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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:

    • The SLA package for GAP:

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.