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gptkbp:instanceOf
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gptkb:mathematical_concept
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gptkbp:automorphismGroup
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automorphism group
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gptkbp:Cartan_matrix
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gptkb:Weyl_group
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gptkbp:centralTo
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trivial center
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gptkbp:characterizedBy
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having no nonzero solvable ideals
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gptkbp:class
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gptkb:Cartan_classification
gptkb:Dynkin_diagrams
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gptkbp:complexification
|
complexification
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gptkbp:definedIn
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Lie algebra that is a direct sum of simple Lie algebras
|
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gptkbp:example
|
gptkb:sl(n,_C)
gptkb:sp(2n,_C)
so(n, C)
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gptkbp:field
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algebraically closed field of characteristic zero
|
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gptkbp:hasBorelSubalgebra
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gptkb:Borel_subalgebra
|
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gptkbp:hasCartanInvolution
|
gptkb:Cartan_involution
|
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gptkbp:hasCartanSubalgebra
|
Cartan subalgebra
|
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gptkbp:hasCasimirElement
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gptkb:Casimir_element
|
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gptkbp:hasChevalleyBasis
|
gptkb:Chevalley_basis
|
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gptkbp:hasDecomposition
|
gptkb:Cartan_decomposition
root space decomposition
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gptkbp:hasDimensionFormula
|
dimension formula in terms of roots
|
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gptkbp:hasHighestWeightTheory
|
highest weight theory
|
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gptkbp:hasIsomorphismClasses
|
isomorphism classes classified by Dynkin diagrams
|
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gptkbp:hasKillingForm
|
gptkb:Killing_form
|
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gptkbp:hasProperty
|
Killing form is non-degenerate
|
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gptkbp:hasRootLattice
|
gptkb:Weyl_group
|
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gptkbp:hasRootSpaceDecomposition
|
root space decomposition
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gptkbp:hasSimpleComponents
|
simple Lie algebras
|
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gptkbp:hasStructureConstants
|
structure constants
|
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gptkbp:hasUniversalEnvelopingAlgebra
|
universal enveloping algebra
|
|
gptkbp:hasWeightLattice
|
weight lattice
|
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gptkbp:hasWeightSpaceDecomposition
|
weight space decomposition
|
|
gptkbp:hasWeylCharacterFormula
|
gptkb:Weyl_character_formula
|
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gptkbp:hasWeylDimensionFormula
|
gptkb:Weyl_dimension_formula
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
semisimple Lie algebras
|
|
gptkbp:realForm
|
real forms
|
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gptkbp:relatedTo
|
gptkb:Lie_groups
reductive Lie algebras
simple Lie algebras
|
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gptkbp:represents
|
finite-dimensional irreducible representations
|
|
gptkbp:studiedBy
|
gptkb:Élie_Cartan
|
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gptkbp:studiedIn
|
representation theory
|
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gptkbp:subclassOf
|
gptkb:Lie_group
|
|
gptkbp:type
|
gptkb:root
|
|
gptkbp:usedIn
|
gptkb:theoretical_physics
differential geometry
algebraic groups
|
|
gptkbp:Weyl_group
|
gptkb:Weyl_group
|
|
gptkbp:bfsParent
|
gptkb:Weyl_character_formula
gptkb:Weyl_group
|
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gptkbp:bfsLayer
|
5
|