Combinatorial bases for multilinear parts of free algebras with double   compatible brackets
  Let X be an ordered alphabet. Lie_2(n) (and P_2(n) respectively) are the multilinear parts of the free Lie algebra (and the free Poisson algebra respectively) on X with a pair of compatible Lie brackets. In this paper, we prove the dimension formulas for these two algebras conjectured by B. Feigin by constructing bases for Lie_2(n) (and P_2(n)) from combinatorial objects. We also define a complementary space Eil_2(n) to Lie_2(n), give a pairing between Lie_2(n) and Eil_2(n), and show that the pairing is perfect. 
Fu Liu