|
gptkbp:instanceOf
|
gptkb:Lie_group
complex Lie algebra
|
|
gptkbp:adjoint_representation
|
3-dimensional
|
|
gptkbp:appearsIn
|
gptkb:quantum_field_theory
|
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gptkbp:automorphismGroup
|
gptkb:PGL(2,C)
|
|
gptkbp:basisFor
|
{E, F, H}
|
|
gptkbp:Borel_subalgebra
|
upper triangular traceless matrices
|
|
gptkbp:Cartan_subalgebra
|
1-dimensional
|
|
gptkbp:Casimir_element
|
exists
|
|
gptkbp:centralTo
|
{0}
|
|
gptkbp:Chevalley_basis
|
{E, F, H}
|
|
gptkbp:commutation_relation
|
[E, F] = H
[H, E] = 2E
[H, F] = -2F
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gptkbp:defining_property
|
traceless 2x2 complex matrices
|
|
gptkbp:dimensions
|
3
|
|
gptkbp:Dynkin_diagram
|
single node
|
|
gptkbp:enveloping_algebra
|
U(sl(2,C))
|
|
gptkbp:field
|
complex numbers
|
|
gptkbp:highest_weight_representations
|
classified
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
Lie algebra sl(2,C)
|
|
gptkbp:isomorphicTo
|
so(3,C)
sp(2,C)
|
|
gptkbp:isSemisimple
|
true
|
|
gptkbp:isSimple
|
true
|
|
gptkbp:Killing_form
|
non-degenerate
|
|
gptkbp:notation
|
gptkb:sl(2,C)
|
|
gptkbp:parabolic_subalgebra
|
gptkb:Borel_subalgebra
|
|
gptkbp:rank
|
1
|
|
gptkbp:realForm
|
gptkb:sl(2,R)
|
|
gptkbp:relatedGroup
|
gptkb:SL(2,C)
|
|
gptkbp:relatedTo
|
gptkb:Lorentz_group
gptkb:SU(2)
|
|
gptkbp:representation_theory
|
well-studied
|
|
gptkbp:root_space_decomposition
|
exists
|
|
gptkbp:Serre_relations
|
satisfied
|
|
gptkbp:structure_constants
|
standard for A1 type
|
|
gptkbp:subalgebra_of
|
gl(2,C)
|
|
gptkbp:type
|
A1
|
|
gptkbp:universalCover
|
gptkb:SL(2,C)
|
|
gptkbp:used_in
|
differential geometry
mathematical physics
quantum mechanics
representation theory
theory of Lie groups
|
|
gptkbp:weight_lattice
|
Z
|
|
gptkbp:Weyl_group
|
gptkb:Z/2Z
|
|
gptkbp:bfsParent
|
gptkb:Lie_algebra_su(2)
|
|
gptkbp:bfsLayer
|
7
|