Infinite dimensional Lie algebras

[Simon Wood]

Hello everyone

I'm new to Sage and unfortunately still completely clueless. I'm
hoping to be able to use Sage for some computations involving infinite
dimensional Lie algebras and their representations. Does Sage have
useful tools for defining algebras with an infinite number of
generators? I see there is an "AlgebrasWithBasis" function, but as far
as I can tell it requires the number of generators to be finite.

What would be the best way to say for example construct the Virasoro
algebra?

[John H Palmieri]

My first inclination would be to base it on CombinatorialFreeModule: if you know a basis, this is a good way to go.  It will take care of addition and scalar multiplication, so you'll have to write a bracket method.  If you're feeling ambitious, you could write something for the Sage categories framework, implementing "Lie algebras with basis", by imitating "algebras with basis" (sage/categories/algebras_with_basis.py).  Then for any particular example, you should just have to define a method "bracket_on_basis" which takes a pair of basis elements and computes their bracket; the category framework (which you will set up separately) will then define a "bracket" method for dealing with arbitrary elements of your Lie algebra.

I used the existing framework for Hopf algebras for the current implementation (as of prerelease versions of Sage 4.7.1) of the Steenrod algebra -- see <http://trac.sagemath.org/sage_trac/ticket/10052>.  I basically only had to define products, antipodes, and coproducts on basis elements, and the framework took care of everything else.  (Note, by the way, that the Steenrod algebra is infinite-dimensional, and this is not a problem for working with CombinatorialFreeModules.)

--
John