|
gptkbp:instanceOf
|
gptkb:Lie_group
|
|
gptkbp:adjointGroup
|
gptkb:PSL(2,_C)
|
|
gptkbp:automorphismGroup
|
gptkb:PGL(2,_C)
|
|
gptkbp:basisFor
|
{E, F, H}
|
|
gptkbp:Cartan_subalgebra_dimension
|
1
|
|
gptkbp:CasimirElement
|
exists
|
|
gptkbp:centralTo
|
{0}
|
|
gptkbp:commutatorSubalgebra
|
gptkb:sl(2,_C)
|
|
gptkbp:complexification
|
gptkb:sl(2,_R)
|
|
gptkbp:definingRelation
|
trace zero 2x2 complex matrices
|
|
gptkbp:dimensions
|
3
|
|
gptkbp:field
|
complex numbers
|
|
gptkbp:fullName
|
special linear Lie algebra of 2x2 complex matrices
|
|
gptkbp:highestWeightTheory
|
applies
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
sl(2, C)
|
|
gptkbp:isomorphicTo
|
so(3,1)
sp(2, C)
|
|
gptkbp:isSemisimple
|
true
|
|
gptkbp:isSimple
|
true
|
|
gptkbp:Killing_form
|
non-degenerate
|
|
gptkbp:notation
|
𝔰𝔩(2, ℂ)
|
|
gptkbp:realForm
|
gptkb:sl(2,_R)
|
|
gptkbp:relatedGroup
|
gptkb:SL(2,_C)
|
|
gptkbp:representationTheory
|
well-studied
|
|
gptkbp:rootMultiplicity
|
1
|
|
gptkbp:structure_constants
|
[H,E]=2E, [H,F]=-2F, [E,F]=H
|
|
gptkbp:type
|
A1
|
|
gptkbp:universalEnvelopingAlgebra
|
U(sl(2, C))
|
|
gptkbp:usedIn
|
gptkb:quantum_field_theory
differential geometry
mathematical physics
quantum mechanics
representation theory
|
|
gptkbp:Weyl_group
|
gptkb:Z/2Z
|
|
gptkbp:bfsParent
|
gptkb:SL(2,_C)
|
|
gptkbp:bfsLayer
|
7
|