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.
with⁡DifferentialGeometry:with⁡LieAlgebras:
Example 1.
Define the standard representation and the adjoint representation for sl2. Then form the direct sum representation. First, setup the representation spaces.
DGsetup⁡x1,x2,V1:
DGsetup⁡y1,y2,y2,V2:
DGsetup⁡z1,z2,z3,z4,z5,W1:
DGsetup⁡z1,z2,z3,z4,z5,z6,W2:
Define the standard representation.
M1≔Matrix⁡0,1,0,0,Matrix⁡1,0,0,−1,Matrix⁡0,0,1,0
M1:=0100,100−1,0010
L≔LieAlgebraData⁡M1,sl2
L:=e1,e2=−2⁢e1,e1,e3=e2,e2,e3=−2⁢e3
DGsetup⁡L
Lie algebra: sl2
ρ1≔Representation⁡sl2,V1,M1
ρ1:=e1,0100,e2,100−1,e3,0010
Define the adjoint representation.
ρ2≔Representation⁡sl2,V2,Adjoint⁡
ρ2:=e1,0−20001000,e2,20000000−2,e3,000−100020
Define the direct sum representation of ρ1and ρ2
φ1≔DirectSumOfRepresentations⁡ρ1,ρ2,W1
φ1:=e1,0100000000000−200000100000,e2,100000−100000200000000000−2,e3,00000100000000000−10000020
Query⁡φ1,Representation
true
Define the direct sum of 3 copies of ρ1.
φ2≔DirectSumOfRepresentations⁡ρ1,ρ1,ρ1,W2
φ2:=e1,010000000000000100000000000001000000,e2,1000000−10000001000000−10000001000000−1,e3,000000100000000000001000000000000010
Query⁡φ2,Representation
See Also
DifferentialGeometry
LieAlgebras
Representation
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