check if a subalgebra, subspace pair defines a reductive pair in a Lie algebra - Maple Help

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Query[ReductivePair] - check if a subalgebra, subspace pair defines a reductive pair in a Lie algebra

Calling Sequences

     Query(S, M, "ReductivePair")

     Query(S, M, parm, "ReductivePair")

Parameters

     S       - a list of independent vectors which defines a subalgebra in a Lie algebra g

     M       - a list of independent vectors which defines a complementary subspace to S in g

     parm    - (optional) a set of parameters appearing in the list of vectors S

 

Description 

Examples

Description 

• 

Let 𝔤 be a Lie algebra, S  𝔤 a subalgebra and M  𝔤 a subspace. Then the subalgebra, subspace pair S, M  is called a reductive pair if [i] 𝔤 = S  M (vector space direct sum) and [ii] x,y  M  for all x  S and y in M. The subspace M is called a reductive complement for the subalgebra S.

• 

Query(S, M, "ReductivePair") returns true if the subspace M defines a reductive complement to the subalgebra S.

• 

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

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:

 

We see that spane3, e4 is not a reductive complement for spane1, e2 but spane1, e2 is a reductive complement for spane3, e4.

Alg > 

S1e1,e2:M1e3,e4:

Alg > 

QueryS1,M1,ReductivePair

false

(2.2)
Alg > 

S2e3,e4:M2e1,e2:

Alg > 

QueryS2,M2,ReductivePair

true

(2.3)

 

Now we look for the most general reductive complement M3 for the subalgebra S3= span e3, e4.

Alg > 

S3e3,e4:M3evalDGe1+a1e3+a2e4,e2+a3e3+a4e4:

Alg > 

TF,EQ,SOL,redPairQueryS3,M3,a1,a2,a3,a4,ReductivePair

TF,EQ,SOL,redPair:=true,0,a2,a1,a2,2a3,a4,a1+a4,a1=0,a2=0,a3=0,a4=0,e3,e4,e1,e2

(2.4)

 

The only possibility is spane1, e2.

Alg > 

redPair1

e3,e4,e1,e2

(2.5)

 

Note that the ComplementaryBasis command can be used to generate the most general complementary subspace. This helps to calculate reductive complements for subalgebras.

Alg > 

S4e4:

Alg > 

M4ComplementaryBasisS4,e1,e2,e3,e4,a

M4:=e1+a1e4,e2+a2e4,e3+a3e4,a1,a2,a3

(2.6)
Alg > 

QueryS4,M4,ReductivePair

true,0,a3,a2,a1=a1,a2=0,a3=0,e4,e1+a1e4,e2,e3

(2.7)

See Also

DifferentialGeometry

LieAlgebras

ComplementaryBasis

Query

 


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