find the coroot of a root vector for a semi-simple Lie algebra - Maple Help

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LieAlgebras[CoRoot] - find the coroot of a root vector for a semi-simple Lie algebra

Calling Sequences

     CoRoot(α, CSA, option)

Parameters

     α        - a vector, defining a root vector for a semi-simple Lie algebra

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

     option   - an r ×r non-singular matrix, defining the restriction of the Killing form to the Cartan subalgebra

  

 

Description

Examples

Description

• 

 Let 𝔤 be a semi-simple Lie algebra, 𝔥 a Cartan subalgebra, and Δ the associated set of roots. Let B be the Killing form. If α  Δ, then the coroot of α is the unique vector Tα 𝔥 such that αx = BTα,x. Let h1, h2 , ... , hr be a basis for 𝔥, αhi = ai,and bij = Bhi, hj with inverse bij. Then Tα= tihi, where ti = bijaj.

• 

The calling sequence CoRoot(α, CSA) returns the vector Tα.

• 

In a situation involving repeated calls to CoRoot, efficiency can be dramatically improved by using the optional 3rd argument to specify the restriction of the Killing form.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

We use the command SimpleLieAlgebraData to retrieve the structure equations for the rank 3 Lie algebra sl4, we initialize this algebra, and we calculate the coroots of several root vectors.

 

LDSimpleLieAlgebraDatasl(4),sl4

LD:=e1,e4=e4,e1,e5=e5,e1,e6=2e6,e1,e7=e7,e1,e9=e9,e1,e10=e10,e1,e12=e12,e1,e13=2e13,e1,e14=e14,e1,e15=e15,e2,e4=e4,e2,e6=e6,e2,e7=e7,e2,e8=e8,e2,e9=2e9,e2,e11=e11,e2,e12=e12,e2,e13=e13,e2,e14=2e14,e2,e15=e15,e3,e5=e5,e3,e6=e6,e3,e8=e8,e3,e9=e9,e3,e10=e10,e3,e11=e11,e3,e12=2e12,e3,e13=e13,e3,e14=e14,e3,e15=2e15,e4,e7=e1e2,e4,e8=e5,e4,e9=e6,e4,e10=e11,e4,e13=e14,e5,e7=e8,e5,e10=e1e3,e5,e11=e4,e5,e12=e6,e5,e13=e15,e6,e7=e9,e6,e10=e12,e6,e13=e1,e6,e14=e4,e6,e15=e5,e7,e11=e10,e7,e14=e13,e8,e10=e7,e8,e11=e2e3,e8,e12=e9,e8,e14=e15,e9,e11=e12,e9,e13=e7,e9,e14=e2,e9,e15=e8,e10,e15=e13,e11,e15=e14,e12,e13=e10,e12,e14=e11,e12,e15=e3

(2.1)

DGsetupLD

Lie algebra: sl4

(2.2)

 

We obtain the Cartan subalgebra and the positive roots using SimpleLieAlgebraProperties

M > 

PSimpleLieAlgebraPropertiessl4:

sl4 > 

CSAPCartanSubalgebra

CSA:=e1,e2,e3

(2.3)
sl4 > 

ΔPPositiveRoots

Δ:=110,011,112,101,121,211

(2.4)

 

Calculate the coroot for the first root Δ1.

sl4 > 

αΔ1

α:=110

(2.5)
sl4 > 

CoRootα,CSA

18e118e2

(2.6)

 

Calculate the coroot for the last root Δ6.

sl4 > 

βΔ1

β:=211

(2.7)
sl4 > 

CoRootβ,CSA

18e1

(2.8)

 

Example 2.

We repeat the calculation the first coroot from Example 1 using the optional calling sequence. The restriction of the Killing form to the Cartan subalgebra is needed.

sl4 > 

BKillinge1,e2,e3

B:=168881688816

(2.9)
sl4 > 

CoRootα,CSA,B

18e118e2

(2.10)

 

Here is the same computation in components.

sl4 > 

TB1.α

T:=18180

(2.11)

 

See Also

DifferentialGeometry

LieAlgebras

Killing

SimpleLieAlgebraData

SimpleLieAlgebraProperties

RootSpaceDecomposition

PositiveRoots

 


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