find the induced representation on an invariant subspace of the representation space - Maple Programming Help

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LieAlgebras[SubRepresentation] - find the induced representation on an invariant subspace of the representation space

Calling Sequences

SubRepresentation(${\mathbf{ρ}}$$,$S, W)

Parameters

$\mathrm{ρ}$       - a representation of a Lie algebra $\mathrm{𝔤}$ on a vector space $V$

S       - a list of vectors in whose span defines a $\mathrm{ρ}$-invariant subspace of $V$

W       - a Maple name or string, giving the frame name for the representation space for the subrepresentation

Description

 • If  is a representation of a Lie algebra $𝔤$ on a vector space $V$, then is a $\mathrm{ρ}$-invariant subspace of if for all  and .
 • The command SubRepresentation(${\mathbf{ρ}}$$,$S,W) returns the representation $\mathrm{φ}$ of $\mathrm{𝔤}$ on the vector space defined by $\mathrm{φ}\left(x\right)\left(y\right)$ =$\mathrm{\rho }\left(x\right)\left(Y\right)$, where  and .

Examples

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

Example 1.

We shall define a 4-dimensional representation of a 4-dimensional Lie algebra taken from the DifferentialGeometry Library, find an invariant subspace $S$ of $\mathrm{ρ}$, and calculate the subrepresentation of $\mathrm{ρ}$ on $S$.

 > $L≔\mathrm{Retrieve}\left("Winternitz",1,\left[4,7\right],\mathrm{Alg1}\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}\right]$ (2.1)

Initialize the Lie algebra Alg1.

 V > $\mathrm{DGsetup}\left(L\right):$

Initialize the representation space $V$.

 Alg1 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],V\right):$

Define the matrices which specify a representation of Alg1 on $V$.

 V > $M≔\left[\mathrm{Matrix}\left(\left[\left[0,0,0,2\right],\left[0,0,0,0\right],\left[0,0,0,0\right],\left[0,0,0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0,1,0\right],\left[0,0,0,1\right],\left[0,0,0,0\right],\left[0,0,0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,-1,0,0\right],\left[0,0,0,1\right],\left[0,0,0,1\right],\left[0,0,0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[-2,0,0,0\right],\left[0,-1,-1,0\right],\left[0,0,-1,0\right],\left[0,0,0,0\right]\right]\right)\right]:$

Define the representation with the Representation command.

 V > $\mathrm{\rho }≔\mathrm{Representation}\left(\mathrm{Alg1},V,M\right)$
 ${\mathrm{ρ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {2}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrr}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrrr}{-}{2}& {0}& {0}& {0}\\ {0}& {-}{1}& {-}{1}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.2)

Define a subspace of $V$.

 Alg1 > $S≔\left[\mathrm{D_x1},\mathrm{D_x2},\mathrm{D_x3}\right]$
 ${S}{:=}\left[{\mathrm{D_x1}}{,}{\mathrm{D_x2}}{,}{\mathrm{D_x3}}\right]$ (2.3)

We can use the Query command to check that S is a $\mathrm{ρ}$-invariant subspace.

 V > $\mathrm{Query}\left(\mathrm{\rho },S,"InvariantSubspace"\right)$
 ${\mathrm{true}}$ (2.4)

Define a frame for the induced representation of on $S.$

 V > $\mathrm{DGsetup}\left(\left[\mathrm{y1},\mathrm{y2},\mathrm{y3}\right],W\right):$
 W > $\mathrm{\phi }≔\mathrm{SubRepresentation}\left(\mathrm{\rho },S,W\right)$
 ${\mathrm{φ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrr}{0}& {-}{1}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrr}{-}{2}& {0}& {0}\\ {0}& {-}{1}& {-}{1}\\ {0}& {0}& {-}{1}\end{array}\right]\right]\right]$ (2.5)
 Alg1 > $\mathrm{Query}\left(\mathrm{\phi },"Representation"\right)$
 ${\mathrm{true}}$ (2.6)