Maria, If your proposal is good we will find a mentor. Just focus on making the proposal strong.
Jason moorepants.info +01 530-601-9791 On Sat, Mar 25, 2017 at 7:07 AM, Kalevi Suominen <jks...@gmail.com> wrote: > 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 sympy+unsubscr...@googlegroups.com. > To post to this group, send email to sympy@googlegroups.com. > 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 > <https://groups.google.com/d/msgid/sympy/0641a57d-56e1-4c53-8692-08275288cd6c%40googlegroups.com?utm_medium=email&utm_source=footer> > . > > For more options, visit https://groups.google.com/d/optout. > -- You received this message because you are subscribed to the Google Groups "sympy" group. 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