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

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LieAlgebras[RootSpace] - find a root space for a semi-simple Lie algebra from a Cartan subalgebra or a root space decomposition

Calling Sequences

     RootSpace(RV, CSA)

     RootSpace(RV, RSD)

Parameters

     RV    - a column vector

     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 theorem of semi-simple Lie algebras asserts that the root spaces Rα are 1-dimensional.

• 

The first call sequence calculates the root space Rα for a given root. If α is not a root, then the zero vector (in 𝔤) is returned.

• 

The second calling sequence simply returns the table entry in the table of root spaces corresponding to the root α.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

Use the command SimpleLieAlgebraData to obtain the Lie algebra data for the simple Lie algebra su4. This is the 15-dimensional Lie algebra of trace-free, skew-Hermitian matrices.

LDSimpleLieAlgebraDatasu(4),su4,labelformat=gl,labels=U,η

LD:=e1,e4=e10,e1,e6=e12,e1,e7=e13,e1,e8=e14,e1,e9=2e15,e1,e10=e4,e1,e12=e6,e1,e13=e7,e1,e14=e8,e1,e15=2e9,e2,e4=e10,e2,e5=e11,e2,e6=e12,e2,e8=2e14,e2,e9=e15,e2,e10=e4,e2,e11=e5,e2,e12=e6,e2,e14=2e8,e2,e15=e9,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=2e12e2,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=2e22e3,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=2e12e3,e7,e15=e12,e8,e9=e4,e8,e10=e15,e8,e11=e12,e8,e12=e11,e8,e14=2e2,e8,e15=e10,e9,e10=e14,e9,e12=e13,e9,e13=e12,e9,e14=e10,e9,e15=2e1,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,Ui11,Ui22,Ui33,U12,U23,U34,U13,U24,U14,Ui12,Ui23,Ui34,Ui13,Ui24,Ui14,etai11,etai22,etai33,η12,η23,η34,η13,η24,η14,etai12,etai23,etai34,etai13,etai24,etai14

(2.1)

 

Initialize the Lie algebra su4.

DGsetupLD

Lie algebra: su4

(2.2)

 

The command StandardRepresentation will produce the actual matrices defining su4. (This command only applies to Lie algebras constructed by the SimpleLieAlgebraData  procedure.)

su4 > 

StandardRepresentationsu4

I00000000000000I,00000I000000000I,0000000000I0000I,0100100000000000,0000001001000000,0000000000010010,0010000010000000,0000000100000100,0001000000001000,0I00I00000000000,000000I00I000000,00000000000I00I0,00I00000I0000000,0000000I00000I00,000I00000000I000

(2.3)

 

The Lie algebra elements corresponding to the complex diagonal matrices define a Cartan subalgebra.

su4 > 

CSAUi11,Ui22,Ui33

CSA:=Ui11,Ui22,Ui33

(2.4)

 

We check this is indeed a Cartan subalgebra using the Query command

su4 > 

QueryCSA,CartanSubalgebra

true

(2.5)

 

 

Here is the root space corresponding to the root <I, I, -I>.

su4 > 

XRootSpaceI&comma;I&comma;2I&comma;CSA

X:=U34IUi34

(2.6)

 

We check that the X is an eigenvector for the elements of the Cartan subalgebra.

su4 > 

BseqLieBracketh&comma;X&comma;h=CSA

B:=IU34&plus;Ui34&comma;IU34&plus;Ui34&comma;2IU34&plus;2Ui34

(2.7)
su4 > 

GetComponentsB&comma;X

I&comma;I&comma;2I

(2.8)

 

The column vector <I, I, I> is not a root

su4 > 

RootSpaceI&comma;I&comma;I&comma;CSA

0Ui11

(2.9)

 

Example 2.

Here is the full root space decomposition for the Lie algebra su4from Example 1.

su4 > 

RSDRootSpaceDecompositionCSA

RSD:=table2I&comma;I&comma;I&equals;U14IUi14&comma;I&comma;I&comma;0&equals;U12&plus;IUi12&comma;I&comma;2I&comma;I&equals;U24IUi24&comma;I&comma;0&comma;I&equals;U13IUi13&comma;0&comma;I&comma;I&equals;U23IUi23&comma;I&comma;I&comma;2I&equals;U34&plus;IUi34&comma;0&comma;I&comma;I&equals;U23&plus;IUi23&comma;I&comma;2I&comma;I&equals;U24&plus;IUi24&comma;2I&comma;I&comma;I&equals;U14&plus;IUi14&comma;I&comma;0&comma;I&equals;U13&plus;IUi13&comma;I&comma;I&comma;2I&equals;U34IUi34&comma;I&comma;I&comma;0&equals;U12IUi12

(2.10)

 

The second calling sequence for RootSpace simply converts the given root vector to a list and extracts the corresponding root space from the root space decomposition table.

su4 > 

RootSpaceI&comma;I&comma;2I&comma;RSD

U34IUi34

(2.11)

See Also

DifferentialGeometry

CartanSubalgebra

GetComponents

Query

RootSpaceDecomposition

SimpleLieAlgebraData

SimpleLieAlgebraProperties

StandardRepresentation