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LieAlgebras[CompactRoots] - find the compact roots in a root system for a non-compact semi-simple real Lie algebra

Calling Sequences

     CompactRoots( Δ,A, CSA)

Parameters

      Δ       - a list of column vectors, defining the root system, positive roots or simple roots of a non-compact semi-simple Lie algebra

       A       - a list of vectors in a Lie algebra, defining a subalgebra of the Cartan subalgebra on which the Killing form is negative-definite

  CSA     - a list of vectors, defining the Cartan subalgebra of a non-compact semi-simple Lie algebra

 

Description

Examples

Description

• 

Let g be a semi-simple real Lie algebra. Then g is called compact if the Killing form of g is negative-definite, otherwise g is called non-compact.  

• 

Every non-compact semi-simple real Lie algebra g admits a Cartan decomposition g = tp . Here t is a subalgebra, p a subspace, [t, p] ⊆ p and [p, p] ⊆ t, that is, t and p define a symmetric pair. Moreover, the Killing form is negative-definite on t and positive-definite on p.

• 

Let h be a Cartan subalgebra for g and let Δ be the associated root system. Set a = hp. Then the set of compact roots is defined to be

Δc = { α Δ | α|𝔞 = 0}.

This means that if we choose a basis a1, a2 , .. . ,asfor a and extend to a basis a1, a2 , .. as, hs+1, ... , hm for h, then the components of a compact root α in the a1, a2 , ... ,as directions are 0. If x  𝔤 determines the root space for α, then adaix = 0 for i = 1... s.  With respect to the standard Cartan algebras for the non-compact, simple matrix algebras we consider here, the compact roots are precisely those which are purely imaginary complex numbers.

• 

In the Satake diagram for a non-compact semi-simple real Lie algebra, the compact roots are given a different color from the other roots.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

We find the compact roots for su5, 2.  First we use the command SimpleLieAlgebraData to initialize the Lie algebra su5, 2.

LDSimpleLieAlgebraDatasu(5, 2),su52,labelformat=gl,labels='E','θ':

DGsetupLD

Lie algebra: su52

(2.1)

 

For this example we use the command SimpleLieAlgebraProperties to generate the various properties of su5, 2 that we need.

su52 > 

Properties_su52SimpleLieAlgebraPropertiessu52:

 

Here is the Cartan subalgebra.

su52 > 

CSAProperties_su52CartanSubalgebra

CSA:=E11,E22,Ei11,Ei22,Ei55,Ei66

(2.2)

 

Here is the Cartan subalgebra decomposition

su52 > 

T,AProperties_su52CartanSubalgebraDecomposition

T,A:=Ei11,Ei22,Ei55,Ei66,E11,E22

(2.3)

 

We check that the restriction of the Killing form to the diagonal matrices T  with imaginary entries is negative-definite. The restriction of the Killing form to the diagonal matrices A with real entries is positive-definite.

su52 > 

K1KillingT

K1:=5628014285601400281414141428

(2.4)
su52 > 

LinearAlgebra:-IsDefiniteK1,query='negative_definite'

true

(2.5)
su52 > 

K2KillingA

K2:=280028

(2.6)

 

The second list of vectors in (2.3)  is therefore our subalgebra 𝔞, as described above.  Next we find the positive roots.

su52 > 

PTProperties_su52PositiveRoots

PT:=11II00,01I2II0,00IIII,00III2I,01I2III,11II00,102III0,010I0I,00002II,010I0I,102IIII,10I00I,01I2III,01I2II0,10I00I,102IIII,020000,102III0,11II00,11II00,200000

(2.7)

 

The compact roots are:

su52 > 

CompactRootsPT,A,CSA

00IIII,00III2I,00002II

(2.8)

2I2I2II00,00I2I00,2I2III00

(2.9)

Note that these roots all have purely imaginary components.

 

Example 2.

We find the compact roots for sp4, 4.  First we use the command SimpleLieAlgebraData to initialize the Lie algebra sp4, 4.

 

LDSimpleLieAlgebraDatasp(4, 4),sp44,labelformat=gl,labels='S','σ':

DGsetupLD

Lie algebra: sp44

(2.10)

 

We use the command SimpleLieAlgebraProperties to generate the various properties of sp4,4 that we need.

sp44 > 

Properties_sp44SimpleLieAlgebraPropertiessp44:

 

Here is the Cartan subalgebra.

sp44 > 

CSAProperties_sp44CartanSubalgebra

CSA:=S13,S24,Si11+Si33,Si22+Si44

(2.11)

 

Here is the Cartan subalgebra decomposition

sp44 > 

T,AProperties_sp44CartanSubalgebraDecomposition

T,A:=Si11+Si33,Si22+Si44,S13,S24

(2.12)

 

The restriction of the Killing form to the diagonal matrices T with imaginary entries is negative-definite. The restriction of the Killing form to the diagonal matrices A with real entries is positive-definite.

sp44 > 

K1KillingT

K1:=400040

(2.13)
sp44 > 

LinearAlgebra:-IsDefiniteK1,query='negative_definite'

true

(2.14)
sp44 > 

K2KillingA

K2:=400040

(2.15)

 

The second list of vectors in (2.3)  is therefore our subalgebra 𝔞, as described above.

 

Next we find the positive roots.

sp44 > 

PTProperties_sp44PositiveRoots

PT:=002I0,11II,0002I,0202I,11II,11II,0200,11II,11II,0202I,11II,11II,11II,202I0,2000,202I0

(2.16)

 

The compact roots are:

sp44 > 

CompactRootsPT,A,CSA

002I0,0002I

(2.17)

See Also

DifferentialGeometry

CartanDecomposition

Cartan Involution

CartanSubalgebra

DynkinDiagram

PositiveRoots

RootSpaceDecomposition

RestrictedRootSpaceDecomposition

SatakeDiagram

SimpleLieAlgebraProperties

SimpleRoots

 


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