# Infinite dimensional Lie algebras in Sage

Dear Sage community,

I'm considering giving Sage a spin. Having the scripting possibilities that python offers at one's disposal seems very appealing.

But I would first like to know if - short of writing the module I need in python - Sage is currently capable of addressing the kind of problems I am interested in. I mainly work with infinite dimensional Lie algebras such as the Virasoro algebra. Is there an easy way to implement such algebras in Sage, by specifying structure constants or something like that?

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Hi @Gonneman,

Glad to hear you're thinking about Sage :) I am not aware of a general constructor for arbitrary Lie algebras, although there is substantial functionality related to certain Lie group computations. Beyond that, the best way to answer your question is probably to browse the reference manual (e.g. the section on Rings) and see if there is enough functionality for you to do the computations you're interested in. There is also documentation for various basic constructions in Sage.

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Thanks a lot for the help Niles and Benjaminfjones. Now I know where to start!

( 2011-08-08 18:14:18 -0500 )edit

As far as I know, facilities for doing Lie theoretic computations directly in a Lie algebra in Sage are sparse. As @niles mentions, there are facilities for dealing with roots, weights, and Weyl groups, but that's probably not what you're looking for if you are working in the Virasoro algebra.

Of course there is fantastic support for all kinds of linear algebra, which most Lie algebra computations boil down to anyway ...

Here are some other resources, though:

• Sage has a CombinatorialFreeModule class which is inherited from to construct many non-commutative / non-associative algebras (e.g. see the documentation and source code for IwahoriHeckeAlgebraT). AFAIK, this class is restricted to finite rank modules.

• GAP (which is included in Sage) has several packages (both standard and optional) for doing Lie algebra calculations. See:

Also, the folks on the sage-combinat-devel and sage-algebra google groups would probably be interested in this topic if you post your question there.

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CombinatorialFreeModule is usable with infinite rank modules, as well as finite rank.

( 2011-08-12 05:22:23 -0500 )edit

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