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.

$\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right)\:$

$V≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x\,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x\,y\right)\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$V≔\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{E2}≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$L≔\mathrm{LAVF}\left(V\,\mathrm{E2}\right)$

$L≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{IsLieAlgebra}\left(L\right)$

$\mathrm{true}$

$\mathrm{Cijk}≔\mathrm{StructureConstants}\left(L\right)$

$\mathrm{Cijk}≔\begin{array}{c}\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& \mathrm{-1}\\ 0& 1& 0\end{array}\right]\\ \hfill \text{slice of 3 × 3 × 3 Array}\end{array}$

$\mathrm{DisplayStructure}\left(\mathrm{Cijk}\,'X'\right)$

$\left[\begin{array}{ccc}0& 0& X_2\\ 0& 0& -X_1\\ -X_2& X_1& 0\end{array}\right]$

$\mathrm{Sinf}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)=0\,\mathrm{\eta }\left(x\,y\right)=y\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)\right]\,\mathrm{indep}=\left[x\,y\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{Sinf}≔\left[{\mathrm{\xi }}_y=0\,\mathrm{\eta }=y\left({\mathrm{\xi }}_x\right)\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{Linf}≔\mathrm{LAVF}\left(V\,\mathrm{Sinf}\right)$

$\mathrm{Linf}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_x=\frac{\mathrm{\eta }}{y}\,{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=\frac{\mathrm{\eta }}{y}\right]\right\}$

$\mathrm{SolutionDimension}\left(\mathrm{Linf}\right)$

$\mathrm{\infty }$

$\mathrm{IsLieAlgebra}\left(\mathrm{Linf}\right)$

$\mathrm{true}$

$C,A≔\mathrm{StructureCoefficients}\left(\mathrm{Linf}\right)$

$C,A≔\begin{array}{c}\left[\begin{array}{cc}0& \frac{1}{y}\\ -\frac{1}{y}& 0\end{array}\right]\\ \hfill \text{slice of 2 × 2 × 2 Array}\end{array},\begin{array}{c}\left[\begin{array}{c}0\\ 0\end{array}\right]\\ \hfill \text{slice of 2 × 1 × 2 Array}\end{array}$

$\mathrm{DisplayStructure}\left(C\,A\,'X'\,'Z'\,'\mathrm{format}'=''commutatorList''\right)$

$\left[\left[X_1\,X_2\right]=\frac{X_1}{y}\,\left[X_1\,Z_1\right]=-X_2\right]$

$\mathrm{DisplayStructure}\left(C\,A\,'\mathrm{\omega }'\,'\mathrm{pi}'\,'\mathrm{format}'=''cartan''\right)$

$\left[d{\mathrm{\omega }}_1=-\frac{{\mathrm{\omega }}_1\wedge {\mathrm{\omega }}_2}{y}\,d{\mathrm{\omega }}_2=-{\mathrm{\omega }}_1\wedge {\mathrm{\π}}_1\right]$

The StructureConstants and StructureCoefficients commands were introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.