calculate the adjoint representation of a LAVF object.

 Calling Sequence AdjointMatrix( L, output = out) AdjointMatrix( L, M, output = out) AdjointMatrix( L, M, N, output = out)

Parameters

 L, M, N - a LAVF object of finite type (see IsFiniteType for more detail) out - (optional) a string: either "matrix" or "basis"

Description

 • In the first calling sequence, AdjointMatrix(L) returns the adjoint representation matrix of L.
 • For AdjointMatrix(L) to make sense, the LAVF object L must be a Lie algebra (i.e. IsLieAlgebra(L) returns true. See IsLieAlgebra for more detail).
 • In the second calling sequence,  AdjointMatrix(L,M) returns a matrix representation of the Lie algebra L on the invariant subspace M. (i.e IsInvariant(M,L) returns true. See IsInvariant for more detail).
 • The third calling sequence is the general form of the method. AdjointMatrix(L,M,N) returns a matrix representing the action of L on M in N.
 • For AdjointMatrix(L, M, N) to make sense, L must commute with M modulo N (i.e. AreCommuting(L,M,N) returns true. See AreCommuting for more detail).
 • By specifying output = "basis", the output will be returned in terms of basis.
 • This method is associated with the LAVF object. For more detail, see Overview of the LAVF object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)\right]\right):$
 > $V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x,y\right)\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${V}{≔}{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (2)

Construct a LAVF for theEuclidean Lie algebra E(2).

 > $L≔\mathrm{LAVF}\left(V,\mathrm{E2}\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (3)
 > $\mathrm{IsLieAlgebra}\left(L\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{AdjointMatrix}\left(L\right)$
 $\left[\begin{array}{ccc}{0}& {{\mathrm{\xi }}}_{{y}}& {-}{\mathrm{\xi }}\\ {-}{{\mathrm{\xi }}}_{{y}}& {0}& {\mathrm{\eta }}\\ {0}& {0}& {0}\end{array}\right]$ (5)
 > $\mathrm{AdjointMatrix}\left(L,'\mathrm{output}'="basis"\right)$
 $\left[\left[\begin{array}{ccc}{0}& {1}& {0}\\ {-1}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{ccc}{0}& {0}& {0}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{ccc}{0}& {0}& {-1}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]$ (6)

Compatibility

 • The AdjointMatrix command was introduced in Maple 2020.