 Derivation - Maple Help

Query[Derivation] - check if a matrix defines a derivation on a Lie algebra

Calling Sequences

Query(Alg, A, "Derivation")

Query(Alg, A, parm, "Derivation")

Parameters

Alg     - the name of an initialized Lie algebra $\mathrm{𝔤}$

A       - an n x n Matrix, where n is the dimension of the Lie algebra $\mathrm{𝔤}$; a transformation mapping $\mathrm{𝔤}$ to $\mathrm{𝔤}$

parm    - a set of parameters appearing in the Matrix A or in the Lie algebra $\mathrm{𝔤}$ Description

 • A matrix $A$ is a derivation for a Lie algebra  if the associated linear transformation mapping   satisfies  ${L}_{A}$( for all .
 • Query(Alg, A, "Derivation") returns true if the matrix A or transformation defines a derivation for the Lie algebra g and false otherwise.
 • Query(Alg, A, parm, "Derivation") returns a 4-tuple TF, Eq, Soln, B.  Here TF is true if Maple finds a set of values for the parameters for which the Matrix or transformation A is a derivation; Eq is the defining set of equations for the parameters parm in order that the matrix A be a derivation; Soln is a list of solutions to the equations Eq; and B is the list of Matrices obtained by evaluating A on the solutions in the list Soln.
 • The command Query is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...). Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

First initialize a Lie algebra and display the Lie bracket multiplication table.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[4\right]\right],\left[\left[\left[1,4,1\right],1\right],\left[\left[2,3,1\right],1\right],\left[\left[2,4,2\right],1\right],\left[\left[3,4,1\right],0\right]\right]\right]\right)$
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$
 Alg1 > $\mathrm{MultiplicationTable}\left("LieBracket"\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$ (2.2)

Because of the Jacobi identities,  is always a derivation for any in the Lie algebra.  For example:

 Alg1 > $A≔\mathrm{Adjoint}\left(\mathrm{e1}-2\mathrm{e3}+\mathrm{e4}\right)$
 ${A}{:=}\left[\begin{array}{rrrr}{-}{1}& {2}& {0}& {1}\\ {0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]$ (2.3)
 Alg1 > $\mathrm{Query}\left(\mathrm{Alg1},A,"Derivation"\right)$
 ${\mathrm{true}}$ (2.4)

Example 2.

To illustrate the second use of Query with keyword "derivation", we find all the derivations of the above Lie algebra of the special form given by the following matrix $A$, which depends upon 3 parameters

 Alg1 > $A≔\mathrm{Matrix}\left(\left[\left[0,0,\mathrm{a1},\mathrm{a2}\right],\left[0,0,0,\mathrm{a3}\right],\left[0,0,0,0\right],\left[0,0,0,0\right]\right]\right)$
 ${A}{:=}\left[\begin{array}{cccc}{0}& {0}& {\mathrm{a1}}& {\mathrm{a2}}\\ {0}& {0}& {0}& {\mathrm{a3}}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]$ (2.5)
 Alg1 > $T≔\mathrm{Transformation}\left(\mathrm{Alg1},\mathrm{Alg1},A\right)$
 ${T}{≔}\left[\left[{\mathrm{e1}}{,}{0}{}{\mathrm{e1}}\right]{,}\left[{\mathrm{e2}}{,}{0}{}{\mathrm{e1}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{a1}}{}{\mathrm{e1}}\right]{,}\left[{\mathrm{e4}}{,}{\mathrm{a2}}{}{\mathrm{e1}}{+}{\mathrm{a3}}{}{\mathrm{e2}}\right]\right]$ (2.6)
 Alg1 > $\mathrm{TF},\mathrm{Eq},\mathrm{Soln},\mathrm{Der}≔\mathrm{Query}\left(\mathrm{Alg1},T,\left\{\mathrm{a1},\mathrm{a2},\mathrm{a3}\right\},"Derivation"\right)$
 ${\mathrm{TF}}{,}{\mathrm{Eq}}{,}{\mathrm{Soln}}{,}{\mathrm{Der}}{≔}{\mathrm{true}}{,}\left\{{0}{,}{-}{\mathrm{a3}}{+}{\mathrm{a1}}\right\}{,}\left[\left\{{\mathrm{a1}}{=}{\mathrm{a3}}{,}{\mathrm{a2}}{=}{\mathrm{a2}}{,}{\mathrm{a3}}{=}{\mathrm{a3}}\right\}\right]{,}\left[\left[\left[{\mathrm{e1}}{,}{0}{}{\mathrm{e1}}\right]{,}\left[{\mathrm{e2}}{,}{0}{}{\mathrm{e1}}\right]{,}\left[{\mathrm{e3}}{,}{\mathrm{a3}}{}{\mathrm{e1}}\right]{,}\left[{\mathrm{e4}}{,}{\mathrm{a2}}{}{\mathrm{e1}}{+}{\mathrm{a3}}{}{\mathrm{e2}}\right]\right]\right]$ (2.7)

We conclude that there is a 2-parameter family of derivations of the type $A$ and these are given by ${a}_{1}={a}_{3}$.  We can confirm this result with another call to Query.

 Alg1 > $\mathrm{Query}\left(\mathrm{Alg1},{\mathrm{Der}}_{1},"Derivation"\right)$
 ${\mathrm{true}}$ (2.8)