Query[Derivation] - check if a matrix defines a derivation on a Lie algebra
Query(Alg, A, "Derivation")
Query(Alg, A, parm, "Derivation")
Alg - the name of an initialized Lie algebra 𝔤
A - an n x n Matrix, where n is the dimension of the Lie algebra 𝔤; a transformation mapping 𝔤 to 𝔤
parm - a set of parameters appearing in the Matrix A or in the Lie algebra 𝔤
A matrix A is a derivation for a Lie algebra 𝔤 if the associated linear transformation mapping LA: 𝔤 → 𝔤 satisfies LA(x, y) = LAx, y +x, LAy for all x, y ∈ 𝔤.
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(...).
First initialize a Lie algebra and display the Lie bracket multiplication table.
L1 ≔ _DG⁡LieAlgebra,Alg1,4,1,4,1,1,2,3,1,1,2,4,2,1,3,4,1,0
L1 ≔ e1,e4=e1,e2,e3=e1,e2,e4=e2
Because of the Jacobi identities, A = adx is always a derivation for any x in the Lie algebra. For example:
A ≔ Adjoint⁡e1−2⁢e3+e4
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 a1, a2, a3.
A ≔ Matrix⁡0,0,a1,a2,0,0,0,a3,0,0,0,0,0,0,0,0
T ≔ Transformation⁡Alg1,Alg1,A
T ≔ e1,0⁢e1,e2,0⁢e1,e3,a1⁢e1,e4,a2⁢e1+a3⁢e2
TF,Eq,Soln,Der ≔ Query⁡Alg1,T,a1,a2,a3,Derivation
TF,Eq,Soln,Der ≔ true,0,−a3+a1,a1=a3,a2=a2,a3=a3,e1,0⁢e1,e2,0⁢e1,e3,a3⁢e1,e4,a2⁢e1+a3⁢e2
We conclude that there is a 2-parameter family of derivations of the type A and these are given by a1=a3. We can confirm this result with another call to Query.
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