LieAlgebras[DirectSumOfRepresentations] - form the direct sum representation for a list of representations of a Lie algebra
Calling Sequences
DirectSumOfRepresentations(R, W)
Parameters
R - a list of representations ρ1, ρ2, ... of a Lie algebra 𝔤 on vector spaces V1, V2, ... .
W - a Maple name or string, the name of the frame for the representation space for the direct sum representation
Description
Examples
Let 𝔤 be a Lie algebra and let ρi : 𝔤 → Vi , i = 1, 2, ..., p be a sequence of representations of 𝔤. Then the direct sum representation of the representationsρi is the representation σ : 𝔤 → W, where W = V1⊕V2 ⊕⋅⋅⋅⊕ Vp and
σxY = ρ1xY1 + ρ2xY2 +⋅⋅⋅+ ρpxYp for Y = Y1 + Y2 + ⋅⋅⋅ + Yp with Yi ∈Vi.
The command DirectSumOfRepresentations(R, W) returns the representation σ.
withDifferentialGeometry:withLieAlgebras:
Example 1.
Define the standard representation and the adjoint representation for sl2. Then form the direct sum representation. First, setup the representation spaces.
DGsetupx1,x2,V1:
DGsetupy1,y2,y2,V2:
DGsetupz1,z2,z3,z4,z5,W1:
DGsetupz1,z2,z3,z4,z5,z6,W2:
Define the standard representation.
M1≔Matrix0,1,0,0,Matrix1,0,0,−1,Matrix0,0,1,0
L≔LieAlgebraDataM1,sl2
L:=e1,e2=−2e1,e1,e3=e2,e2,e3=−2e3
DGsetupL
Lie algebra: sl2
ρ1≔Representationsl2,V1,M1
Define the adjoint representation.
ρ2≔Representationsl2,V2,Adjoint
Define the direct sum representation of ρ1and ρ2
φ1≔DirectSumOfRepresentationsρ1,ρ2,W1
Queryφ1,Representation
true
Define the direct sum of 3 copies of ρ1.
φ2≔DirectSumOfRepresentationsρ1,ρ1,ρ1,W2
Queryφ2,Representation
See Also
DifferentialGeometry
LieAlgebras
Representation
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