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Let $(L,d)$ be a Differential Graded Lie Algebra ($L=\bigoplus L_i$ and $d:L_i \to L_{i-1}$ satisfying the graded Leibniz rule).

On the algebra $\mathrm{Der}L$ of derivations of $L$ define a grading such that $|\delta| = n$ if $\delta$ takes every homogeneous element $a\in L_i$ to a homogeneous element $\delta(a)\in L_{i+n}$.

Extend $(\mathrm{Der}L, d)$ to a chain complex by defining $d(\delta) = [d,\delta] = d\delta-(-1)^{|\delta|}\delta d$ (Since $d$ has grading $-1$).

Is there any relation between the homology of $L$ and the homology of $\mathrm{Der}L$ ? I would also be happy to get links to any relevant sources about this.

P.S. - I have found a similar question on math.stackexchange, but sadly it has no replies: https://math.stackexchange.com/q/1057780/214568

P.P.S. - It is my first time posting here, so if I have made any big mistakes / no-no's - I apologize in advance, and please let me know.

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  • $\begingroup$ If anyone is interested - I have managed to show that if $L$ has a trivial homology then $\mathrm{Der}L$ has homology only in degrees 0 and -1. It is not hard to show with a little bit of diagram chasing. As to a complete answer - I have yet to find one. $\endgroup$ – Itai Mar 3 '15 at 22:54

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