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LieAlgebras[AscendingIdealsBasis] - find a basis for a solvable Lie algebra which defines an ascending chain of ideals

Calling Sequences

     AscendingIdealsBasis(Alg)

Parameters

     Alg        - (optional) Maple name or string, the name of an initialized Lie algebra

 

Description

Examples

Description

• 

Every (complex) solvable Lie algebra admits a basis e1, e2, ... , en  such that the subspace spane1, e2, ... , ek form an ideal in spane1, e2, ... , ek+1. The command AscendingIdealsBasis calculates such a basis. This basis can be quite useful in a situation where the matrix exponentials of the adjoint matrices are needed.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First we initialize a 5-dimensional Lie algebra.

L_DGLieAlgebra,Alg1,5,1,2,1,22,1,2,2,11,1,2,3,21,1,2,4,1,1,2,5,11,1,3,1,3,1,3,2,2,1,3,3,4,1,4,1,1,1,4,5,2,1,5,1,12,1,5,2,7,1,5,3,13,1,5,4,1,1,5,5,3,2,3,1,12,2,3,2,6,2,3,3,11,2,3,4,1,2,3,5,6,2,4,1,19,2,4,2,9,2,4,3,19,2,4,4,1,2,4,5,11,2,5,1,16,2,5,2,8,2,5,3,16,2,5,5,8,3,4,1,2,3,4,3,1,3,4,4,1,3,4,5,4,3,5,1,8,3,5,2,5,3,5,3,9,3,5,4,1,3,5,5,1,4,5,1,7,4,5,2,4,4,5,3,7,4,5,4,1,4,5,5,2:

DGsetupL

Lie algebra: Alg1

(2.1)

 

We can use the command Query/"Solvable" to check that this is a solvable Lie algebra.

Alg1 > 

QuerySolvable

true

(2.2)

 

Now we calculate a basis with the ascending ideals property.

Alg1 > 

BAscendingIdealsBasis

B:=e12e5,e22e4+3e5,e3e4,e1,e2

(2.3)

 

The following two commands check, for example, that  span B1..3 is an ideal in span B1..4.

Alg1 > 

CBracketOfSubspacesB1..3,B1..4

C:=24e114e2+26e3+2e4+6e5,60e1+32e260e34e424e5,2e12e2+4e32e5

(2.4)
Alg1 > 

GetComponentsC,B1..3,trueorfalse=on

true

(2.5)

 

The command  Query/"AscendingIdealsBasis" will verify that the basis B has the ascending ideals property.

Alg1 > 

QueryB,AscendingIdealsBasis

true

(2.6)

 

The ascending ideals property becomes apparent if we re-initialize the Lie algebra using the basis B (using the command LieAlgebraData).

Alg1 > 

L2LieAlgebraDataB,alg2

L2:=e1,e4=24e114e2+26e3,e1,e5=10e1+5e211e3,e2,e4=60e1+32e260e3,e2,e5=10e16e2+10e3,e3,e4=2e12e2+4e3,e3,e5=7e1+3e28e3,e4,e5=22e111e2+21e3

(2.7)
Alg1 > 

DGsetupL2

Lie algebra: alg2

(2.8)
alg2 > 

MultiplicationTableLieTable

| e1e2e3e4e5------------------------e1| 00024e114e2+26e310e1+5e211e3e2| 00060e1+32e260e310e16e2+10e3e3| 0002e12e2+4e37e1+3e28e3e4| 24e1+14e226e360e132e2+60e32e1+2e24e3022e111e2+21e3e5| 10e15e2+11e310e1+6e210e37e13e2+8e322e1+11e221e30

(2.9)

See Also

DifferentialGeometry

LieAlgebras

BracketOfSubspaces

GetComponents

MultiplicationTable

Query

 


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