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What is actually known about the (co-) homology of free nilpotent Lie Algebras over $\mathbb{C}$ and coefficients also in $\mathbb{C}$? I.e. the nilpotent Lie Algebras that one gets by a quotient of a free Lie algebra by some part of its descending central series.

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There are several results on the homology of free-nilpotent Lie algebras with trivial coefficients, in particular for free two-step nilpotent Lie algebras. Stefan Sigg has determined the homology in this case by working out the structure of the homology as a module under the general linear group. The main tool is a Laplacian for the free two-step nilpotent Lie algebras, which turns out to be closely related to the Casimir operator of the general linear group. In addition, Grassberger, King, Tirao give an explicit formula for the total dimension of the homology of a free 2-step nilpotent Lie algebra.

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