SO(3) and SU(2) lie algebras

[Ian George]

Hello,

So, first, I'm just getting started with Sympy, so if I'm missing something obvious, forgive me.

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.

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...

[Kalevi Suominen]

I could not find it explicitly pointed out, but it seems to me too that only complex simple Lie algebras have been dealt with. It would be useful to have an implementation of real simple Lie algebras as well. You are welcome to contribute.

Kalevi Suominen

[Francesco Bonazzi]

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.

[Ian George]

That sounds like a good, achievable idea. I'll look into it. My group theory is a little rusty, but not too bad.

[Ian George]

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.