find a root or the roots for a semi-simple Lie algebra from a root space and the Cartan subalgebra; or from a root space decomposition - Maple Programming Help

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LieAlgebras[LieAlgebraRoots] - find a root or the roots for a semi-simple Lie algebra from a root space and the Cartan subalgebra; or from a root space decomposition

Calling Sequences

     LieAlgebraRoots(X, CSA)

     LieAlgebraRoots(RSD )

Parameters

     X     - a vector in a Lie algebra, defining a root space

     CSA   - a list of vectors in a semi-simple Lie algebra, defining a Cartan subalgebra  

     RSD   - a table, defining a root space decomposition of a semi-simple Lie algebra

     

 

Description

Examples

Description

• 

Let g be a Lie algebra and h a Cartan subalgebra. Let h1, h2, ... , hm be a basis for 𝔥. A root for g with respect to this basis is a non-zero m-tuple of complex numbers α  = α1, α2, ... , αm such that adhix =  αi x  (*)  for some x 𝔤.

• 

The set of  x 𝔤 which satisfy (*) is called the root space of g defined by α and denoted by Rα.  A basic theorem in the structure theory of semi-simple Lie algebras asserts that the root spaces  Rα are 1-dimensional.

• 

The first calling sequence calculates the root α for the given root space X. If X is not a root space, then an empty vector is returned.

• 

The second calling sequence simply returns the indices, as column vectors, for the table defining the root space decomposition.

 

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

Use the command SimpleLieAlgebraData to initialize the simple Lie algebra su4. This is a 15-dimensional Lie algebra of skew-Hermitian matrices.

LDSimpleLieAlgebraDatasu(4),su4,labelformat=gl,labels=E,θ

LD:=e1,e4=2e10,e1,e5=e11,e1,e7=e13,e1,e8=e14,e1,e9=e15,e1,e10=2e4,e1,e11=e5,e1,e13=e7,e1,e14=e8,e1,e15=e9,e2,e4=e10,e2,e5=2e11,e2,e6=e12,e2,e7=e13,e2,e8=e14,e2,e10=e4,e2,e11=2e5,e2,e12=e6,e2,e13=e7,e2,e14=e8,e3,e5=e11,e3,e6=2e12,e3,e7=e13,e3,e8=e14,e3,e9=e15,e3,e11=e5,e3,e12=2e6,e3,e13=e7,e3,e14=e8,e3,e15=e9,e4,e5=e7,e4,e7=e5,e4,e8=e9,e4,e9=e8,e4,e10=2e1,e4,e11=e13,e4,e13=e11,e4,e14=e15,e4,e15=e14,e5,e6=e8,e5,e7=e4,e5,e8=e6,e5,e10=e13,e5,e11=2e2,e5,e12=e14,e5,e13=e10,e5,e14=e12,e6,e7=e9,e6,e8=e5,e6,e9=e7,e6,e11=e14,e6,e12=2e3,e6,e13=e15,e6,e14=e11,e6,e15=e13,e7,e9=e6,e7,e10=e11,e7,e11=e10,e7,e12=e15,e7,e13=2e12e2,e7,e15=e12,e8,e9=e4,e8,e10=e15,e8,e11=e12,e8,e12=e11,e8,e14=2e22e3,e8,e15=e10,e9,e10=e14,e9,e12=e13,e9,e13=e12,e9,e14=e10,e9,e15=2e12e22e3,e10,e11=e7,e10,e13=e5,e10,e14=e9,e10,e15=e8,e11,e12=e8,e11,e13=e4,e11,e14=e6,e12,e13=e9,e12,e14=e5,e12,e15=e7,e13,e15=e6,e14,e15=e4,Ei11,Ei22,Ei33,E12,E23,E34,E13,E24,E14,Ei12,Ei23,Ei34,Ei13,Ei24,Ei14,thetai11,thetai22,thetai33,θ12,θ23,θ34,θ13,θ24,θ14,thetai12,thetai23,thetai34,thetai13,thetai24,thetai14

(2.1)

DGsetupLD

Lie algebra: su4

(2.2)

 

The explicit matrices defining su4are given by the StandardRepresentation command.

su4 > 

StandardRepresentationsu4

I0000I0000000000,00000I0000I00000,0000000000I0000I,0100100000000000,0000001001000000,0000000000010010,0010000010000000,0000000100000100,0001000000001000,0I00I00000000000,000000I00I000000,00000000000I00I0,00I00000I0000000,0000000I00000I00,000I00000000I000

(2.3)

 

The diagonal matrices determine a Cartan subalgebra.

su4 > 

CSAEi11,Ei22,Ei33

CSA:=Ei11,Ei22,Ei33

(2.4)

 

We use the Query command to check that (2.4) is a Cartan subalgebra.

su4 > 

QueryCSA,CartanSubalgebra

true

(2.5)

 

Find the root for the root space E14 I Ei14.

LieAlgebraRootsE14IEi14,CSA

I0I

(2.6)

 

Note that the command RootSpace  performs the inverse operation to LieAlgebraRoot  - given a root, the command RootSpace returns the corresponding root space.

su4 > 

RootSpaceI,0,I,CSA

E14IEi14

(2.7)

 

Example 2.

If the complete root space decomposition is given as a table, then the command LieAlgebraRoots returns the indices of that table as column vectors.

su4 > 

RSDRootSpaceDecompositionCSA

RSD:=tableI,I,I=E24+IEi24,0,I,2I=E34IEi34,0,I,2I=E34+IEi34,I,2I,I=E23IEi23,I,2I,I=E23+IEi23,I,0,I=E14+IEi14,I,0,I=E14IEi14,I,I,I=E24IEi24,2I,I,0=E12IEi12,I,I,I=E13+IEi13,I,I,I=E13IEi13,2I,I,0=E12+IEi12

(2.8)
su4 > 

LieAlgebraRootsRSD

III,0I2I,0I2I,I2II,I2II,I0I,I0I,III,2II0,III,III,2II0

(2.9)

 

See Also

DifferentialGeometry

CartanSubalgebra

Query

RootSpace

RootSpaceDecomposition