check if a list of vectors defines a Lie subalgebra - Maple Programming Help

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Query[Subalgebra] - check if a list of vectors defines a Lie subalgebra

Calling Sequences

     Query(S, "Subalgebra")

     Query(S, parm, "Subalgebra")

Parameters

     S       - a list of independent vectors in a Lie algebra 𝔤

     parm    - (optional) a set of parameters appearing in the list of vectors S.  It is assumed that the set of vectors S is well-defined when the parameters vanish.

 

Description 

Examples

Description 

• 

A list of vectors S defines a basis for a Lie subalgebra if  x, y  spanS for all x,y S. 

• 

Query(S, "Subalgebra") returns true if the set S defines a subalgebra.

• 

Query(S, parm, "Subalgebra") returns a sequence TF, Eq, Soln, SubAlgList.  Here TF is true if Maple finds parameter values for which S is a subalgebra and false otherwise; Eq is the set of equations (with the variables in parm as unknowns) which must be satisfied for S to be a subalgebra; Soln is the list of solutions to the equations Eq; and SubAlgList is the list of subalgebras obtained from the parameter values given by the different solutions in Soln.

• 

The program calculates the defining equations Eq for S to be a subalgebra as follows.  First the list of vectors S is evaluated with the parameters set to zero to obtain a set of vectors S0.  The program ComplementaryBasis is then used to calculate a complement C to S0  The list of vectors B= S, C then gives a basis for the entire Lie algebra 𝔤.  For each x,y S  the bracket x,y is calculated and expressed as a linear combination of the vectors in the basis B. The components of [x,y[ in C must all vanish for S to be a Lie subalgebra.

• 

We remark that the equations Eq, which the parameters must satisfy in order for S to be a subalgebra, will in general be a system of coupled quadratic equations.  Maple may not be able to solve these equations or may not solve them in full generality.

• 

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

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First initialize a Lie algebra.

L_DGLieAlgebra,Alg,4,1,4,1,0,2,3,1,1,2,4,2,1,3,4,3,1

L:=e2,e3=e1,e2,e4=e2,e3,e4=e3

(2.1)

DGsetupL:

 

The vectors S1 = e2, e3, e4 do not determine a subalgebra while the vectors S2 = e1, e3, e4 do.

Alg > 

S1e2,e3,e4:

Alg > 

QueryS1,Subalgebra

false

(2.2)
Alg > 

S2e1,e3,e4:

Alg > 

QueryS2,Subalgebra

true

(2.3)

 

We find the values of the parameters  a1, a2 for which S3 = e2, e1+a1e3 +a2e4 determines a Lie subalgebra.

Alg > 

S3evalDGe2,e1+a1e3+a2e4:

Alg > 

QueryS3,a1,a2,Subalgebra

true,0,a12,a1a2,a1=0,a2=a2,e2,a2e4+e1

(2.4)

 

There are no values of the parameters  a1, a2  for which S4 = e2, e3+a1e2 +a2e4 determines a Lie subalgebra.

Alg > 

S4evalDGe2,e3+a1e2+a2e4:

Alg > 

QueryS4,a1,a2,Subalgebra

false,0,1

(2.5)

See Also

DifferentialGeometry

LieAlgebras

Query

 


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