find the sequence of roots through a given root of a semi-simple Lie algebra - Maple Programming Help

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LieAlgebras[RootString] - find the sequence of roots through a given root of a semi-simple Lie algebra

Calling Sequences

     RootString(α, β, Δ, option)

Parameters

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

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

     Δ       - a list of vectors, defining a list of root vectors for a semi-simple Lie algebra and containing α and β

     option       - the keyword argument output = "stringlengths"

  

  

 

Description

Examples

Description

• 

Let 𝔤 be a semi-simple Lie algebra, 𝔥 a Cartan subalgebra, and Δ the associated set of roots. If α, β  Δ, then the α-string through β is the maximal sequence of roots of the form

β  p α,  ... ,  β  2 α, β  α, β, β+α,  β+2 α, ..., β + q α   where p,q are non-negative integers.

• 

The calling sequence RootString(α, β, Δ) returns the α-string of roots through β. The calling sequence RootString(α, β, Δ, output = "stringlengths") returns the list of non-negative integers p, q.

Examples

 with(DifferentialGeometry): with(LieAlgebras):

 

Example 1.

We initialize the split real form of the exceptional Lie algebra g2 and retrieve the root space decomposition and the list of all roots. We then calculate some root strings. The structure equations for g2 are obtained using SimpleLieAlgebraData.

LD := SimpleLieAlgebraData("g(2,Split)", g2);

LD:=e1,e3=2e3,e1,e4=3e4,e1,e5=e5,e1,e6=e6,e1,e7=3e7,e1,e9=2e9,e1,e10=3e10,e1,e11=e11,e1,e12=e12,e1,e13=3e13,e2,e3=e3,e2,e4=2e4,e2,e5=e5,e2,e7=e7,e2,e8=e8,e2,e9=e9,e2,e10=2e10,e2,e11=e11,e2,e13=e13,e2,e14=e14,e3,e4=e5,e3,e5=2e6,e3,e6=3e7,e3,e9=e1,e3,e11=3e10,e3,e12=2e11,e3,e13=e12,e4,e7=e8,e4,e10=e2,e4,e11=e9,e4,e14=e13,e5,e6=3e8,e5,e9=3e4,e5,e10=e3,e5,e11=e13e2,e5,e12=2e9,e5,e14=e12,e6,e9=2e5,e6,e11=2e3,e6,e12=2e13e2,e6,e13=e9,e6,e14=e11,e7,e9=e6,e7,e12=e3,e7,e13=e1e2,e7,e14=e10,e8,e10=e7,e8,e11=e6,e8,e12=e5,e8,e13=e4,e8,e14=e12e2,e9,e10=e11,e9,e11=2e12,e9,e12=3e13,e10,e13=e14,e11,e12=3e14

(2.1)

 

Initialize the Lie algebra with DGsetup.

DGsetup(LD);

Lie algebra: g2

(2.2)

 

The root space decomposition is retrieved, without calculation, using SimpleLieAlgebraProperties

P := SimpleLieAlgebraProperties(g2):

 

Here is the root space decomposition and the list of all positive roots.

RSD := eval(P["RootSpaceDecomposition"]);

RSD:=table1,1=e5,3,2=e4,0,1=e8,3,1=e7,2,1=e3,1,0=e6,0,1=e14,2,1=e9,3,1=e13,3,2=e10,1,0=e12,1,1=e11

(2.3)
g2 > 

Delta := LieAlgebraRoots(RSD);

Δ:=11,32,01,31,21,10,01,21,31,32,10,11

(2.4)

 

Define two roots α and β.

g2 > 

alpha := <2, -1>;

&alpha;:=21

(2.5)
g2 > 

beta := <-3, 2>;

&beta;:=32

(2.6)

 

Calculate the &alpha;-string through &beta;&period;

g2 > 

RootString(alpha, beta, Delta);

32&comma;11&comma;10&comma;31

(2.7)

 

With the optional keyword argument output = "stringlengths", we obtain the lengths of the &alpha;-string through &beta; in the negative and positive directions.

g2 > 

RootString(alpha, beta, Delta, output = "stringlengths");

0&comma;3

(2.8)

 

Thus the &alpha;-string through &beta; in (2.7) is given explicitly by.

g2 > 

[beta, beta + alpha, beta +2*alpha, beta + 3*alpha];

32&comma;11&comma;10&comma;31

(2.9)

Example 2.

Here is another example of a root string for the exceptional Lie algebra g2.

g2 > 

alpha := <2, -1>;

&alpha;:=21

(2.10)
g2 > 

beta := <1, 0>;

&beta;:=10

(2.11)

 

The root &alpha;-string through &beta; is now

g2 > 

RootString(alpha, beta, Delta);

32&comma;11&comma;10&comma;31

(2.12)

 

and the string lengths are

g2 > 

RootString(alpha, beta, Delta, output = "stringlengths");

2&comma;1

(2.13)

Thus, the root string (2.12) is explicitly given by

g2 > 

[beta -2*alpha, beta - alpha, beta, beta + alpha];

32&comma;11&comma;10&comma;31

(2.14)

 

Example 3.

Root strings can also be calculated for abstract roots systems, that is, a set of vectors satisfying the standard axioms of a root system and not explicitly defined from the root space decomposition of a semi-simple Lie algebra. The positive roots of an abstract root systems can be calculated with the PositiveRoots command.

 

g2 > 

AbstractRoots := PositiveRoots("B", 3);

AbstractRoots:=100&comma;010&comma;001&comma;110&comma;011&comma;111&comma;012&comma;112&comma;122

(2.15)
g2 > 

Delta := [seq(-v, v = AbstractRoots), seq(v, v = AbstractRoots)];

&Delta;:=100&comma;010&comma;001&comma;110&comma;011&comma;111&comma;012&comma;112&comma;122&comma;100&comma;010&comma;001&comma;110&comma;011&comma;111&comma;012&comma;112&comma;122

(2.16)

 

Here are are 2 roots and their root string.

g2 > 

alpha := Delta[12];

&alpha;:=001

(2.17)
g2 > 

beta := Delta[11];

&beta;:=010

(2.18)
g2 > 

RootString(alpha, beta, Delta);

010&comma;011&comma;012

(2.19)

 

See Also

DifferentialGeometry

LieAlgebras

CartanSubalgebra

PositiveRoots

SimpleLieAlgebraData

SimpleLieAlgebraProperties

SimpleRoots

RootSpaceDecomposition