There is a module on complex Lie algebras in SymPy that also deals with Dynkin diagrams. The possibility of adding real Lie algebras was discussed in this thread <https://groups.google.com/forum/#!topic/sympy/lbTwNgehbWk> some time ago.
Kalevi Suominen On Saturday, March 25, 2017 at 4:49:31 AM UTC+2, Maria Zameshina wrote: > > 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. > -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/0641a57d-56e1-4c53-8692-08275288cd6c%40googlegroups.com. For more options, visit https://groups.google.com/d/optout.
