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Question: How to find all ideals in this?

Question:How to find all ideals in this?

mskalsi 290 Maple 18
radical series ideal differential-geometry liealgebra nilradical
May 18 2016
2

Dear All

Using Lie algebra package in Maple we can easily find nilradical for given abstract algebra, but how we can find all the ideal in lower central series by taking new basis as nilradical itself?

Please see following;

 

> 

with(DifferentialGeometry); with(LieAlgebras)
> 

DGsetup([x, y, t, u, v])

`frame name: Euc`

 (1)
Euc > 

VectorFields := evalDG([D_v, D_v*x+D_y*t, 2*D_t*t-2*D_u*u-D_v*v+D_y*y, t*D_v, D_v*y+D_u, D_t, D_x, D_x*t+D_u, 2*D_v*x+D_x*y, -D_t*t+2*D_u*u+2*D_v*v+D_x*x, D_y])

[_DG([["vector", "Euc", []], [[[5], 1]]]), _DG([["vector", "Euc", []], [[[2], t], [[5], x]]]), _DG([["vector", "Euc", []], [[[2], y], [[3], 2*t], [[4], -2*u], [[5], -v]]]), _DG([["vector", "Euc", []], [[[5], t]]]), _DG([["vector", "Euc", []], [[[4], 1], [[5], y]]]), _DG([["vector", "Euc", []], [[[3], 1]]]), _DG([["vector", "Euc", []], [[[1], 1]]]), _DG([["vector", "Euc", []], [[[1], t], [[4], 1]]]), _DG([["vector", "Euc", []], [[[1], y], [[5], 2*x]]]), _DG([["vector", "Euc", []], [[[1], x], [[3], -t], [[4], 2*u], [[5], 2*v]]]), _DG([["vector", "Euc", []], [[[2], 1]]])]

 (2)
Euc > 

L1 := LieAlgebraData(VectorFields)

_DG([["LieAlgebra", "L1", [11]], [[[1, 3, 1], -1], [[1, 10, 1], 2], [[2, 3, 2], -1], [[2, 5, 4], 1], [[2, 6, 11], -1], [[2, 7, 1], -1], [[2, 8, 4], -1], [[2, 9, 5], -1], [[2, 9, 8], 1], [[2, 10, 2], 1], [[3, 4, 4], 3], [[3, 5, 5], 2], [[3, 6, 6], -2], [[3, 8, 8], 2], [[3, 9, 9], 1], [[3, 11, 11], -1], [[4, 6, 1], -1], [[4, 10, 4], 3], [[5, 10, 5], 2], [[5, 11, 1], -1], [[6, 8, 7], 1], [[6, 10, 6], -1], [[7, 9, 1], 2], [[7, 10, 7], 1], [[8, 9, 4], 2], [[8, 10, 8], 2], [[9, 10, 9], 1], [[9, 11, 7], -1]]])

 (3)
Euc > 

DGsetup(L1)

`Lie algebra: L1`

 (4)
L1 > 

MultiplicationTable("LieTable"):
L1 > 

N := Nilradical(L1)

[_DG([["vector", "L1", []], [[[1], 1]]]), _DG([["vector", "L1", []], [[[2], 1]]]), _DG([["vector", "L1", []], [[[4], 1]]]), _DG([["vector", "L1", []], [[[5], 1]]]), _DG([["vector", "L1", []], [[[6], 1]]]), _DG([["vector", "L1", []], [[[7], 1]]]), _DG([["vector", "L1", []], [[[8], 1]]]), _DG([["vector", "L1", []], [[[9], 1]]]), _DG([["vector", "L1", []], [[[11], 1]]])]

 (5)
L1 > 

Query(N, "Nilpotent")

true

 (6)
L1 > 

Query(N, "Solvable")

true

 (7)

Taking N as new basis , how we can find all ideals in lower central series of this solvable ideal N?

 

Download [944]_Structure_of_Lie_algebra.mw

Regards
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