(n) gptkb:symplectic_group_Sp(n) gptkb:special_unitary_group_SU(n) gptkb:unitary_group_U(n) gptkb:n-dimensional_torus gptkb:exceptional_Lie_group_F4 gptkb:additive_group_of_real_numbers gptkb:circle_group gptkb:exceptional_Lie_group_E6 gptkb:exceptional_Lie_group_E7 gptkb:exceptional_Lie_group_E8 gptkb:exceptional_Lie_group_G2 nilpotent Lie algebra semisimple Lie algebra solvable Lie algebra universal enveloping algebra orthogonal group matrix Lie algebra vector fields with commutator group algebra multiplicative group of nonzero real numbers quantum group coordinate ring of an algebraic group finite group (as a discrete Lie group) |
| gptkbp:field |
gptkb:algebra
gptkb:mathematics differential geometry |
| gptkbp:firstDescribed |
1878
|
| gptkbp:generalizes |
gptkb:associative_algebra
gptkb:Lie_group quaternions complex numbers enveloping algebra group algebra exterior algebra finite-dimensional simple Lie algebra |
| gptkbp:hasApplication |
gptkb:gauge_theory
gptkb:geometry gptkb:quantum_field_theory gptkb:theoretical_physics gptkb:topology gptkb:C*-algebra control theory harmonic analysis mathematical physics modular forms number theory partial differential equations particle physics quantum mechanics combinatorics knot theory symmetry groups in mathematics soliton equations |
| gptkbp:hasComponent |
gptkb:Clifford_module
gptkb:Clifford_bundle gptkb:algebra gptkb:military_unit gptkb:Clifford_group multiplication coalgebra antipode comultiplication counit |
| gptkbp:hasDual |
gptkb:Langlands_dual_affine_Lie_algebra
dual Hopf algebra |
| gptkbp:hasHomomorphism |
Lie algebra homomorphism
|
| gptkbp:hasInvariant |
gptkb:fundamental_group
gptkb:Weyl_group gptkb:center gptkb:Killing_form gptkb:root gptkb:Haar_measure gptkb:universal_covering_group Lie bracket |
| gptkbp:hasProperty |
gptkb:Weyl_group
gptkb:center gptkb:knowledge_representation gptkb:Peter–Weyl_theorem gptkb:associative_algebra gptkb:semisimple_Lie_group gptkb:Lie's_first_theorem gptkb:Lie's_second_theorem gptkb:Lie's_third_theorem gptkb:Lie_group gptkb:root gptkb:reductive_Lie_group gptkb:linear_Lie_group gptkb:solvable_Lie_group gptkb:Baker–Campbell–Hausdorff_formula gptkb:Cartan_subgroup gptkb:maximal_torus gptkb:universal_covering_group gptkb:Hausdorff representation theory compact Lie bracket adjoint representation automorphism exponential map homomorphism structure constants structure theory continuous symmetry infinite-dimensional lattice non-abelian associativity identity element connected derivation direct product finite-dimensional flag manifold homogeneous space normal subgroup second-countable Borel subgroup symmetric space abelian (may or may not be abelian) classification of simple Lie groups closure coadjoint representation compactness (may or may not be compact) connectedness (may or may not be connected) differentiable group operations discrete subgroup every compact Lie group is a real Lie group every compact Lie group is locally path-connected every compact Lie group is metrizable every compact Lie group is unimodular every representation is completely reducible group operations are smooth inverse element simply connected (may or may not be simply connected) locally Euclidean matrix group (may or may not be a matrix group) nilpotent Lie group non-abelian (may or may not be non-abelian) non-compact Lie group quotient group real or complex structure semidirect product semisimple (may or may not be semisimple) simple (may or may not be simple) smooth structure every compact Lie group is a real analytic manifold finite center no nontrivial connected normal subgroups antipode is an involution bialgebra with antipode coassociative coalgebra every closed subgroup is compact every compact Lie group has a finite center every compact Lie group is second-countable vector space over a field every compact Lie group is isomorphic to a closed subgroup of U(n) for some n (Peter–Weyl theorem) every compact Lie group is connected iff its identity component is the whole group central extension |
| gptkbp:hasQuantumDeformation |
gptkb:quantum_affine_algebra
|
| gptkbp:hasSubalgebra |
gptkb:Heisenberg_algebra
|
| gptkbp:hasSubfield |
gptkb:Lie_group
gptkb:Grassmann_algebra abstract algebra geometric algebra spinor algebra |
| gptkbp:hasType |
classical simple Lie group
exceptional simple Lie group |
| https://www.w3.org/2000/01/rdf-schema#label |
Lie group
|
| gptkbp:introduced |
gptkb:Victor_Kac
gptkb:Robert_Moody |
| gptkbp:introducedIn |
1941
1968 |
| gptkbp:isCentralExtensionOf |
loop algebra
|
| gptkbp:isQuotientOf |
quotient Lie algebra
|
| gptkbp:namedAfter |
gptkb:William_Kingdon_Clifford
gptkb:Sophus_Lie gptkb:Heinz_Hopf |
| gptkbp:namedFor |
gptkb:Élie_Cartan
|
| gptkbp:operator |
Lie bracket
|
| gptkbp:property |
gptkb:Lie_group
Jacobi identity compact alternativity bilinearity group structure locally Euclidean smooth structure abelian |
| gptkbp:relatedTo |
gptkb:Dirac_matrices
gptkb:Pin_group gptkb:algebra gptkb:geometry gptkb:topology gptkb:K-theory gptkb:knowledge_representation gptkb:symmetry gptkb:associative_algebra gptkb:semisimple_Lie_group gptkb:Killing_form gptkb:Lie's_third_theorem gptkb:Lie_group gptkb:category_theory gptkb:root Pauli matrices representation theory Lie group action Lie subgroup adjoint representation automorphism center of a Lie algebra exponential map structure constants universal enveloping algebra orthogonal group orthogonal transformations quadratic form tensor algebra differential forms Lie bialgebra Lie coalgebra Lie ring Lie superalgebra Lie triple system Poisson algebra commutator derivation enveloping algebra universal property Lie group homomorphism group algebra quantum group coalgebra bialgebra multivector loop algebra |
| gptkbp:represents |
highest weight representation
integrable representation |
| gptkbp:structure |
gptkb:group_of_people
gptkb:topology gptkb:Riemannian_manifold |
| gptkbp:studiedBy |
gptkb:Hermann_Weyl
gptkb:Sophus_Lie gptkb:Moss_Sweedler gptkb:Heinz_Hopf gptkb:Wilhelm_Killing gptkb:Élie_Cartan gptkb:Jean-Louis_Koszul gptkb:Pierre_Cartier harmonic analysis representation theory |
| gptkbp:studiedIn |
gptkb:Lie_theory
gptkb:mathematics gptkb:theoretical_physics abstract algebra differential geometry mathematical physics representation theory |
| gptkbp:subclassOf |
gptkb:topology
gptkb:Lie_group gptkb:Kac–Moody_algebra abelian group |
| gptkbp:subunit |
gptkb:Lie_group
gptkb:Cartan_subgroup gptkb:maximal_torus Lie subgroup ideal Lie subalgebra |
| gptkbp:theory |
gptkb:Bott_periodicity_theorem
gptkb:Weyl_character_formula gptkb:Borel–Weil_theorem gptkb:Peter–Weyl_theorem gptkb:Lie's_first_theorem gptkb:Lie's_second_theorem gptkb:Lie's_third_theorem gptkb:Ado's_theorem gptkb:Baker–Campbell–Hausdorff_formula gptkb:Cartan's_theorem gptkb:Cartan_decomposition gptkb:Iwasawa_decomposition gptkb:Levi_decomposition Malcev's theorem |
| gptkbp:type |
gptkb:affine_root_system
|
| gptkbp:usedIn |
gptkb:geometry
gptkb:quantum_field_theory gptkb:theoretical_physics gptkb:topology gptkb:string_theory gptkb:category_theory differential geometry noncommutative geometry physics representation theory quantum group theory spinors |
| gptkbp:Weyl_group |
gptkb:affine_Weyl_group
|
| gptkbp:bfsParent |
gptkb:algebra
gptkb:group_of_people |
| gptkbp:bfsLayer |
4
|