GSoC 2017: Lie groups in SymPy
Maria Zameshina
Hello everyone!
I have a project in mind for construction of a module in SymPy for being able to distinguish and classify Lie groups efficiently. In general, a project in these lines evokes quite some interests for me, as I like the idea of geometric viewpoints for groups very much, and this feature especially manifests in the context of Lie groups. I have attended the Lie groups course in Independent University of Moscow and I liked it. So, I would be very happy if I can get a chance to work on the following schematic proposals. I would also be extremely glad to get feedbacks from you in this regard.
As is known, all the Lie groups can be classified starting from SU(2). For building Lie groups we will use a standard gadget, the so-called Dynkin diagrams.Starting from the SU(2) which is typically represented by a circle, one can build up higher Lie groups by attaching such circles using various lines. For example, SU(3) is represented by two such circles attached by a single line.
In fact, removing a few lines and circles, one can identify the subgroups thereof. Furthermore, from the symmetries of the diagrams, it is simpler to identify the (outer)- automorphisms of the groups. For example, one of the most symmetric representations come from the dihedral group, from the diagram of which, it becomes quite clear that it has order 6 automorphism given by permutation of 3 letters, or S_3.
Of course, these are rather simple examples. One needs to find such representations, for more complicated groups. But, given such diagrams, one can keep on retaining tracks of building them up, and from find appropriate isomorphisms for example.
This is important, because up to isomorphism, all simple Lie groups can be classified into categories called, classical Lie algebras and exceptional Lie algebras. So, identifying isomorphisms is an important problem on its own. To achieve this, Dynkin diagrams come in
as very useful objects.
I have a somewhat sketchy ideas about executing these tasks. So, first of all, we will have to realise conception of Dynkin diagrams, and after it we will build up all the Lie groups on it. Then we should build up Dynkin diagrams for known classical and exceptional cases. We should check, if our algorithms really work at these stage by checking against these cases. Then, we should find how to identify isomorphisms. Standard techniques of removing/attaching nodes and lines exist. One should finds ways to efficiently implement them or find better algorithms than the existing ones. Once these have been achieved, the goal will be to identify automorphisms by looking at the symmetry of diagrams.
Finally, if there is time, I would also like to work on complex Lie (semi-simple) algebras (the real Lie algebras are determined as the real forms of them). These are classified by Satake diagrams which are further generalizations of Dynkin diagrams. In a nutshell, one also attaches some filled (black) circles and add arrowed edges according to some specified rules.
I would also like to know if anyone in the work group is interested in mentoring this project. I could not find anyone in the list. It would be very helpful, if you could direct me to someone who is willing to be a mentor for this project, or any similar project related to this.
Kalevi Suominen
Kalevi Suominen
Jason Moore
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