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Question: How do I convert LieTable into a matrix?

Question:How do I convert LieTable into a matrix?

nrussoniello 10 Maple 2019
matrix rank commutator liealgebra
July 20 2020
2

I am trying to compute the rank of the Commutator Matrix of a Lie algebra. That is, I wish to construct a matrix version of the multiplication table for a given matrix Lie algebra, and then compute the rank of this matrix.

 

Download CommutatorExample.mw
 

with(DifferentialGeometry); with(LieAlgebras)

``

``

M := [Matrix([[1, 0], [0, 0]]), Matrix([[0, 0], [0, 1]]), Matrix([[0, 1], [0, 0]])]

[Matrix(%id = 18446744078224010646), Matrix(%id = 18446744078224010766), Matrix(%id = 18446744078224003190)]

 (1)

L := LieAlgebraData(M, Alg1)

_DG([["LieAlgebra", Alg1, [3, table( [ ] )]], [[[1, 3, 3], 1], [[2, 3, 3], -1]]])

 (2)

``

DGsetup(L)

`Lie algebra: Alg1`

 (3)

T := MultiplicationTable("LieTable")

Matrix(%id = 18446744078223968366)

 (4)

``

``

``

``

 

 

I have the above, but I run into two issues:

1. It doesn't have matrix format, so Rank is undefined (I have tried convert(T,matrix) to no avail),

2. I need to be able to remove the first and second rows and columns from T (because these rows/columns are occupied by the Lie algebra's basis elements, and separating lines, respectively).

 

I believe that if I can convert T into a matrix somehow, I can simply use SubMatrix to remove the things I don't want, and then Rank should work.

 

Any help would be greatly appreciated!

 

P.S. Thanks to dharr1338 for the suggestion of including the worksheet, I'm very new to Maple and MaplePrimes, so I appreciate the patience.
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