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

Abstract

Let X be an ordered alphabet. L i e 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 L i e 2 ( n ) (and P 2 ( n ) ) from combinatorial objects. We also define a complementary space E i l 2 ( n ) to L i e 2 ( n ) , give a pairing between L i e 2 ( n ) and E i l 2 ( n ) , and show that the pairing is perfect.