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Query[NaturallyReductivePair] - check if a subalgebra, subspace pair is naturally reductive with respect to an inner product on the subspace

Calling Sequences

     Query(S, M, B, "NaturallyReductivePair")

     Query(S, M, B, parm, "NaturallyReductivePair")

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 reductive complement to S in g

     B       - a symmetric m x m matrix, which defines an inner product on M with respect to the given basis

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

 

Description 

Examples

Description 

• 

 Let 𝔤 be a Lie algebra, S  𝔤 a subalgebra, and M 𝔤 a subspace. Let B be a non-degenerate inner product on M.  Then the subalgebra, subspace pair S, M is called naturally reductive with respect to the inner product B if [i] the subspace M defines a reductive complement to the subalgebra S, and [ii] the inner product B is 𝔤  invariant, that is, Bx,yM ,z + By, x,zM = 0 for all x 𝔤 and y,z M.  Here x,yM denotes the Mcomponent of x,y with respect to the decomposition 𝔤= S M .

• 

Query(S, M, B, "NaturallyReductivePair") returns true if S, M is naturally reductive with respect to the inner product B, and false otherwise.

• 

Query(S, M, B, parm, "NaturallyReductivePair") returns a sequence TF, Eq, Soln, NatRedPair.  Here TF is true if Maple finds parameter values for which the pair S, M is naturally reductive and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for S, M to be naturally reductive; Soln is the list of solutions to the equations Eq; and NatRedPair is the list of naturally reductive subspaces and inner products 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.

L1_DGLieAlgebra,Alg1,3,1,2,1,1,1,3,2,2,2,3,3,1:

DGsetupL1:

 

Define a subspace S1, a complement M1, and an inner product B1 on M1.

Alg1 > 

S1e2:M1e1,e3:B1Matrix0,1,1,0

B1:=0110

(2.1)

 

Check that S1, M1  is naturally reductive with respect to B1.

Alg1 > 

QueryS1,M1,B1,NaturallyReductivePair

true

(2.2)

 

Naturally reductive means that [i] the symmetric tensor g defined by B is invariant with respect to the vectors in S1 and [ii] the Lie derivative of B with respect to the vectors in M1 vanishes on pairs of vectors from M1. Thus, for the above example we have:

Alg1 > 

gevalDGθ1 &t θ3+θ3 &t θ1

g:=θ1θ3+θ3θ1

(2.3)
Alg1 > 

LieDerivativee2,g

0θ1θ1

(2.4)
Alg1 > 

LieDerivativee1,g

θ2θ3θ3θ2

(2.5)
Alg1 > 

LieDerivativee3,g

θ1θ2+θ2θ1

(2.6)

 

Example 2.

In this example we consider a Lie algebra containing a parameter b.  We find that a certain subspace S2 admits a naturally reductive complement M2 when b =0.

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

Alg1 > 

L2_DGLieAlgebra,Alg2,4,1,4,1,1,2,4,2,b,3,4,2,1,2,4,3,1,3,4,3,b

L2:=e1,e4=e1,e2,e4=be2e3,e3,e4=e2+be3

(2.7)
Alg1 > 

DGsetupL2:

 

For S2 we have that M2 is a reductive complement.  We let the inner product B2 be arbitrary.

Alg2 > 

S2e1,e4:M2e2,e3:B2Matrixr,s,s,t

B2:=rsst

(2.8)
Alg2 > 

QueryS2,M2,B2,NaturallyReductivePair

false

(2.9)
Alg2 > 

TF,EQ,SOLN,natRedPairQueryS2,M2,B2,b,r,s,t,NaturallyReductivePair

TF,EQ,SOLN,natRedPair:=true,0,2s2bt,2br+2s,2bsr+t,b=0,r=t,s=0,t=t,b=b,r=0,s=0,t=0,b=RootOf_Z2+1,r=t,s=RootOf_Z2+1t,t=t,e1,e4,e2,e3,t00t

(2.10)

 

We see that the that M2 =span e2, e3  is naturally reductive only when b = 0.  To check this we substitute b =0 into the Lie algebra data structure for L2 and change the name of the algebra to Alg3.

Alg2 > 

L3subsb=0,L2

L3:=e1,e4=e1,e2,e4=e3,e3,e4=e2

(2.11)
Alg2 > 

DGsetupL3

Lie algebra: Alg2

(2.12)
Alg2 > 

S3natRedPair11;M3natRedPair12;B3natRedPair13

S3:=e1,e4

M3:=e2,e3

B3:=t00t

(2.13)
Alg2 > 

QueryS3,M3,B3,NaturallyReductivePair

true

(2.14)

See Also

DifferentialGeometry

LieAlgebras

LieDerivative

Query

 


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