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LieAlgebras[MinimalIdeal] - find the smallest ideal containing a given set of vectors

Calling Sequences

     MinimalIdeal(S)

Parameters

     S        - a list of vectors in a Lie algebra

 

Description

Examples

Description

• 

MinimalIdeal(S) calculates the smallest ideal J containing the list of vectors S in an Lie algebra 𝔤.

• 

A list of vectors giving a basis for J is returned.

• 

The command MinimalIdeal is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form MinimalIdeal(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-MinimalIdeal(...).

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First initialize a Lie algebra and display the multiplication table.

L1_DGLieAlgebra,Alg1,5,1,5,1,2,2,3,1,1,2,5,2,1,2,5,3,1,3,5,3,1,4,5,4,2:

Alg1   > 

DGsetupL1:

MultiplicationTableLieBracket

e1,e5=2e1,e2,e3=e1,e2,e5=e2+e3,e3,e5=e3,e4,e5=2e4

(2.1)

 

Find the minimal ideal containing the vectors e1, e3.

Alg1 > 

S1e1,e3:

Alg1 > 

I1MinimalIdealS1

I1:=e1,e3

(2.2)

 

Find the minimal ideal containing the vectors e2, e4.

Alg1   > 

S2e2,e4:

Alg1 > 

I2MinimalIdealS2

I2:=e1,e2,e3,e4

(2.3)
Alg1 > 

QueryS2,Ideal

false

(2.4)
Alg1 > 

QueryI2,Ideal

true

(2.5)

See Also

DifferentialGeometry

LieAlgebras

MinimalSubalgebra

MultiplicationTable

Query[Ideal]