The cohomology of a~finite-dimensional coalgebra over a finitely generated quadratic operad, with coefficients in itself, is defined and is shown to have the structure of a graded Lie algebra. The cohomology of such a coalgebra is isomorphic to the cohomology of its linear dual as graded Lie algebras.
"Cohomology and duality for coalgebras over a quadratic operad." J. Gen. Lie Theory Appl. 3 (2) 131 - 148, May 2009. https://doi.org/10.4303/jglta/S090204