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

TL;DR

This paper constructs combinatorial bases for the multilinear parts of free Lie and Poisson algebras with compatible brackets, proving their dimensions and establishing a perfect pairing with a complementary space.

Contribution

It provides explicit combinatorial bases and confirms the conjectured dimension formulas for these algebras, advancing understanding of their structure.

Findings

Dimension formulas for Lie_2(n) and P_2(n) confirmed

Constructed bases from combinatorial objects

Established a perfect pairing with a complementary space

Abstract

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.