This means that if we chose a basis for a and extend to a basis  for h, then the components of a complex root in the directions are 0.  If    determines the root space for  then  for  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.

We call the Satake associate of .In the Satake diagram one draws a line connecting each root  to its associate.

Example 1.

Here are a few examples of Satake diagrams.

Example 2.

We make a detailed study of the root structure and the Satake diagram for We shall calculate the simple roots and check that these roots have the properties indicated by the Satake Diagram.

If we ignore the coloring of the dots and the red lines we see that the Dynkin diagram of  coincides with the Dynkin diagram of root type .

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

First we use the command SimpleLieAlgebraData to initialize the Lie algebra

The Lie algebra elements corresponding to the diagonal matrices in the standard representation of define a Cartan subalgebra.

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.

The subalgebra (2.5)  is therefore our subalgebra  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.

The compact roots are:

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.

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.

Here are the simple roots.

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

These are the simple roots we want. First, they give the Cartan matrix in standard form for .

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

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

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.

DifferentialGeometry,  CartanDecomposition, Cartan Involution, CartanSubalgebra, DynkinDiagram, CompactRoots, PositiveRoots, RootSpaceDecomposition, RestrictedRootSpaceDecomposition, SatakeAssociate, SimpleRoots