LieAlgebras[SatakeDiagram] - display the Satake diagram for a non-compact, real, simple matrix algebra
Calling Sequences
SatakeDiagram(AT)
Parameters
AT - a string, specifying the type of a classical, non-compact, real simple matrix algebra
Description
Examples
The Satake diagram for a non-compact, real, semi-simple algebra g is a refinement of Dynkin diagram for the associated complex Lie algebra. We describe the construction of the Satake diagram as follows.
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 ,real, semi-simple algebra g admits a Cartan decomposition g = t ⊕p. In this vector space decomposition t is a subalgebra, p a subspace, [t, p] ⊆ p and [p, p] ⊆ t, that is, t and p define a symmetry 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 chose a basis a1, a2 , ... , asfor a and extend to a basis a1, a2 , ... , as, hs+1, ... , hm for h, then the components of a complex root in the a1, a2 , ... , as directions are 0. If x ∈ 𝔤 determines the root space for α, then adaix = 0 for i = 1,2, ... ,s. With respect to the standard Cartan algebras for the non-compact, real, simple matrix algebras we consider here, the compact roots are precisely those which are pure imaginary.
In a Satake diagram, the complex roots are designed by a solid black circle, the other roots are designed by a circle. Sometimes, the compact roots appear adjacent to one-another; for other algebras the compact roots alternate with the non-compact roots.
There is one more important piece of information encoded in the Satake diagrams. For this one needs to choose a set of positive roots Δ+such that non-compact positive roots Δ+/Δc+ are closed under complex conjugation. Let Δ0 be the corresponding simple roots and put Δ0 c = Δ0 ⋂Δc . Then for each root α∈Δ0 /Δ0 c, there is a unique root α' such that
α‾ − α'∈spanΔ0.
We call α' the Satake associate of α In the Satake diagram one draws a line connecting each root α∈Δ0 /Δ0 c to its associate.
The command plots the Satake diagram for each of the following real simple Lie algebras: sln, sup, q, su∗n, sop, q, so∗n, spn, ℝ, spp, q, spn.
See Details for Satake Diagrams for a complete list of the different types of all Satake diagrams.
withDifferentialGeometry:withLieAlgebras:
Example 1.
Here are a few examples of Satake diagrams.
SatakeDiagramsu(9, 4)
SatakeDiagramsu(6, 6)
SatakeDiagramsp(10, 6)
SatakeDiagramso(12, 4)
Example 2.
We make a detailed study of the root structure and the Satake diagram for su6, 2. We shall calculate the simple roots and check that these roots have the properties indicated by the Satake Diagram.
SatakeDiagramsu(6, 2)
If we ignore the coloring of the dots and the red lines we see that the Dynkin diagram of su6, 2 coincides with the Dynkin diagram of root type A7.
DynkinDiagramA,7
According to the Satake diagram we see that there are 3 compact roots which appear adjacent to each. Each non-compact root has a Satake associate different from itself.
Let us verify these facts by explicitly constructing the simple roots for su6, 2.
First we use the command SimpleLieAlgebraData to initialize the Lie algebra su6, 2.
LD≔SimpleLieAlgebraDatasu(6, 2),su62,labelformat=gl,labels=E,θ:
DGsetupLD
Lie algebra: su62
The Lie algebra elements corresponding to the diagonal matrices in the standard representation of su6,2define a Cartan subalgebra.
CSA≔E11,E22,Ei11,Ei22,Ei55,Ei66,Ei77
CSA:=E11,E22,Ei11,Ei22,Ei55,Ei66,Ei77
The restriction of the Killing form to the diagonal matrices with imaginary entries is negative-definite. The restriction of the Killing form to the diagonal matrices with real entries is positive-definite.
K1≔KillingEi11,Ei22,Ei55,Ei66,Ei77
K1:=−64−3201616−32−640161600−32−16−161616−16−32−161616−16−16−32
LinearAlgebra:-IsDefiniteK1,query=negative_definite
true
A≔E11,E22
A:=E11,E22
K2≔KillingA
K2:=320032
The subalgebra (2.5) is therefore our subalgebra 𝔞, as described above. Note that we have listed the elements of a first in the basis for the Cartan subalgebra.
Next we find the root space decomposition, the root system, and the positive roots. The root space decomposition is computed using the command RootSpaceDecomposition. The root system is then obtained using the LieAlgebraRoots command. For efficiency, we have saved the result we need.
# RSD := RootSpaceDecomposition(CSA):
RSD≔mapevalDG,table0,1,0,I,0,0,I=E28−IEi28,1,1,I,−I,0,0,0=E14+IEi14,1,0,3I,2I,−I,0,0=E15−IEi15,0,−1,0,−I,0,−I,I=E47+IEi47,0,0,2I,2I,−I,−I,I=E57−IEi57,0,0,0,0,I,−I,−I=E68−IEi68,−1,−1,I,−I,0,0,0=E32+IEi32,0,−1,0,I,I,−I,0=E46−IEi46,2,0,0,0,0,0,0=Ei13,0,−1,0,−I,−I,I,0=E46+IEi46,0,0,0,0,0,−I,2I=E78+IEi78,−1,0,−3I,−2I,I,0,0=E35+IEi35,−1,−1,−I,I,0,0,0=E32−IEi32,0,0,0,0,−I,I,I=E68+IEi68,−1,0,−I,0,0,−I,I=E37+IEi37,1,−1,−I,I,0,0,0=E12+IEi12,0,2,0,0,0,0,0=Ei24,−1,1,−I,I,0,0,0=E21−IEi21,0,0,2I,2I,−I,0,−I=E58−IEi58,0,−1,0,I,0,I,−I=E47−IEi47,0,0,0,0,I,−2I,I=E67−IEi67,0,1,0,−I,0,0,−I=E28+IEi28,0,0,0,0,0,I,−2I=E78−IEi78,−1,0,I,0,I,−I,0=E36−IEi36,−1,0,−I,0,−I,I,0=E36+IEi36,1,0,I,0
