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Query[SymmetricPair] - check if a subalgebra, subspace pair defines a symmetric pair in a Lie algebra

Calling Sequences

     Query(M, S, "SymmetricPair")

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

Parameters

     M      - a list of independent vectors which defines a reductive complement to S in a Lie algebra 𝔤

     S      - a list of independent vectors which form a subalgebra in 𝔤

     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. The subalgebra, subspace pair S, M is a symmetric pair if [i] 𝔤= S M,  [ii] x, y M for xS and y M ,  and [iii] x, y   S for xM and y M . Note that [i] and [ii] imply that S,M define a  reductive pair. If  S,M  is a symmetric pair then  S,M  is a naturally reductive pair for any inner product on M.  If  S,M  is a symmetric pair, then M  is called a symmetric complement to the subalgebra S.

• 

Query(M, S, "SymmetricPair") returns true if the subspace M defines a symmetric complement to the subalgebra S, and false otherwise.

• 

Query(M, S, parm, "SymmetricPair") returns a sequence TF, Eq, Soln, symmetricList. Here TF is true if Maple finds parameter values for which S is a symmetric complement and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for S to be a symmetric complement; Soln is the list of solutions to the equations Eq; and symmetricList is the list of symmetric complements 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 and display the Lie bracket multiplication table.

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 can check that the subspace M1 =spane2, e3 defines a symmetric complement for the subalgebra S1 =spane1, e4.

Alg > 

S1e1,e4:M1e2,e3:

Alg > 

QueryS1,M1,SymmetricPair

true

(2.2)

 

In fact, we can show that M1 =spane2, e3 is the only symmetric complement to S1 by constructing the general complement M2= span{e2 +a1e1 + a2e4,   e3+a3e1 + a3e4}.

Alg > 

S2e1,e4:M2e2+a1e1+a2e4,e3+a3e1+a4e4:

Alg > 

TF,EQ,SOLN,symPairQueryS2,M2,a1,a2,a3,a4,ReductivePair

TF,EQ,SOLN,symPair:=true,0,a3,a4,a1,a2,a1=0,a2=0,a3=0,a4=0,e1,e4,e2,e3

(2.3)

 

SOLN shows that all the parameters  a1, a2, a3, a4 must be zero in order for S2, M2 to define a symmetric pair.

Alg > 

SOLN

a1=0,a2=0,a3=0,a4=0

(2.4)

 

Next we show that the subalgebra  S3 = spane4 does not admit a symmetric complement at all.

Alg > 

S3e4:

Alg > 

M3evalDGe1+a1e4,e2,e3:

Alg > 

QueryS3,M3,ReductivePair

true

(2.5)
Alg > 

QueryS3,M3,a1,SymmetricPair

false,0,1,a1,a1

(2.6)

See Also

DifferentialGeometry

LieAlgebras

Query

 


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