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@Alexander Thank you for your comment! Yes I think Chern character map may be the answer and there are a lot of interesting theory on it (for example the paper by J. Block and E. Getzler "Equivariant cyclic homology and equivariant differential forms"). I will think more carefully about this.
Thank you for your references! By details I mean how to define sum, difference, multiplication. I think $2$-term chain complexes is enough for sum and difference but may have difficulty to define mutiplications.
Yes it is very natural from the viewpoint of coadjoint orbits. Maybe I should think more carefully before asking this question. Nevertheless, thank you all!
Thank you very much! I will look at paper you suggested. By the way, since the equivalent class of Maurer-Cartan elements forms an $\infty$-groupoid, does it means that the "composition" of two equivalences is not unique?