Subject: GSoC Proposal: Implement Representations of Lie Algebras in SageMath

[kshipra wadikar]

Hi SageMath Team,

I am Dr. Kshipra Wadikar, and I have a PhD in Noncommutative Algebra. I am interested in contributing to SageMath’s Lie Algebra and Quantum Group module for GSoC 2025. I have experience in Python.

I have reviewed SageMath’s existing Lie algebra implementation and found that representation theory can be extended. Below is a short summary of my proposal:

1. Define a framework for Lie algebra representations (modules, weight spaces, tensor products).
2.  Implement fundamental and irreducible representations (Verma modules, highest weight representations).
3.  Develop algorithms for weight multiplicities and branching rules.
4. Introduce quantum groups (Drinfeld-Jimbo definition) and their representations.

Would this be a good project for SageMath? I’d love to get your feedback before submitting the full proposal.

Thank you!

Best,
Kshipra Wadikar

[tcscrims]

Dear Kshipra,

   Thank you for your interest in doing GSoC with SageMath.

Please be aware that Verma modules and simple modules (in BBG category O) for simple Lie algebras have already been implemented in SageMath. In principle, that implementation also works for affine Lie algebras (I think it might just need to be enabled; I haven't actually tried testing it yet). Manipulating branching rules and multiplicities for simple Lie algebras is done by the WeylCharacterRing. Quantum groups and their irreps for simple Lie algebras are available through GAP's QuaGroup package.

The problem is that these implementations are fairly slow and heavy-handed for the finite dimensional simples (even for small rank/dimensional cases). Some of this I know how to deal with (the PBW basis is slow due to how it currently handles ordering elements). Furthermore, the class structure of all of these representations is not really connected and has code duplication.

Some of the things I would like to see, beyond fixing the aforementioned problems, would be

- parabolic Verma modules and morphisms between them

- Kirillov-Reshetikhin modules (or at least fundamental) and fusion products

- simples for the Virasoro algebra

There's a lot of math involved with all of these, and there are parts that are not well developed with an algorithmic approach. So that is something to be careful about.

Anyways, it is your project and your proposal. So please write your proposal with what you would want to do and think would benefit SageMath (and its users).

Best,

Travis

[kshipra wadikar]

Dear SageMath Team,

 Thank you for your detailed feedback on my initial proposal. Based on your suggestions, I am refining my focus to address key areas that would improve SageMath’s existing implementation, particularly in performance, structure, and integration with GAP.   

 Enhancing Integration with GAP’s QuaGroup 

 Goal: Improve SageMath’s interface with GAP’s QuaGroup package for quantum groups.
🔹 Implement wrappers in SageMath for defining quantum groups, computing irreducible representations, and tensor product decompositions.
🔹 Optimize performance and reduce redundancy in existing implementations.
🔹 Improve documentation to make these tools more accessible to users.

 Parabolic Verma Modules & Morphisms

Goal: Extend SageMath’s current Verma module implementation to support parabolic Verma modules.
🔹 Define the structure of parabolic Verma modules in SageMath.
🔹 Implement morphisms between Verma modules.
🔹 Optimize computation efficiency (as current Verma modules are slow).

Would this be a good direction for SageMath? I’d love to hear your feedback before submitting the full proposal.

Best regards,
Kshipra Wadikar

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[tcscrims]

Dear Kshipra,

   This could work, but it will be highly dependent on the details. The general statements are still fairly far from a good proposal. Also, let me say a bit more clearly that these wrappers to QuaGroup are already in SageMath as I recall.

Best,

Travis