DifferentialGeometry/LieAlgebras/Query/NilRepresentation - Maple Help

Query[NilRepresentation] - check if a representation of a Lie algebra is nilpotent

Calling Sequences

Query(

Parameters

rho       - a representation of a Lie algebra

Description

 • Let g be a Lie algebra, a vector space and  a representation.  This query returns true if for each , the matrix is nilpotent, that is,  for some positive integer r.

Examples

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

Example 1.

Retrieve the structure equations for a Lie algebra from the DifferentialGeometry library.

 > $L≔\mathrm{Retrieve}\left("Winternitz",1,\left[4,1\right],\mathrm{alg1}\right)$
 ${L}{:=}\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$ (2.1)

Initialize the Lie algebra and create a 4-dimensional representation space.

 > $\mathrm{DGsetup}\left(L\right):$
 alg1 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],V\right)$
 ${\mathrm{frame name: V}}$ (2.2)

 V > $\mathrm{\rho }≔\mathrm{Representation}\left(\mathrm{alg1},V,\mathrm{Adjoint}\left(\mathrm{alg1}\right)\right)$
 ${\mathrm{ρ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.3)
 alg1 > $\mathrm{Query}\left(\mathrm{\rho },"NilRepresentation"\right)$
 ${\mathrm{true}}$ (2.4)