Many thanks, Jim! My question was mainly for good references, to find out how well this spaces are understood.

Thank you very much, Allen and Robert, your explanations are very helpful for me!

processes has only one parameter. Of course we already know that this is true more generally for simple compact Lie groups, it has to be a multiple of the Laplace-Beltrami operator.

Yes, thank you! I was hoping for a more "geometric" description of this set. Say we are interested in convolution semigroups of central probability measures (or, equivalently, conjugate invariant Levy processes). For this I have to study how I can move move away from the identity. The descriptions above for SU(2) and SU(3) show that there is only one possible direction to leave the identity without jumping (the identity is the boundary point to the right, i.e. 2 for SU(2) and (3,0) for SU(3)). This explains intuitively why the diffusion part in the generator of conjugate invariant Levy

Too bad I missed the party. I think this is great question! And I can't understand the reaction of some people here and on Physics.SE. Every serious mathematician, physicist, or chemist is aware of the merits of Mendeleev's table and its experimental validation. And every serious mathematician, physicist, or chemist knows similarly well that Quantum Mechanics is an idealisation. Furthermore we can not even give a closed solution for the helium atom. So I think it is a good test of QM and our mathmatical or computational skills, if we can derive the properties of M's table from SE.