find the grading of a semi-simple Lie algebra defined by a set of simple roots or restricted simple roots - Maple Programming Help

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LieAlgebras[GradeSemiSimpleLieAlgebra] - find the grading of a semi-simple Lie algebra defined by a set of simple roots or restricted simple roots

Calling Sequences

    GradeSemiSimpleLieAlgebra(Σ , T1)  

    GradeSemiSimpleLieAlgebra(Σ , T2, method = "non-compact")

Parameters

     Σ       - a list or set of column vectors, defining a subset of the simple roots or a subset of the restricted simple roots

   T1      - a table, with indices that include "RootSpaceDecomposition", "CartanSubalgebra", "SimpleRoots", "PositiveRoots"

     T2      - a table, with indices that include "RestrictedRootSpaceDecomposition", "CartanSubalgebra", "RestrictedSimpleRoots", "RestrictedPositiveRoots"

 

 

Description

Examples

See Also

Description

• 

Let g be a Lie algebra. A grading of g is a (vector space) direct sum decomposition g =  k ℤ𝔤k where 𝔤k , 𝔤l   𝔤k +l . Gradings of semi-simple Lie algebras can easily be constructed from the root space decomposition. Let h be a Cartan subalgebra and 𝔤 = 𝔥 α  ΔRα the associated root space decomposition Let Δ+ be a choice of positive roots and let Δ0  Δ+ be a set of simple roots. Every root α is a sum of simple roots, say α = Σi=1m ai αi , and one defines the height of the root α as htα = Σi=1m ai .

• 

Now let ΣΔ0 be a collection of simple roots and define the Σ height of α as ht Σα = Σi ai , where the sum is taken over those i such that αΣ . Then the subspaces

𝔤t  =α : htΣα =t Rα     and   𝔤0 = 𝔥 α : htΣα =0 Rα

define a (symmetric) grading g =t = k k 𝔤t. 

• 

For real Lie algebras, real gradings can be similarly constructed using the restricted root space decomposition.

• 

The command Query/"Gradation" will test if a given decomposition of a Lie algebra is graded.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

We calculate the various gradations for sl4. We use the command SimpleLieAlgebraData to initialize the Lie algebra.

LDSimpleLieAlgebraDatasl(4),sl4,labelformat=gl,labels=E,ω:

DGsetupLD

Lie algebra: sl4

(2.1)
sl4 > 

PSimpleLieAlgebraPropertiessl4:

 

We use the command SimpleLieAlgebraProperties to create a table T containing the structure properties of sl4.

TSimpleLieAlgebraPropertiessl4:

sl4 > 

SRTSimpleRoots

SR:=110,011,112

(2.2)

 

Here are the possible subsets of the set of simple roots.

sl4 > 

Σ,SR1..1,SR2..2,SR3..3,SR1..2,SR2..3,SR1,SR3,SR

Σ:=,110,011,112,110,011,011,112,110,112,110,011,112

(2.3)

 

Here are the gradings defined by each subset of the simple roots.

sl4 > 

Σ1,GradeSemiSimpleLieAlgebraΣ1,P

,table0=E11,E22,E33,E12,E23,E34,E13,E24,E14,E21,E32,E43,E31,E42,E41

(2.4)
sl4 > 

Σ2,GradeSemiSimpleLieAlgebraΣ2,P

110,table0=E11,E22,E33,E23,E34,E24,E32,E43,E42,1=E12,E13,E14,1=E21,E31,E41

(2.5)
sl4 > 

Σ3,GradeSemiSimpleLieAlgebraΣ3,P

011,table0=E11,E22,E33,E12,E34,E21,E43,1=E23,E13,E24,E14,1=E32,E31,E42,E41

(2.6)
sl4 > 

Σ4,GradeSemiSimpleLieAlgebraΣ4,P

112,table0=E11,E22,E33,E12,E23,E13,E21,E32,E31,1=E34,E24,E14,1=E43,E42,E41

(2.7)
sl4 > 

Σ5,GradeSemiSimpleLieAlgebraΣ5,P

110,011,table0=E11,E22,E33,E34,E43,1=E12,E23,E24,2=E13,E14,2=E31,E41,1=E21,E32,E42

(2.8)
sl4 > 

Σ6,GradeSemiSimpleLieAlgebraΣ6,P

011,112,table0=E11,E22,E33,E12,E21,1=E23,E34,E13,2=E24,E14,2=E42,E41,1=E32,E43,E31

(2.9)
sl4 > 

Σ7,GradeSemiSimpleLieAlgebraΣ7,P

110,112,table0=E11,E22,E33,E23,E32,1=E12,E34,E13,E24,2=E14,2=E41,1=E21,E43,E31,E42

(2.10)
sl4 > 

Σ8,GradeSemiSimpleLieAlgebraΣ8,P

110,011,112,table0=E11,E22,E33,1=E12,E23,E34,2=E13,E24,3=E14,3=E41,2=E31,E42,1=E21,E32,E43

(2.11)
sl4 > 

Σ2,GradeSemiSimpleLieAlgebraΣ2,P

110,table0=E11,E22,E33,E23,E34,E24,E32,E43,E42,1=E12,E13,E14,1=E21,E31,E41

(2.12)

 

The Query command can be used to check that each of these define a grading of sl4.

sl4 > 

G7GradeSemiSimpleLieAlgebraΣ7,P

G7:=table0=E11,E22,E33,E23,E32,1=E12,E34,E13,E24,2=E14,2=E41,1=E21,E43,E31,E42

(2.13)
sl4 > 

QueryG7,Gradation

true

(2.14)

 

Example 2.

We calculate the various gradings for so5,3. We use the command SimpleLieAlgebraData to initialize the Lie algebra.

sl4 > 

LD2SimpleLieAlgebraDataso(5,3),so53,labelformat=gl,labels=R,θ:

sl4 > 

DGsetupLD2

Lie algebra: so53

(2.15)

We use the command SimpleLieAlgebraProperties to calculate the restricted root space decomposition, restricted simple roots, etc.

so53 > 

TSimpleLieAlgebraPropertiesso53:

so53 > 

RSRTRestrictedSimpleRoots

RSR:=110,011,001

(2.16)

 

The subsets of the restricted simple roots are:

so53 > 

ΣRSR,RSR1..2,RSR2..3,RSR1,RSR3,RSR1..1,RSR2..2,RSR3..3,

 

Here are the possible gradings for so5,3.  

so53 > 

Σ1,GradeSemiSimpleLieAlgebraΣ1,T,method=non-compact

110,011,001,table0=R22,R11,R78,R33,1=R12,R23,R37,R38,2=R13,R27,R28,3=R17,R18,R26,5=R15,4=R16,5=R42,4=R43,3=R47,R48,R53,2=R31,R57,R58,1=R21,R32,R67,R68

(2.17)
so53 > 

Σ2,GradeSemiSimpleLieAlgebraΣ2,T,method=non-compact

110,011,table0=R78,R33,R22,R11,R37,R38,R67,R68,1=R26,R27,R28,R12,R23,2=R16,R13,R17,R18,3=R15,3=R42,2=R43,R31,R47,R48,1=R53,R57,R58,R21,R32

(2.18)
so53 > 

Σ3,GradeSemiSimpleLieAlgebraΣ3,T,method=non-compact

011,001,table0=R78,R33,R22,R11,R12,R21,1=R13,R23,R37,R38,2=R17,R18,R27,R28,3=R16,R26,4=R15,4=R42,3=R43,R53,2=R47,R48,R57,R58,1=R31,R32,R67,R68

(2.19)
so53 > 

Σ4,GradeSemiSimpleLieAlgebraΣ4,T,method=non-compact

110,001,table0=R78,R33,R22,R11,R23,R32,1=R13,R27,R28,R12,R37,R38,2=R17,R18,R26,3=R16,R15,3=R43,R42,2=R47,R48,R53,1=R31,R57,R58,R21,R67,R68

(2.20)
so53 > 

Σ5,GradeSemiSimpleLieAlgebraΣ5,T,method=non-compact

110,table0=R78,R33,R22,R11,R26,R27,R28,R23,R37,R38,R53,R57,R58,R32,R67,R68,1=R16,R13,R17,R18,R12,R15,1=R43,R31,R47,R48,R21,R42

(2.21)
so53 > 

Σ6,GradeSemiSimpleLieAlgebraΣ6,T,method=non-compact

011,table0=R78,R33,R22,R11,R12,R37,R38,R21,R67,R68,1=R16,R13,R17,R18,R26,R27,R28,R23,2=R15,2=R42,1=R43,R31,R47,R48,R53,R57,R58,R32

(2.22)
so53 > 

Σ7,GradeSemiSimpleLieAlgebraΣ7,T,method=non-compact

001,table0=R78,R33,R22,R11,R13,R12,R23,R31,R21,R32,1=R17,R18,R27,R28,R37,R38,2=R16,R26,R15,2=R43,R53,R42,1=R47,R48,R57,R58,R67,R68

(2.23)
so53 > 

Σ8,GradeSemiSimpleLieAlgebraΣ8,T,method=non-compact

,table0=R78,R33,R22,R11,R16,R13,R17,R18,R26,R27,R28,R12,R23,R15,R37,R38,R43,R31,R47,R48,R53,R57,R58,R21,R32,R42,R67,R68

(2.24)

 

The Query command can be used to check that each of these define a grading of so5,3.

so53 > 

G1GradeSemiSimpleLieAlgebraΣ1,T,method=non-compact

G1:=table0=R78,R33,R22,R11,1=R12,R23,R37,R38,2=R13,R27,R28,3=R17,R18,R26,5=R15,4=R16,5=R42,4=R43,3=R47,R48,R53,2=R31,R57,R58,1=R21,R32,R67,R68

(2.25)
so53 > 

QueryG1,Gradation

true

(2.26)

 

See Also

DifferentialGeometry,  CartanSubalgebra, KillingForm, LieAlgebras, PositiveRoots, Query, SimpleRoots, RootSpaceDecomposition, RestrictedRootSpaceDecomposition,  SimpleLieAlgebraData, SimpleLieAlgebraProperties