DisplayStructure - Maple Help

LieAlgebrasOfVectorFields

 DisplayStructure
 display the structure constants of a LAVF object

 Calling Sequence DisplayStructure( L, u, order = n, format = x ) DisplayStructure( C, u, format = x )

Parameters

 L - a LAVF object C - an array with three indices representing the structure constants of a LAVF (see method StructureConstants) u - a name or a list of names n - (optional) a positive integer or a string "involution" x - (optional) a string: either "commutatorTable", "commutatorList", or "cartan"

Description

 • The command DisplayStructure(...) displays the structure constants of a LAVF object L as with respect to L's current initial data basis, formatted as a commutator table with the second input argument u as the printed name(s).
 • The second calling sequence gives alternative way to display the structure constants from the input argument C directly, without needed to know which LAVF object is C from. The structure constants of a LAVF object L can be got by calling StructureConstants(L).
 • If the second input argument u is a name,then the basis of solution vector fields are printed as ${X}_{1},{X}_{2},\dots ,{X}_{r}$ where $r$ is the dimension of the Lie algebra. If $u$ is provided as list of names $\left[A,B,C,\dots \right]$ then these will be taken as names of basis elements. The number of entries in the list must be equivalent to $r$.
 • By specifying the option format = "commutatorList" or "cartan", the structure constants cijk can be displayed as commutator lists or cartan format respectively,
 • This command is part of the LieAlgebrasOfVectorFields package. For more detail, see Overview of the LieAlgebrasOfVectorFields package.
 • This command can be used in the form DisplayStructure(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used in the form :-LieAlgebrasOfVectorFields:-DisplayStructure(...).

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)
 > $L≔\mathrm{LAVF}\left(V,\mathrm{E2}\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}\right]\right\}$ (3)
 > $\mathrm{DisplayStructure}\left(L,'X'\right)$
 $\left[\begin{array}{ccc}{0}& {0}& {{X}}_{{2}}\\ {0}& {0}& {-}{{X}}_{{1}}\\ {-}{{X}}_{{2}}& {{X}}_{{1}}& {0}\end{array}\right]$ (4)

Or the above step is equivalent to the following two steps.

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

Alternative way to display cijk as commutator lists.

 > $\mathrm{DisplayStructure}\left(\mathrm{Cijk},'X',\mathrm{format}="commutatorList"\right)$
 $\left[\left[{{X}}_{{1}}{,}{{X}}_{{3}}\right]{=}{{X}}_{{2}}{,}\left[{{X}}_{{2}}{,}{{X}}_{{3}}\right]{=}{-}{{X}}_{{1}}\right]$ (7)

Compatibility

 • The LieAlgebrasOfVectorFields[DisplayStructure] command was introduced in Maple 2020.