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

Calling Sequences

     RootSpaceDecomposition(CSA)

Parameters

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

 

Description

Examples

Description

• 

 Let g be a semi-simple Lie algebra and h a Cartan subalgebra. Let h1, h2, ... ,hm be a basis for 𝔥 . The linear transformations adhi are simultaneously diagonalizable over C - if x ∈ g is a common eigenvector for all these transformations, then adhix = hi , x = αi x . The m-tuples α = α1, α2, ... , αm are called the roots of 𝔤 with respect to the Cartan sub-algebra 𝔥 and 𝔤 = 𝔥  α  Δ Rα the root space decomposition of g with respect to h.

• 

The roots and root space decomposition enjoy the following basic properties.

1. 

 The eigenspaces or root spaces Rα are each 1-dimensional.

2. 

 If α is a root, then so is α .

3. 

 If x Rα and y Rβ, then x, y Rα+β if α + β is a root; otherwise x, y =0.

4. 

 If x Rα and y Rα, then x, y  𝔥. The vectors x, y, x,y define a 3-dimensional Lie algebra isomorphic to sl2.

5. 

If x Rα and y Rβ and α  ±β, then x, y =0, where ,  is the Killing form.

6. 

The Killing form is non-degenerate on h.

7. 

 The number of linearly independent roots is m.

• 

The command RootSpaceDecomposition returns a table describing the root space decomposition of g with respect to h. The indices of the table are the roots α and the table entries are vectors in g defining the root spaces Rα .

• 

The command Query/"RootSpaceDecomposition" will check that a given table defines a root space decomposition.

• 

The commands SimpleLieAlgebraData and SimpleLieAlgebraProperties can be used to quickly obtain the root space decomposition for any simple classical matrix algebra.

Examples

 

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

In this example we initialize the simple Lie algebra sl3 (of 3×3 trace-free matrices), calculate a Cartan subalgebra and a root space decomposition. We then illustrate the above properties of the root space decomposition.

 

First, we use the program SimpleLieAlgebraData to generate the Lie algebra data for sl3.

LDSimpleLieAlgebraDatasl(3),sl3,labelformat=gl,labels=E,θ

LD:=e1,e3=e3,e1,e4=2e4,e1,e5=e5,e1,e6=e6,e1,e7=2e7,e1,e8=e8,e2,e3=e3,e2,e4=e4,e2,e5=e5,e2,e6=2e6,e2,e7=e7,e2,e8=2e8,e3,e5=e1e2,e3,e6=e4,e3,e7=e8,e4,e5=e6,e4,e7=e1,e4,e8=e3,e5,e8=e7,e6,e7=e5,e6,e8=e2,E11,E22,E12,E13,E21,E23,E31,E32,θ11,θ22,θ12,θ13,θ21,θ23,θ31,θ32

(2.1)

DGsetupLD

Lie algebra: sl3

(2.2)

 

The program CartanSubalgebra calculates a Cartan subalgebra for sl3.

sl3 > 

CSACartanSubalgebrasl3

CSA:=E11,E22

(2.3)

 

Now compute the root space decomposition. We see that each root space is 1-dimensional (Property 1).

sl3 > 

RSDRootSpaceDecompositionCSA

RSD:=table2,1=E31,2,1=E13,1,2=E23,1,1=E12,1,1=E21,1,2=E32

(2.4)

 

The roots are the indices for this table, given as column vectors. It is easy to see that the negative of any root is a root (Property 2).

sl3 > 

RT1,2,1,2,1,1,2,1,2,1,1,1

RT:=12,12,11,21,21,11

(2.5)

 

Here are the eigenvectors or root spaces.

sl3 > 

RSseqRSDconvertv,list,v=RT

RS:=E32,E23,E21,E13,E31,E12

(2.6)

 

The 2nd and 6th roots add to give the 4th root. This means that the Lie bracket of the 3rd and 4th vectors in (2.6) should be a multiple of the 1st vector (Property 3).

sl3 > 

LieBracketRS2,RS6

E13

(2.7)

 

The 1st and 2nd roots are negatives of each other so the Lie bracket of the 1st and 2nd vectors in (2.6) should belong to the Cartan subalgebra (Property 4).

sl3 > 

HLieBracketRS1,RS2

H:=E22

(2.8)

 

The vectors RS1, RS2, H  form a 3-dimensional Lie algebra (Property 4).

sl3 > 

LieAlgebraDataH,RS1,RS2

e1,e2=2e2,e1,e3=2e3,e2,e3=e1

(2.9)

 

The Killing form restricted to the root spaces of the 2nd, 4rd and 6th roots is diagonal (Property 5).

sl3 > 

KillingRS2,RS4,RS6

000000000

(2.10)

 

Example 2.

We repeat the analysis of Example 1 using the Lie algebra so4, 3. This is a 21-dimensional Lie algebra of 7×7 matrices which preserve the quadratic form 0I30I300001.  First, we use the program SimpleLieAlgebraData to generate the Lie algebra data for sl3.

LDSimpleLieAlgebraDataso(4,3),so43,labelformat=gl,labels=R,σ

LD:=e1,e2=e2,e1,e3=e3,e1,e4=e4,e1,e7=e7,e1,e10=e10,e1,e11=e11,e1,e13=e13,e1,e14=e14,e1,e16=e16,e1,e19=e19,e2,e4=e1e5,e2,e5=e2,e2,e6=e3,e2,e7=e8,e2,e12=e11,e2,e14=e15,e2,e17=e16,e2,e19=e20,e3,e4=e6,e3,e7=e1e9,e3,e8=e2,e3,e9=e3,e3,e12=e10,e3,e13=e15,e3,e18=e16,e3,e19=e21,e4,e5=e4,e4,e8=e7,e4,e11=e12,e4,e15=e14,e4,e16=e17,e4,e20=e19,e5,e6=e6,e5,e8=e8,e5,e10=e10,e5,e12=e12,e5,e13=e13,e5,e15=e15,e5,e17=e17,e5,e20=e20,e6,e7=e4,e6,e8=e5e9,e6,e9=e6,e6,e11=e10,e6,e13=e14,e6,e18=e17,e6,e20=e21,e7,e9=e7,e7,e10=e12,e7,e15=e13,e7,e16=e18,e7,e21=e19,e8,e9=e8,e8,e10=e11,e8,e14=e13,e8,e17=e18,e8,e21=e20,e9,e11=e11,e9,e12=e12,e9,e14=e14,e9,e15=e15,e9,e18=e18,e9,e21=e21,e10,e13=e1e5,e10,e14=e6,e10,e15=e3,e10,e19=e17,e10,e20=e16,e11,e13=e8,e11,e14=e1e9,e11,e15=e2,e11,e19=e18,e11,e21=e16,e12,e13=e7,e12,e14=e4,e12,e15=e5e9,e12,e20=e18,e12,e21=e17,e13,e16=e20,e13,e17=e19,e14,e16=e21,e14,e18=e19,e15,e17=e21,e15,e18=e20,e16,e17=e10,e16,e18=e11,e16,e19=e1,e16,e20=e2,e16,e21=e3,e17,e18=e12,e17,e19=e4,e17,e20=e5,e17,e21=e6,e18,e19=e7,e18,e20=e8,e18,e21=e9,e19,e20=e13,e19,e21=e14,e20,e21=e15,R11,R12,R13,R21,R22,R23,R31,R32,R33,R15,R16,R26,R42,R43,R53,R17,R27,R37,R47,R57,R67,σ11,σ12,σ13,σ21,σ22,σ23,σ31,σ32,σ33,σ15,σ16,σ26,σ42,σ43,σ53,σ17,σ27,σ37,σ47,σ57,σ67

(2.11)

DGsetupLD

Lie algebra: so43

(2.12)

 

The program CartanSubalgebra calculates a Cartan subalgebra for so4,3.

so43 > 

CSACartanSubalgebraso43

CSA:=R11,R22,R33

(2.13)

 

Now compute the root space decomposition. We see that each root space is 1-dimensional (Property 1).

so43 > 

RSDRootSpaceDecompositionCSA

RSD:=table1,0,0=R17,1,1,0=R21,0,1,0=R57,0,1,0=R27,0,1,1=R32,1,0,1=R31,0,0,1=R37,0,0,1=R67,1,1,0=R42,0,1,1=R26,1,0,1=R16,1,1,0=R15,1,0,0=R47,0,1,1=R23,1,1,0=R12,1,0,1=R43,1,0,1=R13,0,1,1=R53

(2.14)

RSD:=table1,0,1=R43,0,1,0=R27,0,1,1=R23,0,1,0=R57,1,0,0=R47,1,1,0=R12,0,1,1=R26,1,1,0=R15,0,1,1=R53,1,1,0=R21,0,0,1=R67,1,1,0=R42,0,1,1=R32,1,0,1=R16,1,0,1=R31,1,0,0=R17,0,0,1=R37,1,0,1=R13

(2.15)

 

The roots, given as column vectors, are obtained using LieAlgebraRoots. It is easy to see that the negative of any root is a root (Property 2).

so43 > 

RTmapVector,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0,1,0,1,1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,1,0,1,1,0,1,1,1,0,0,1,1,0,1,0,0,1,1

RT:=011,100,101,110,010,001,011,110,101,110,100,001,101,101,110,011,010,011

(2.16)

 

Here are the eigenvectors or root spaces.

so43 > 

RSseqRSDconvertv,list,v=RT

RS:=R23,R17,R13,R15,R57,R67,R26,R12,R43,R42,R47,R37,R16,R31,R21,R32,R27,R53

(2.17)

 

The 2nd and 12th roots add to give the 13th root. This means that the Lie bracket of the 1st and 7th vectors should be a multiple of the 10th vector (Property 3).

so43 > 

RT2+RT12,RT13

101,101

(2.18)
so43 > 

LieBracketRS2,RS12,RS13

R16,R16

(2.19)

 

The Lie bracket of any root and its negative belongs to the Cartan subalgebra (Property 4).

so43 > 

HLieBracketRSD1,0,1,RSD1,0,1

H:=R11R33

(2.20)

 

 

The Killing form restricted to the positive root space is diagonal (Property 5).

so43 > 

PRPositiveRootsRT,5,3,1

PR:=011,100,101,110,011,110,001,101,010

(2.21)
so43 > 

PSseqRSDconvertv,list,v=PR

PS:=R23,R17,R13,R15,R26,R12,R37,R16,R27

(2.22)
sl3 > 

KillingPS

000000000000000000000000000000000000000000000000000000000000000000000000000000000

(2.23)

See Also

DifferentialGeometry

CartanSubalgebra

LieAlgebraRoots

PositiveRoots

SimpleLieAlgebraData

SimpleLieAlgebraProperties