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LieAlgebras[ChevalleyBasis] - find the Chevalley basis for a real, split semi-simple Lie algebra

Calling Sequences

     ChevalleyBasis(CSA, RSD, PosRts)

   

Parameters

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

     RSD      - a table, specifying the root space decomposition of the Lie algebra with respect to the Cartan subalgebra CSA

     PosRts   - a list of vectors, specifying a choice of positive roots for the root space decomposition

     option   - the keyword argument Algebratype = [A, r] where A is a string "A", "B", "C", "D", "E", "F", or "G" and r is the rank of the Lie algebra.

  

Description

• 

A Chevalley basis is a special choice of basis for a real, split semi-simple Lie algebra. It is adapted to the root space decomposition. In a Chevalley basis, a Cartan subalgebra, the root space decomposition, the Cartan matrix, the simple roots, and the root pattern can be determined by inspection. The structure constants are all integers.

• 

The command ChevalleyBasis(CSA, RSD, PosRts) returns a list of vectors defining a Chevalley basis ℬ =h1, h2 , ... , hr , x1, x2, ... , xℓ, y1, y2, ... , yℓ. The structure equations of this basis are

hi, hj = 0,  hi,xj = aijxj ,  hi,yj = aijxj  ,     i,j = 1,2, ..., r,

xa , xb =  ±q+1xc ,  a, b=1,2,...,ℓ.

Here aij is the Cartan matrix for 𝔤. The roots for xa, xb, and xc are αa, αb, and αc=αa+αb.  The integer q is the largest positive integer such that αa q αb is not a root. See ChevalleyBasisDetails for the algorithm used to construct this basis.

• 

Note that in the Chevalley basis all the structure constants are integers and the transformation hi  hi  , xi yi,  yi  xi   is a Lie algebra automorphism.

• 

The Chevalley basis is used by the command SplitAndCompactForms to find the split and compact forms of a general semi-simple Lie algebra.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

We calculate a Chevalley basis for the rank 2 Lie algebra so3,2. We begin with the basis provided by the command SimpleLieAlgebraData.

 

LD:=SimpleLieAlgebraDataso(3,2),so32:

DGsetupLD

Lie algebra: so32

(2.1)

 

We will use the choices of the Cartan subalgebra, root space decomposition, and positive roots for so3,2 contained in SimpleLieAlgebraProperties. (For Lie algebras not created by the SimpleLieAlgebraData command, use CartanSubalgebra, RootSpaceDecomposition, PositiveRoots.)

so32 > 

P:=SimpleLieAlgebraPropertiesso32:

so32 > 

CSA:=PCartanSubalgebra

CSA:=e1,e4

(2.2)
so32 > 

RSD:=evalPRootSpaceDecomposition

RSD:=table1,0=e9,1,1=e3,0,1=e10,1,0=e7,1,1=e5,1,1=e2,0,1=e8,1,1=e6

(2.3)
so32 > 

PosRts:=PPositiveRoots

PosRts:=11,01,10,11

(2.4)

 

The Chevalley basis for so3,2 determined by this Cartan subalgebra and choice of positive roots is:

so32 > 

CB:=ChevalleyBasisCSA,RSD,PosRts

CB:=e1e4,2e4,e2,2e8,2e7,2e5,e3,e10,e9,12e6

(2.5)

 

We calculate the structure equations for so3,2 in the Chevalley basis and initialize the Lie algebra in this new basis.

so32 > 

newLD:=LieAlgebraDataCB,so32CB

newLD:=e1,e3=2e3,e1,e4=e4,e1,e5=e5,e1,e7=2e7,e1,e8=e8,e1,e9=e9,e2,e3=2e3,e2,e4=2e4,e2,e6=2e6,e2,e7=2e7,e2,e8=2e8,e2,e10=2e10,e3,e4=e5,e3,e7=e1,e3,e9=e8,e4,e5=2e6,e4,e8=e2,e4,e9=2e7,e4,e10=e9,e5,e7=e4,e5,e8=2e3,e5,e9=2e1e2,e5,e10=e8,e6,e8=e5,e6,e9=e4,e6,e10=e1e2,e7,e8=e9,e8,e9=2e10

(2.6)
so32 > 

DGsetupnewLD,'e1,e2,x1,x2,x3,x4,y1,y2,y3,y4','ω'

Lie algebra: so32CB

(2.7)

 

To display the multiplication table for this Lie algebra we use interface to increase the maximum inline array display size.

so32CB > 

interfacertablesize=15

10

(2.8)
so32CB > 

M:=MultiplicationTableLieTable

M:=| e1e2x1x2x3x4y1y2y3y4--------------------------------------------e1| 002x1x2x302y1y2y30e2| 002x12x202x42y12y202y4x1| 2x12x10x300e10y20x2| x22x2x302x400e22y1y3x3| x3002x400x22x12e1e2y2x4| 02x400000x3x2e1e2y1| 2y12y1e10x200y300y2| y22y20e22x1x3y302y40y3| y30y22y12e1+e2x202y400y4| 02y40y3y2e1+e20000

(2.9)

 

Let us focus in on various parts of the multiplication table. From the first two rows

so32CB > 

LinearAlgebra:-SubMatrixM,1..4,1..1

| e1e2x1x2x3x4y1y2y3y4--------------------------------------------e1| 002x1x2x302y1y2y30e2| 002x12x202x42y12y202y4

(2.10)

 

it is clear that e1, e2 act diagonally and so form a Cartan subalgebra. From the 3rd and 4th columns we can read off the Cartan matrix for so3,2 as the coefficients of:

so32CB > 

LinearAlgebra:-SubMatrixM,1..4,1..2,5..6

| x1x2------------e1| 2x1x2e2| 2x12x2

(2.11)

The vectors x1, x2, x3, x4 correspond to the roots α1, α2, α3, α4 with α1 , α2 being the simple roots. Therefore, from the multiplication table

so32CB > 

LinearAlgebra:-SubMatrixM,1..2,5..8,1..2,5..8

| x1x2x3x4--------------------x1| 0x300x2| x302x40x3| 02x400x4| 0000

(2.12)

 

we can read off the root pattern as α1 + α2 = α3, α2+α3 = α4. Finally we note that the vectors y1, y2, y3, y4 satisfy the same structure equations as x1, x2, x3, x4.

so32CB > 

LinearAlgebra:-SubMatrixM,1..2,9..12,1..2,9..12

| y1y2y3y4--------------------y1| 0y300y2| y302y40y3| 02y400y4| 0000

(2.13)

See Also

DifferentialGeometry,  LieAlgebras, CartanDecomposition, CartanMatrix, CartanSubalgebra, PositiveRoots, RootSpaceDecomposition, SimpleLieAlgebraData, SimpleLieAlgebraProperties, SplitAndCompactForms


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