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

Details Combinatorial bases for multilinear parts of free algebras with double compatible brackets Combinatorial bases for multilinear parts of free algebras with double compatible brackets

2008-08-26

arxiv.org/abs/0808.3439v1

Mathematics

Combinatorics

38 pages; 10 figures

Scientific paper

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.

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