Let Δ be the root system for a non-compact, simple Lie algebra. Let be a set of positive roots,  the compact roots,  be the simple roots and the compact simple roots. We chose the positive roots to be closed under complex conjugation. Then for each root  there is a unique root such that The root  is called the Satake associate of .

Example 1.

Here is the Satake diagram for and the corresponding simple roots.

All the roots are non-compact so that the Satake associate is just the complex conjugate, for example,

The root is its own associate.

Example 2

Here is the Satake diagram for and the corresponding simple roots.

Roots and are compact.  The root is real and is therefore its own Satake associate.  The root satisfies

and is therefore also its own Satake associate.

Example 3.

Here is the Satake diagram for and the corresponding simple roots.

There are no compact roots. The roots and are real and therefore are their own Satake associates.  Because there are no compact roots the Satake associate of is its complex conjugate which is

Example 4.

Here is the Satake diagram for and the corresponding simple roots.

The roots and are compact.  Since

the Satake associate of  is itself. Since

the Satake associate of  is

These calculations agree with the output of the command SatakeAssociate.

DifferentialGeometry, CompactRoots, Details for Satake Diagram, DynkinDiagram, LieAlgebras, PositiveRoots, Simple Roots, SatakeDiagram, SimpleLieAlgebraData, SimpleLieAlgebraProperties