CommutatorTable

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\vspace*{5mm}
\centerline{\fbox{\parbox{9.6cm}{\Large\bf UserFunction CommutatorTable} }}
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\vspace{5mm}\noindent{\bf CommutatorTable}$(u:LIEVEC(C,v,O))$.
The single argument $u$ is of type $LIEVEC$ which abbreviates {\em Lie-Algebra of Vector Fields}. The meaning of its parameters is as follows.

$C:$ Coefficient type.

$v:$ Vector field variables.

$O=LEX|GRLEX:$ Term order.

\vspace*{1mm}\noindent{\bf Specification.} Returns commutator table for elements on $u$.


\vspace*{1mm}\noindent{\bf Examples.} The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is returned in a separate window. Symmetry generators of a first-order ode are computed first.

\4{\tt ls:=\{x*D(x)-2*y*D(y),(x**3*y-10*x)*D(x)}

\4\hspace*{50mm}{\tt +(x**4*y**3-x**2*y**2)*D(y)\};}


\4{\tt CommutatorTable(ls|LIEVEC(RATF Q,\{x,y\},LEX)|);}

\vspace{1mm}\hspace*{10mm}
\fbox{\parbox{13cm}{
The Lie vector fields:

\begin{displaymath}\begin{array}{l}U_{1}=x\partial_x-2y\partial_y\\[3mm]U_{2}=(x^{3}y-10x)\partial_x
+(x^{4}y^{3}-x^{2}y^{2})\partial_y\\[3mm]\end{array}\end{displaymath}
Commutator Table:
\begin{displaymath}
\begin{array}{c|cc}
 & U_{1} & U_{2}\\[3mm]
\hline
U_{1} &  0 &0\\[3mm]
U_{2} &  & 0 \\
\end{array}
\end{displaymath}  }}

\vspace*{5mm}
Symmetry generators of a second-order ode.


\4{\tt ls:=\{D(x),D(y),x*D(x)+2/3*y*D(y)\};};

\4{\tt CommutatorTable(ls|LIEVEC(RATF Q,\{x,y\},LEX)|);}

\vspace{1mm}\hspace*{10mm}
\fbox{\parbox{13cm}{
The Lie vector fields

\begin{displaymath}\begin{array}{l}
U_{1}=\partial_x\\[3mm]
U_{2}=\partial_y\\[3mm]U_{3}=x\partial_x+ \Frac{2}{3}{y}\partial_y
\end{array}\end{displaymath} }}

\vspace{1mm}\hspace*{10mm}
\fbox{\parbox{13cm}{
Commutator Table:
\begin{displaymath}
\begin{array}{c|ccc}
 & U_{1} & U_{2} & U_{3}\\[3mm]
\hline
U_{1} &  0 &0 & U_{1}\\[3mm]
U_{2} &  & 0 &\Frac{2}{3}U_{2}\\[3mm]
U_{3} &  &  & 0 \\
\end{array}
\end{displaymath} }}

\vspace*{5mm}
Symmetry generators of Burgers' equation.

\4{\tt gs:=\{D(x),D(t),D(u)+t*D(x),t**2*D(t)+(x-u*t)*D(u)+x*t*D(x)}

\hspace*{30mm}{\tt t/2*D(t)-t/4*u*D(u)+x/2*D(x)\};}

\4{\tt CommutatorTable(gs|LIEVEC(RATF Q,\{u,t,x\},LEX)|);}

\vspace{1mm}\hspace*{10mm}
\fbox{\parbox{13cm}{
The Lie vector fields

\begin{displaymath}
\begin{array}{l}
U_{1}=\partial_x\\[3mm]
U_{2}=\partial_t\\[3mm]
U_{3}=\partial_u+t
\partial_x\\[3mm]
U_{4}=(-tu+x)\partial_u+t^{2}\partial_t+tx\partial_x\\[3mm]
U_{5}=-u\partial_u+2t\partial_t+x\partial_x\\[3mm]
\end{array}\end{displaymath}
The Commutator Table:
\begin{displaymath}
\begin{array}{c|ccccc}
 & U_{1} & U_{2} & U_{3} & U_{4} & U_{5}\\[3mm]
\hline
U_{1} &  0 &0 & 0 & U_{3} & U_{1}\\[3mm]
U_{2} &  & 0 &U_{1} & U_{5} & 2U_{2}\\[3mm]
U_{3} &  & & 0 &0 & -U_{3}\\[3mm]
U_{4} &  & & & 0 &-2U_{4}\\[3mm]
U_{5} &  &  &  &  & 0 \\
\end{array}
\end{displaymath}  }}

 


\vspace*{1mm}\noindent{\bf Related Functions:} Symmetries.

\vspace*{5mm}

{\bf References}

P. Olver, Application of Lie Groups to Differential Equations, Springer, 1986.

G. W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, 1989.


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