# Cohomology ring of a Lie algebra

I would like to compute the cup products in cohomology for certain families of nilpotent Lie algebras over R. So far I could access to the cochain complex and compute the Betti numbers using the method chevalley_eilenberg_complex(). On the other hand, I see that the cup product of the cohomology ring of a cell complex can be computed.

So, is there a way to compute the cup products starting from the Lie algebra?

(Note that the Lie algebras that I consider are not always defined over the rational field, so that the cohomologies that I am interested in is not always that of a space of which I could describe the homotopy type by giving a cell complex.)

EDIT (2020-10-26): Problem not solved, but I found a seemingly related issue: https://trac.sagemath.org/ticket/6100 . Namely, chain_complex().homology(generators=true) uses a basis in which the simplices are listed in an order that seems intractable (especially, not the lexicographic order). If one could guess in which order the simplices are listed, then I would be able to compute the cup products.

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chain_complex().homology(...) uses the order on the chains as given when constructing the chain complex, which in this case is taken care of by chevalley_eilenberg_complex. So you can look at that code to see what order is chosen.