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Question: Normalizer in Abstract Lie Algebra

Question:Normalizer in Abstract Lie Algebra

mskalsi 290 Maple
normalizer dgsetup
April 12 2016
0

Dear All

I have started exploring Lie Algebra package in Maple 18 and it wonderful experience for me. But as I am in learning stage so I feel little difficulty to understand even its basic enviroment.

I am trying to fing quotient algebra for Normalizer of certain element W1 in Lie algebra and I don't how to do it. I have expected answer for quotient of Normlizer but don't know to obtain it. Please see following details:

 

> 

with(DifferentialGeometry):with(LieAlgebras):``
> 

L1 := _DG([["LieAlgebra", Alg1, [6]], [[[1, 3, 1], -1], [[1, 6, 2], -1], [[2, 3, 2], -1], [[2, 4, 1], 1], [[2, 5, 2], 1], [[4, 5, 4], -1], [[4, 6, 3], -1], [[4, 6, 5], -2], [[5, 6, 6], -1]]])

_DG([["LieAlgebra", Alg1, [6]], [[[1, 3, 1], -1], [[1, 6, 2], -1], [[2, 3, 2], -1], [[2, 4, 1], 1], [[2, 5, 2], 1], [[4, 5, 4], -1], [[4, 6, 3], -1], [[4, 6, 5], -2], [[5, 6, 6], -1]]])

 (1)
> 

DGsetup(L1)

`Lie algebra: Alg1`

 (2)
Alg1 > 

MultiplicationTable("LieTable")

"[[[,`| `,e1,e2,e3,e4,e5,e6],[,-`---`,-`---`,-`---`,-`---`,-`---`,-`---`,-`---`],[e1,`| `,0,0,_DG([["vector",Alg1,[]],[[[1],-1]]]),0,0,_DG([["vector",Alg1,[]],[[[2],-1]]])],[e2,`| `,0,0,_DG([["vector",Alg1,[]],[[[2],-1]]]),_DG([["vector",Alg1,[]],[[[1],1]]]),_DG([["vector",Alg1,[]],[[[2],1]]]),0],[e3,`| `,_DG([["vector",Alg1,[]],[[[1],1]]]),_DG([["vector",Alg1,[]],[[[2],1]]]),0,0,0,0],[e4,`| `,0,_DG([["vector",Alg1,[]],[[[1],-1]]]),0,0,_DG([["vector",Alg1,[]],[[[4],-1]]]),_DG([["vector",Alg1,[]],[[[3],-1],[[5],-2]]])],[e5,`| `,0,_DG([["vector",Alg1,[]],[[[2],-1]]]),0,_DG([["vector",Alg1,[]],[[[4],1]]]),0,_DG([["vector",Alg1,[]],[[[6],-1]]])],[e6,`| `,_DG([["vector",Alg1,[]],[[[2],1]]]),0,0,_DG([["vector",Alg1,[]],[[[3],1],[[5],2]]]),_DG([["vector",Alg1,[]],[[[6],1]]]),0]]]"

 (3)
Alg1 > 

W1 := [a*e3+e5]

[a*_DG([["vector", Alg1, []], [[[3], 1]]])+_DG([["vector", Alg1, []], [[[5], 1]]])]

 (4)
Alg1 > 

Nor[W1] := SubalgebraNormalizer(W1)

[_DG([["vector", Alg1, []], [[[5], 1]]]), _DG([["vector", Alg1, []], [[[3], 1]]])]

 (5)

How one can find Quotient alegebra "(Nor[W1])/(W1)....???`? ``The expected answer is {e3}.`"

I am trying verify 4th 2-dimensional subalgebra results given in research artcle by Coggeshall and Meyer-ter-Vehn (See pp. 3592 in article).

Download [670]_Normalizer_in_Abstract_Lie_Algebra.mw

Regards
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