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LieAlgebras[SatakeAssociate] - find the non-compact simple root associated to a given non-compact root in the Satake diagram

Calling Sequences

     SatakeAssociate(α, Δ0, Δc )

Parameters

     α     - a column vector, a non-compact root of a non-compact, simple Lie algebra

     Δ0    - a list of column vectors, the simple roots of a non-compact simple Lie algebra

     Δc    - (optional) a list of column vectors, defining the compact roots of non-compact simple Lie algebra

 

Description

Examples

Description

• 

Let Δ be the root system for a non-compact, simple Lie algebra. Let Δ+be a set of positive roots, Δc the compact roots, Δ0 be the simple roots and Δ0 c the compact simple roots. We chose the positive roots to be closed under complex conjugation. Then for each root αΔ0 /Δ0 c, there is a unique root α' Δ0 /Δ0 csuch that αα'spanΔ0. The root α' is called the Satake associate of α.

• 

The command SatakeAssociate(α, Δ0, Δc ) returns the Satake associate of α.

Examples

with(DifferentialGeometry): with(LieAlgebras):

 

Example 1.

Here is the Satake diagram for su4, 4 and the corresponding simple roots.

 

 

SatakeDiagram("su(4, 4");

alpha1, alpha2, alpha3, alpha4, alpha5, alpha6, alpha7 := seq(Vector(v), v = [[-2*I, I, 0, 1, -1, 0, 0], [I, -2*I, I, 0, 1, -1, 0], [0, I, -2*I, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0, 2], [0, -I, 2*I, 0, 0, 1, -1], [-I, 2*I, -I, 0, 1, -1, 0], [2*I, -I, 0, 1, -1, 0, 0]]);

α1,α2,α3,α4,α5,α6,α7:=2II01100,I2II0110,0I2I0011,0000002,0I2I0011,I2II0110,2II01100

(2.1)

Delta0 := [alpha1, alpha2, alpha3, alpha4, alpha5, alpha6, alpha7]:

 

All the roots are non-compact so that the Satake associate is just the complex conjugate, for example,

su44 > 

alpha1, SatakeAssociate(alpha1, Delta0), alpha7;

2II01100,2II01100,2II01100

(2.2)

 

The root α4 is its own associate.

SatakeAssociate(alpha4, Delta0), alpha4;

0000002,0000002

(2.3)

 

Example 2

Here is the Satake diagram for so7, 2 and the corresponding simple roots.

 

SatakeDiagram("so(7,2)");

alpha1, alpha2, alpha3, alpha4 := seq(Vector(v), v = [[1, -1, 0, 0], [0, 1, -I, 0], [0, 0, I, -I], [0, 0, 0, I]]);

α1,α2,α3,α4:=1100,01I0,00II,000I

(2.4)

Delta0 := [alpha1, alpha2, alpha3, alpha4]:

 

Roots α3 and α4 are compact. The root α1 is real and is therefore its own Satake associate. The root α2 satisfies

map(conjugate, alpha2) - alpha2, 2*alpha3 + 2*alpha4;

002I0,002I0

(2.5)

and is therefore also its own Satake associate.

 

Example 3.

Here is the Satake diagram for so5, 3 and the corresponding simple roots.

SatakeDiagram("so(5, 3)");

alpha1, alpha2, alpha3, alpha4 := seq(Vector(v), v = [[1, -1, 0, 0], [0, 1, -1, 0], [0, 0, 1, -I], [0, 0, 1, I]]);

α1,α2,α3,α4:=1100,0110,001I,001I

(2.6)

Delta0 := [alpha1, alpha2, alpha3, alpha4]:

 

There are no compact roots. The roots α1 and α2 are real and therefore are their own Satake associates. Because there are no compact roots the Satake associate of α3 is its complex conjugate which is α4.

alpha3, SatakeAssociate(alpha3, Delta0), alpha4;

001I,001I,001I

(2.7)

 

Example 4.

Here is the Satake diagram for so*10 and the corresponding simple roots.

SatakeDiagram("so*(10)");

alpha1, alpha2, alpha3, alpha4, alpha5 := seq(Vector(v), v = [[0, 0, 2*I, 0, 0], [1, -1, -I, -I, 0], [0, 0, 0, 2*I, 0], [0, 1, 0, -I, -I], [0, 1, 0, -I, I]]);

α1,α2,α3,α4,α5:=002I00,11II0,0002I0,010II,010II

(2.8)

 

The roots α1 and α3 are compact. Since

map(conjugate, alpha2) - alpha2 = alpha1 + alpha3;

002I2I0=002I2I0

(2.9)

the Satake associate of  α2  is itself. Since

map(conjugate, alpha4) - alpha5 = alpha3;

0002I0=0002I0

(2.10)

the Satake associate of α4 is α5.   

 

These calculations agree with the output of the command SatakeAssociate.

alpha2, SatakeAssociate(alpha2, [alpha1, alpha2, alpha3, alpha4, alpha5]);

11II0,11II0

(2.11)

alpha4, SatakeAssociate(alpha4, [alpha1, alpha2, alpha3, alpha4, alpha5]), alpha5;

010II,010II,010II

(2.12)

 

See Also

DifferentialGeometry

CompactRoots

Details for Satake Diagram

DynkinDiagram

LieAlgebras

PositiveRoots

Simple Roots

SatakeDiagram

SimpleLieAlgebraData

SimpleLieAlgebraProperties