 DirectSumOfRepresentations - Maple Help

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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 , ... of a Lie algebra on vector spaces .

W         - a Maple name or string, the name of the frame for the representation space for the direct sum representation Description

 • Let $\mathrm{𝔤}$ be a Lie algebra and let , be a sequence of representations of $\mathrm{𝔤}$. Then the direct sum representation of the representationsis the representation , where  and

for  with .

 • The command DirectSumOfRepresentations(R, W) returns the representation $\mathrm{σ}$. Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

Define the standard representation and the adjoint representation for $\mathrm{sl}\left(2\right)$. Then form the direct sum representation. First, setup the representation spaces.

 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2}\right],\mathrm{V1}\right):$
 V1 > $\mathrm{DGsetup}\left(\left[\mathrm{y1},\mathrm{y2},\mathrm{y2}\right],\mathrm{V2}\right):$
 V2 > $\mathrm{DGsetup}\left(\left[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4},\mathrm{z5}\right],\mathrm{W1}\right):$
 W1 > $\mathrm{DGsetup}\left(\left[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4},\mathrm{z5},\mathrm{z6}\right],\mathrm{W2}\right):$

Define the standard representation.

 W2 > $\mathrm{M1}≔\left[\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,-1\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0\right],\left[1,0\right]\right]\right)\right]$
 ${\mathrm{M1}}{:=}\left[\left[\begin{array}{rr}{0}& {1}\\ {0}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{1}& {0}\\ {0}& {-}{1}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {0}\\ {1}& {0}\end{array}\right]\right]$ (2.1)
 W2 > $L≔\mathrm{LieAlgebraData}\left(\mathrm{M1},\mathrm{sl2}\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}\right]$ (2.2)
 W2 > $\mathrm{DGsetup}\left(L\right)$
 ${\mathrm{Lie algebra: sl2}}$ (2.3)
 sl2 > $\mathrm{ρ1}≔\mathrm{Representation}\left(\mathrm{sl2},\mathrm{V1},\mathrm{M1}\right)$
 ${\mathrm{ρ1}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rr}{0}& {1}\\ {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rr}{1}& {0}\\ {0}& {-}{1}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rr}{0}& {0}\\ {1}& {0}\end{array}\right]\right]\right]$ (2.4)

 sl2 > $\mathrm{ρ2}≔\mathrm{Representation}\left(\mathrm{sl2},\mathrm{V2},\mathrm{Adjoint}\left(\right)\right)$
 ${\mathrm{ρ2}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrr}{0}& {-}{2}& {0}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrr}{2}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {-}{2}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {-}{1}& {0}& {0}\\ {0}& {2}& {0}\end{array}\right]\right]\right]$ (2.5)

Define the direct sum representation of ${\mathrm{ρ}}_{1}$and ${\mathrm{ρ}}_{2}$

 sl2 > $\mathrm{φ1}≔\mathrm{DirectSumOfRepresentations}\left(\left[\mathrm{ρ1},\mathrm{ρ2}\right],\mathrm{W1}\right)$
 ${\mathrm{φ1}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrrr}{0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{2}& {0}\\ {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrrr}{1}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}\\ {0}& {0}& {2}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{2}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrrr}{0}& {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {2}& {0}\end{array}\right]\right]\right]$ (2.6)
 sl2 > $\mathrm{Query}\left(\mathrm{φ1},"Representation"\right)$
 ${\mathrm{true}}$ (2.7)

Define the direct sum of 3 copies of ${\mathrm{ρ}}_{1}$.

 sl2 > $\mathrm{φ2}≔\mathrm{DirectSumOfRepresentations}\left(\left[\mathrm{ρ1},\mathrm{ρ1},\mathrm{ρ1}\right],\mathrm{W2}\right)$
 ${\mathrm{φ2}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrrrr}{0}& {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrrrr}{1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {0}& {-}{1}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrrrr}{0}& {0}& {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}\end{array}\right]\right]\right]$ (2.8)
 sl2 > $\mathrm{Query}\left(\mathrm{φ2},"Representation"\right)$
 ${\mathrm{true}}$ (2.9)