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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

Description

•	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

$Δ_{c} = {α \in Δ_{} | α_{| &afr;} = 0} &period;$

This means that if we chose a basis $\{a_{1}, a_{2}, .. &period;, a_{s}\}$ for a and extend to a basis $\{a_{1}, a_{2}, .. &period;, a_{s}, h_{s + 1}, .. &period;, h_{m}\}$ for h, then the components of a complex root in the $\{a_{1}, a_{2}, .. &period;, a_{s}\}$ directions are 0. If $x \in &gfr;$ determines the root space for $α_{},$ then $ad (a_{i}) (x) = 0$ for $i = 1, 2, .. &period;, s &period;$ 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 $Δ^{+} / Δ_{c^{}}^{+}$ are closed under complex conjugation. Let $Δ_{0}$ be the corresponding simple roots and put $Δ_{0 c} = Δ_{0} ⋂ Δ_{c} &period;$ Then for each root $α_{} \in Δ_{0} / Δ_{0 c},$ there is a unique root $α'$ such that

$\overline{α} - α' \in span (Δ_{0}) &period;$

We call $α_{}'$ the Satake associate of $α$ In the Satake diagram one draws a line connecting each root $α_{} \in Δ_{0} / Δ_{0 c}$ to its associate.

•	The command plots the Satake diagram for each of the following real simple Lie algebras: $sl (n), su (p, q), su * (n), so (p, q), so * (n), sp (n, &reals;), sp (p, q), sp (n) &period;$

•	See Details for Satake Diagrams for a complete list of the different types of all Satake diagrams.

Examples

>	$with (DifferentialGeometry) &colon;$ $with (LieAlgebras) &colon;$

Example 1.

Here are a few examples of Satake diagrams.

>	$SatakeDiagram (su(9, 4))$

>	$SatakeDiagram (su(6, 6))$

>	$SatakeDiagram (sp(10, 6))$

>	$SatakeDiagram (so(12, 4))$

Example 2.

We make a detailed study of the root structure and the Satake diagram for $su (6, 2) &period;$ We shall calculate the simple roots and check that these roots have the properties indicated by the Satake Diagram.

>	$SatakeDiagram (su(6, 2))$

If we ignore the coloring of the dots and the red lines we see that the Dynkin diagram of $su (6, 2)$ coincides with the Dynkin diagram of root type $A_{7}$ .

>	$DynkinDiagram (A, 7)$

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 $su (6, 2) &period;$

First we use the command SimpleLieAlgebraData to initialize the Lie algebra $su (6, 2) &period;$

>	$LD ≔ SimpleLieAlgebraData (su(6, 2), su62, labelformat = gl, labels = ['E','θ']) &colon;$

>	$DGsetup (LD)$

$Lie algebra: su62$

(2.1)

The Lie algebra elements corresponding to the diagonal matrices in the standard representation of $su (6, 2)$ define a Cartan subalgebra.

su62 >

$CSA ≔ [E11, E22, Ei11, Ei22, Ei55, Ei66, Ei77]$

$CSA := [E11, E22, Ei11, Ei22, Ei55, Ei66, Ei77]$

(2.2)

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.

su62 >

$K1 ≔ Killing ([Ei11, Ei22, Ei55, Ei66, Ei77])$

su62 >

$LinearAlgebra :- IsDefinite (K1, query ='negative_definite')$

$true$

(2.3)

su62 >

$A ≔ [E11, E22]$

$A := [E11, E22]$

(2.4)

su62 >

$K2 ≔ Killing (A)$

The subalgebra (2.4) is therefore our subalgebra $&afr;,$ 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.

su62 >

$# RSD := RootSpaceDecomposition(CSA):$

su62 >

$RSD ≔ map (evalDG, table ([[0, 1, 0, I, 0, 0, I] = E28 - I Ei28, [1, 1, I, - I, 0, 0, 0] = E14 + I Ei14, [1, 0, 3 I, 2 I, - I, 0, 0] = E15 - I Ei15, [0, - 1, 0, - I, 0, - I, I] = E47 + I Ei47, [0, 0, 2 I, 2 I, - I, - I, I] = E57 - I Ei57, [0, 0, 0, 0, I, - I, - I] = E68 - I Ei68, [- 1, - 1, I, - I, 0, 0, 0] = E32 + I Ei32, [0, - 1, 0, I, I, - I, 0] = E46 - I Ei46, [2, 0, 0, 0, 0, 0, 0] = Ei13, [0, - 1, 0, - I, - I, I, 0] = E46 + I Ei46, [0, 0, 0, 0, 0, - I, 2 I] = E78 + I Ei78, [- 1, 0, - 3 I, - 2 I, I, 0, 0] = E35 + I Ei35, [- 1, - 1, - I, I, 0, 0, 0] = E32 - I Ei32, [0, 0, 0, 0, - I, I, I] = E68 + I Ei68, [- 1, 0, - I, 0, 0, - I, I] = E37 + I Ei37, [1, - 1, - I, I, 0, 0, 0] = E12 + I Ei12, [0, 2, 0, 0, 0, 0, 0] = Ei24, [- 1, 1, - I, I, 0, 0, 0] = E21 - I Ei21, [0, 0, 2 I, 2 I, - I, 0, - I] = E58 - I Ei58, [0, - 1, 0, I, 0, I, - I] = E47 - I Ei47, [0, 0, 0, 0, I, - 2 I, I] = E67 - I Ei67, [0, 1, 0, - I, 0, 0, - I] = E28 + I Ei28, [0, 0, 0, 0, 0, I, - 2 I] = E78 - I Ei78, [- 1, 0, I, 0, I, - I, 0] = E36 - I Ei36, [- 1, 0, - I, 0, - I, I, 0] = E36 + I Ei36, [1, 0, I, 0, 0, 0, I] = E18 - I Ei18, [- 1, 0, - I, 0, 0, 0, - I] = E38 + I Ei38, [0, 0, 2 I, 2 I, - 2 I, I, 0] = E56 - I Ei56, [0, - 2, 0, 0, 0, 0, 0] = Ei42, [0, - 1, 2 I, 3 I, - I, 0, 0] = E45 - I Ei45, [- 1, 1, I, - I, 0, 0, 0] = E21 + I Ei21, [0, 1, 0, - I, 0, - I, I] = E27 + I Ei27, [0, 1, 2 I, 3 I, - I, 0, 0] = E25 - I Ei25, [1, 0, - I, 0, 0, - I, I] = E17 + I Ei17, [0, - 1, 0, I, 0, 0, I] = E48 - I Ei48, [- 2, 0, 0, 0, 0, 0, 0] = Ei31, [0, 1, 0, I, 0, I, - I] = E27 - I Ei27, [1, - 1, I, - I, 0, 0, 0] = E12 - I Ei12, [0, 1, - 2 I, - 3 I, I, 0, 0] = E25 + I Ei25, [- 1, 0, I, 0, 0, 0, I] = E38 - I Ei38, [1, 0, - 3 I, - 2 I, I, 0, 0] = E15 + I Ei15, [0, - 1, 0, - I, 0, 0, - I] = E48 + I Ei48, [1, 0, I, 0, 0, I, - I] = E17 - I Ei17, [0, 1, 0, - I, - I, I, 0] = E26 + I Ei26, [0, 1, 0, I, I, - I, 0] = E26 - I Ei26, [0, 0, - 2 I, - 2 I, I, I, - I] = E57 + I Ei57, [- 1, 0, I, 0, 0, I, - I] = E37 - I Ei37, [1, 0, - I, 0, 0, 0, - I] = E18 + I Ei18, [1, 0, - I, 0, - I, I, 0] = E16 + I Ei16, [0, 0, - 2 I, - 2 I, I, 0, I] = E58 + I Ei58, [- 1, 0, 3 I, 2 I, - I, 0, 0] = E35 - I Ei35, [1, 1, - I, I, 0, 0, 0] = E14 - I Ei14, [1, 0, I, 0, I, - I, 0] = E16 - I Ei16, [0, 0, - 2 I, - 2 I, 2 I, - I, 0] = E56 + I Ei56, [0, - 1, - 2 I, - 3 I, I, 0, 0] = E45 + I Ei45, [0, 0, 0, 0, - I, 2 I, - I] = E67 + I Ei67])) &colon;$

su62 >

$RT ≔ LieAlgebraRoots (RSD)$

The compact roots are:

su62 >

$CompactRoots (RT, A, CSA)$

Note that these roots all have pure imaginary components.

Now find a set of positive roots. We use the second calling sequence for the command PositiveRoots. For this we need a basis for the space of roots for which the components are all real.

su62 >

$B ≔ [⟨1, 0, 0, 0, 0, 0, 0⟩, ⟨0, 1, 0, 0, 0, 0, 0⟩, ⟨0, 0, I, 0, 0, 0, 0⟩, ⟨0, 0, 0, I, 0, 0, 0⟩, ⟨0, 0, 0, 0, I, 0, 0⟩, ⟨0, 0, 0, 0, 0, I, 0⟩, ⟨0, 0, 0, 0, 0, 0, I⟩]$

su62 >

$PR ≔ PositiveRoots (RT, B)$

We check (using Query) that the non-compact roots are closed under complex conjugation. This is important if we hope to get a set of simple roots which match with the Satake diagram.

su62 >

$NC ≔ CompactRoots (PR, A, CSA, output = noncompact)$

su62 >

$Query (NC, ClosedUnderConjugation)$

$true$

(2.5)

Here are the simple roots.

su62 >

$SR ≔ SimpleRoots (PR)$

Finally, we have to put the simple roots in the order that renders the Cartan matrix in standard form. For this we use CartanMatrixToStandardForm.

su62 >

$CM ≔ CartanMatrix (SR, RSD)$

su62 >

$newCM, newSR, RootType ≔ CartanMatrixToStandardForm (CM, SR)$

su62 >

$newSR$

These are the simple roots we want. First, they give the Cartan matrix in standard form for $A_{7}$ .

su62 >

$CartanMatrix (newSR, RSD)$

Second, we see that the positive roots are given in terms of the simple roots by the correct set of linear combinations for $A_{7}$ .

su62 >

$C1 ≔ GetComponents (PR, newSR)$

$C1 := [[0, 0, 1, 1, 1, 1, 0], [0, 0, 1, 0, 0, 0, 0], [1, 1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 1, 1, 0, 0], [1, 1, 1, 1, 1, 1, 0], [0, 1, 1, 1, 1, 1, 0], [1, 1, 1, 1, 1, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0], [1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0], [1, 1, 1, 1, 1, 1, 1], [0, 1, 1, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0], [1, 1, 1, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1, 1], [0, 0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1, 0], [0, 0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 0, 1], [0, 1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 1, 0, 0]]$

(2.6)

su62 >

$C2 ≔ map (convert, PositiveRoots (A, 7), list)$

$C2 := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 1, 1], [1, 1, 1, 0, 0, 0, 0], [0, 1, 1, 1, 0, 0, 0], [0, 0, 1, 1, 1, 0, 0], [0, 0, 0, 1, 1, 1, 0], [0, 0, 0, 0, 1, 1, 1], [1, 1, 1, 1, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 1, 0], [0, 0, 0, 1, 1, 1, 1], [1, 1, 1, 1, 1, 0, 0], [0, 1, 1, 1, 1, 1, 0], [0, 0, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 0], [0, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1]]$

(2.7)

su62 >

$is (convert (C1, set) = convert (C2, set))$

$true$

(2.8)

Third, in accordance with the Satake diagram, the middle 3 roots are compact.

su62 >

$newSR [3 .. 5]$

And fourth, the Satake associate of the 1st root is the 7th root and the Satake associate of the 2nd root is the 6th root.

su62 >

$SatakeAssociate (SR [1], SR), SR [7]$

su62 >

$SatakeAssociate (SR [2], SR), SR [6]$

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