find the smallest Lie subalgebra containing a given set of vectors from a Lie algebra, find the smallest matrix algebra containing a given set of matrices - Maple Programming Help

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LieAlgebras[MinimalSubalgebra] - find the smallest Lie subalgebra containing a given set of vectors from a Lie algebra, find the smallest matrix algebra containing a given set of matrices

Calling Sequences

     MinimalSubalgebra(S)

     MinimalSubalgebra(M)

Parameters

     S        - a list of vectors in a Lie algebra

     M        - a list of square matrices

 

Description

Examples

Description

• 

MinimalSubalgebra(S) calculates the smallest Lie subalgebra J containing the list of vectors S from a defined Lie algebra 𝔤. A list of basis vectors for the subalgebra J returned.

• 

MinimalSubalgebra(M) calculates the smallest matrix algebra containing the matrices in the list M.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First we 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

L1:=e1,e5=2e1,e2,e3=e1,e2,e5=e2+e3,e3,e5=e3,e4,e5=2e4

(2.1)

DGsetupL1:

 

Find the minimal subalgebra containing e1, e3.

Alg1 > 

S1e1,e3:

Alg1 > 

A1MinimalSubalgebraS1

A1:=e1,e3

(2.2)

 

Find the minimal subalgebra containing e2, e3.

Alg1 > 

S2e2,e3:

Alg1 > 

A2MinimalSubalgebraS2

A2:=e1,e2,e3

(2.3)
Alg1 > 

QueryS2,Subalgebra

false

(2.4)
Alg1 > 

QueryA2,Subalgebra

true

(2.5)

 

Find the minimal subalgebra containing e2, e5.

Alg1 > 

S3e2,e5:

Alg1 > 

A3MinimalSubalgebraS3

A3:=e1,e2,e3,e5

(2.6)

 

Example 2.

The command MinimalSubalgebra also works with matrices.

Alg1 > 

MMatrix1,0,0,0,1,0,1,0,0,Matrix0,1,0,0,1,1,0,1,0

M:=100010100,010011010

(2.7)
Alg1 > 

NMinimalSubalgebraM

N:=100010100,010011010,000101000,101101101

(2.8)

 

We can use the LieAlgebraData command to verify that the set of matrices N defines a 4-dimensional Lie algebra and to determine the commutator relationships.

Alg1 > 

LieAlgebraDataN

e1,e2=e3,e1,e3=e3,e1,e4=e4,e2,e3=e4,e2,e4=e3+e4

(2.9)
Alg1 > 

 

Here e1,e2, e3, e4 denote the four matrices N[1], N[2], N[3], N[4].

See Also

DifferentialGeometry

LieAlgebras

LieAlgebraData

MinimalIdeal

Query[Subalgebra]