StructureConstants - Maple Help

StructureConstants

calculate the structure constants of a LAVF object.

StructureCoefficients

calculate the structure coefficients for a LAVF of infinite type.

 Calling Sequence StructureConstants( obj) StructureCoefficients( obj)

Parameters

 obj - a LAVF object that is a Lie algebra i.e. IsLieAlgebra(obj) returns true, see IsLieAlgebra.

Description

 • Let L be a LAVF object which is a Lie algebra and is of finite type (see IsFiniteType). Then StructureConstants (and StructureCoefficients) methods return the structure constants of L (namely cijk), as a 3-dim array.
 • If L is not of finite type (i.e. IsFiniteType(L) returns false), then StructureCoefficients(L) returns a sequence of two arrays of structure coefficients cijk, akirho, (k up, irho down)
 • The output structure constants Cijk can then be displayed via the DisplayStructure command of the LieAlgebrasOfVectorFields package. See Overview of the LieAlgebrasOfVectorFields package for more detail.
 • These methods are 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 Euclidean group 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)
 > $\mathrm{Cijk}≔\mathrm{StructureConstants}\left(L\right)$
 ${\mathrm{Cijk}}{≔}\begin{array}{c}\left[\begin{array}{ccc}{0}& {0}& {0}\\ {0}& {0}& {-1}\\ {0}& {1}& {0}\end{array}\right]\\ \hfill {\text{slice of 3 × 3 × 3 Array}}\end{array}$ (5)
 > $\mathrm{DisplayStructure}\left(\mathrm{Cijk},'X'\right)$
 $\left[\begin{array}{ccc}{0}& {0}& {{X}}_{{2}}\\ {0}& {0}& {-}{{X}}_{{1}}\\ {-}{{X}}_{{2}}& {{X}}_{{1}}& {0}\end{array}\right]$ (6)

Now consider a LAVF of infinite type.

 > $\mathrm{Sinf}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)=0,\mathrm{\eta }\left(x,y\right)=y\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${\mathrm{Sinf}}{≔}\left[{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{\mathrm{\eta }}{=}{y}{}\left({{\mathrm{\xi }}}_{{x}}\right)\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (7)
 > $\mathrm{Linf}≔\mathrm{LAVF}\left(V,\mathrm{Sinf}\right)$
 ${\mathrm{Linf}}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{x}}{=}\frac{{\mathrm{\eta }}}{{y}}{,}{{\mathrm{\xi }}}_{{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{y}}{=}\frac{{\mathrm{\eta }}}{{y}}\right]\right\}$ (8)
 > $\mathrm{SolutionDimension}\left(\mathrm{Linf}\right)$
 ${\mathrm{\infty }}$ (9)
 > $\mathrm{IsLieAlgebra}\left(\mathrm{Linf}\right)$
 ${\mathrm{true}}$ (10)
 > $C,A≔\mathrm{StructureCoefficients}\left(\mathrm{Linf}\right)$
 ${C}{,}{A}{≔}\begin{array}{c}\left[\begin{array}{cc}{0}& \frac{{1}}{{y}}\\ {-}\frac{{1}}{{y}}& {0}\end{array}\right]\\ \hfill {\text{slice of 2 × 2 × 2 Array}}\end{array}{,}\begin{array}{c}\left[\begin{array}{c}{0}\\ {0}\end{array}\right]\\ \hfill {\text{slice of 2 × 1 × 2 Array}}\end{array}$ (11)
 > $\mathrm{DisplayStructure}\left(C,A,'X','Z','\mathrm{format}'="commutatorList"\right)$
 $\left[\left[{{X}}_{{1}}{,}{{X}}_{{2}}\right]{=}\frac{{{X}}_{{1}}}{{y}}{,}\left[{{X}}_{{1}}{,}{{Z}}_{{1}}\right]{=}{-}{{X}}_{{2}}\right]$ (12)

More usually, the structure of an infinite Lie pseudogroup is displayed using Cartan structure equations, which are expressed in terms of 1-forms:

 > $\mathrm{DisplayStructure}\left(C,A,'\mathrm{\omega }','\mathrm{pi}','\mathrm{format}'="cartan"\right)$
 $\left[{d}{}{{\mathrm{\omega }}}_{{1}}{=}{-}\frac{{{\mathrm{\omega }}}_{{1}}{\wedge }{{\mathrm{\omega }}}_{{2}}}{{y}}{,}{d}{}{{\mathrm{\omega }}}_{{2}}{=}{-}{{\mathrm{\omega }}}_{{1}}{\wedge }{{\mathrm{π}}}_{{1}}\right]$ (13)

Compatibility

 • The StructureConstants and StructureCoefficients commands were introduced in Maple 2020.