QuotientRepresentation - Maple Help

LieAlgebras[QuotientRepresentation] - find the induced representation defined on the quotient space of the representation space by an invariant subspace

Calling Sequences

QuotientRepresentation(${\mathbf{ρ}}$, S, C, 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 invariant subspace of $V$

C       - a list of vectors in $V$ defining a complementary subspace to $S$

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

Description

 • If   is a representation and is a subspace of $V$, then $S$ is $\mathrm{ρ}$ invariant if for all  and Y . For any let  denote the coset of in the quotient space $V/S$. The induced representation is defined

by

 • The command QuotientRepresentation(rho, S, C, W) returns the representation $\stackrel{‾}{\mathrm{\rho }}$. The coset representatives of the vectors in C in the quotient space give the basis used to calculate the matrices for the linear transformation $\stackrel{‾}{\mathrm{\rho }}$.

Examples

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

Example 1.

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

Initialize the Lie algebra Alg1.

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

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.

 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  and use the Query command to check that it is invariant.

 Alg1 > $S≔\left[\mathrm{D_x1}\right]$
 ${S}{:=}\left[{\mathrm{D_x1}}\right]$ (2.3)
 V > $\mathrm{Query}\left(\mathrm{\rho },S,"InvariantSubspace"\right)$
 ${\mathrm{true}}$ (2.4)

Pick a complement $C=$span$[{\mathrm{D}}_{{x}^{2}},$ ${\mathrm{D}}_{{x}^{4}}$].This complement need not be invariant.

 V > $\mathrm{Query}\left(\mathrm{\rho },\left[\mathrm{D_x2},\mathrm{D_x3},\mathrm{D_x4}\right],"InvariantSubspace"\right)$
 ${\mathrm{false}}$ (2.5)

Define a vector space for the induced representation of on $V/S$.

 V > $\mathrm{DGsetup}\left(\left[\mathrm{y1},\mathrm{y2},\mathrm{y3}\right],W\right)$
 ${\mathrm{frame name: W}}$ (2.6)

Compute the quotient representation. Note that in this example the matrices are just the lower blocks of the matrices in the original representation.

 W > $\mathrm{\phi }≔\mathrm{QuotientRepresentation}\left(\mathrm{\rho },S,\left[\mathrm{D_x2},\mathrm{D_x3},\mathrm{D_x4}\right],W\right)$
 ${\mathrm{φ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrr}{-}{1}& {-}{1}& {0}\\ {0}& {-}{1}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.7)
 Alg1 > $\mathrm{Query}\left(\mathrm{\phi },"Representation"\right)$
 ${\mathrm{true}}$ (2.8)