Combinatorial bases for multilinear parts of free algebras with two compatible brackets

Let X be an ordered alphabet. Lie2(n) (and P2(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 Lie2(n) (and P2(n)) from combinatorial objects. We also define a complementary space Eil2(n) to Lie2(n), give a pairing between Lie2(n) and Eil2(n), and show that the pairing is perfect.