Sorry for my late reply... I think I'll start by contributing something modest: maybe classifying the real Lie algebras, as I mentioned above. While I do that, I will take a deeper look at the physics module and see if there already exists a quick "cookbook" for commonly found representations in physics.
On Saturday, February 11, 2017 at 12:52:24 PM UTC-6, Francesco Bonazzi wrote: > > Hello and welcome, > > On Friday, 10 February 2017 18:54:27 UTC+1, Ian George wrote: >> >> >> The Lie Algebra module only seems to handle the A-G groups. Wouldn't it >> be prudent to add on SO(3), SU(2), U(1), stuff that tends to be used by >> physicists/representation theorists more often? I'm checking out the >> physics module, too, but so far haven't found anything. >> >> > The module *sympy.liealgebras* provides tools for the classification of > complex Lie algebras. They are not very useful for practical applications > in physics, indeed. > > Technically, the exponential already supports matrices, so basically if > you want to work with a Lie algebra, you can currently use matrices. Of > course, it would be nicer to have a real support for Lie algebra objects. > > The other question is how to represent Lie algebras in a computer algebra > system as SymPy: which level of abstraction choose? > > For example, one could: > > 1. just use matrices > 2. use matrices and create supporting tools like Lie brackets, etc. > 3. Use abstract vectors for the base of the Lie algebra generators (no > matrices). > > Also take into account that we already have a differential geometry module > (*sympy.diffgeom*). > > > Lie Groups are manifolds in differential geometry with the addition of the > Lie bracket operations, while Lie algebras are their tangent space. > > > The differential geometry module already supports differentiable manifold > nicely. One point could be to extend it. There was even one paper once > about how to represent a Lie group differentiable manifold in a CAS. > > > Connection from point 3 to points 1,2 could be handled by simple > replacement/substitution operations. > > As a side note, that might be a good first step toward contributing, >> seeing if you can do something with representation theory here. Of course, >> I've got to brush up on that, before I even consider it, having been away >> from physics for a while... >> > > That's that the abstract representation theory hardest approach you could > get towards Lie algebras in a CAS. Furthermore, I think it's the less > useful for end users. > > I guess that complex Lie algebras have already been classified, to extend > it (apart from currently missing feature) one could start classifying the > real Lie algebras associated with the complex ones, then their > representations (I wouldn't do it... doubt end users need this feature). > > Of course representations could also be dealt in a quick way: create a map > associating vectors with matrices and add some recursive relations to > explore all representations. This could be useful, like getting the > rotation generators when needed and stuff like that. I would still try to > reason on how to integrate representations of Lie groups and algebras and > the differential geometry module. > > -- 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/0e8a8416-5353-4e71-8369-c232224f222d%40googlegroups.com. For more options, visit https://groups.google.com/d/optout.
