RestrictedRepresentation - Maple Help

LieAlgebras[RestrictedRepresentation] - find the restriction of a representation of a subalgebra

Calling Sequences

RestrictedRepresentation(${\mathbf{\rho }}$, alg,W)

Parameters

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

alg       - a Maple name or string, giving the frame name of an initialized algebra, corresponding to a subalgebra of $\mathrm{𝔤}$

H         - (optional) a list of vectors in defining a basis for a subalgebra of $\mathrm{𝔤}$

Description

 • If is a representation and $\mathrm{𝔥}$ is a subalgebra of $\mathrm{𝔤}$ , then the restriction of $\mathrm{ρ}$ to $\mathrm{𝔥}$ is the representation defined by $\mathrm{φ}\left(x\right)\left(Y\right)$ =$\mathrm{\rho }\left(x\right)\left(Y\right)$, where  and .
 • The command RestrictedRepresentation(rho, alg, H) returns the restriction of the representation to the subalgebra defined by the vectors in the list $H.$The subalgebra defined by the vectors must be initialized as a Lie algebra in its own right with the name alg.
 • If the basis for $\mathrm{𝔤}$ is adapted to the subalgebra defined by in the sense that then the list need not be specified in the calling sequence for RestrictedRepresentation.

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, define a subalgebra, and calculate the restricted representation of $\mathrm{ρ}$ to the subalgebra..

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

Initialize the Lie algebra Alg1.

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

Initialize the representation space V.

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

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

Define a 2-dimensional abelian subalgebra of Alg1 using the command LieAlgebraData.

 Alg1 > $\mathrm{H1}≔\left[\mathrm{e1},\mathrm{e2}\right]$
 ${\mathrm{H1}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]$ (2.3)
 Alg1 > $\mathrm{L1}≔\mathrm{LieAlgebraData}\left(\mathrm{H1},\mathrm{Alg1}\right)$
 ${\mathrm{L1}}{:=}\left[{}\right]$ (2.4)
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L1},\left['P'\right],\left['p'\right]\right)$
 ${\mathrm{Lie algebra: Alg1}}$ (2.5)
 Alg1 > $\mathrm{ρ1}≔\mathrm{RestrictedRepresentation}\left(\mathrm{\rho },\mathrm{Alg1}\right)$
 ${\mathrm{ρ1}}{:=}\left[\left[{\mathrm{P1}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {2}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{P2}}{,}\left[\begin{array}{rrrr}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.6)
 Alg1 > $\mathrm{Query}\left(\mathrm{ρ1},"Representation"\right)$
 ${\mathrm{true}}$ (2.7)

Example 2.

Define a 2 dimensional solvable subalgebra of Alg1, one that is not adapted to the basis .

 Alg2 > $\mathrm{H2}≔\left[\mathrm{e4}+\mathrm{e2},\mathrm{e2}\right]$
 ${\mathrm{H2}}{:=}\left[{\mathrm{e4}}{+}{\mathrm{e2}}{,}{\mathrm{e2}}\right]$ (2.8)
 Alg1 > $\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{H2},\mathrm{Alg2}\right)$
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e2}}\right]$ (2.9)
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2},\left['Q'\right],\left['q'\right]\right)$
 ${\mathrm{Lie algebra: Alg2}}$ (2.10)
 Alg1 > $\mathrm{ρ2}≔\mathrm{RestrictedRepresentation}\left(\mathrm{\rho },\mathrm{Alg2},\mathrm{H2}\right)$
 ${\mathrm{ρ2}}{:=}\left[\left[{\mathrm{Q1}}{,}\left[\begin{array}{rrrr}{-}{2}& {0}& {1}& {0}\\ {0}& {-}{1}& {-}{1}& {1}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{Q2}}{,}\left[\begin{array}{rrrr}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.11)
 Alg1 > $\mathrm{Query}\left(\mathrm{ρ2},"Representation"\right)$
 ${\mathrm{true}}$ (2.12)