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
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 = t ⊕p . 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 = h ⋂ p. 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.
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.
LD≔SimpleLieAlgebraDatasu(5, 2),su52,labelformat=gl,labels=E,θ:
DGsetupLD
Lie algebra: su52
For this example we use the command SimpleLieAlgebraProperties to generate the various properties of su5, 2 that we need.
Properties_su52≔SimpleLieAlgebraPropertiessu52:
Here is the Cartan subalgebra.
CSA≔Properties_su52CartanSubalgebra
CSA:=E11,E22,Ei11,Ei22,Ei55,Ei66
Here is the Cartan subalgebra decomposition
T,A≔Properties_su52CartanSubalgebraDecomposition
T,A:=Ei11,Ei22,Ei55,Ei66,E11,E22
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.
K1≔KillingT
LinearAlgebra:-IsDefiniteK1,query=negative_definite
true
K2≔KillingA
The second list of vectors in (2.3) is therefore our subalgebra 𝔞, as described above. Next we find the positive roots.
PT≔Properties_su52PositiveRoots
The compact roots are:
CompactRootsPT,A,CSA
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.
LD≔SimpleLieAlgebraDatasp(4, 4),sp44,labelformat=gl,labels=S,σ:
Lie algebra: sp44
We use the command SimpleLieAlgebraProperties to generate the various properties of sp4,4 that we need.
Properties_sp44≔SimpleLieAlgebraPropertiessp44:
CSA≔Properties_sp44CartanSubalgebra
CSA:=S13,S24,Si11+Si33,Si22+Si44
T,A≔Properties_sp44CartanSubalgebraDecomposition
T,A:=Si11+Si33,Si22+Si44,S13,S24
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.
The second list of vectors in (2.3) is therefore our subalgebra 𝔞, as described above.
Next we find the positive roots.
PT≔Properties_sp44PositiveRoots
See Also
DifferentialGeometry
CartanDecomposition
Cartan Involution
CartanSubalgebra
DynkinDiagram
PositiveRoots
RootSpaceDecomposition
RestrictedRootSpaceDecomposition
SatakeDiagram
SimpleLieAlgebraProperties
SimpleRoots
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