Quadri-algebras, preLie algebras, and the Catalan family of Lie idempotents

TL;DR

This paper analyzes the structure of Catalan Lie idempotents, revealing their dependence on tree edges and identifying related preLie algebra substructures, thus advancing understanding of algebraic combinatorics.

Contribution

It introduces a detailed expansion of Catalan Lie idempotents on the PBW basis and identifies their placement within preLie algebra frameworks.

Findings

Coefficient depends only on left and right internal edges of trees

Catalan idempotents form a preLie algebra based on binary trees

Several Lie and preLie subalgebras are identified

Abstract

We compute the expansion of the Catalan family of Lie idempotents introduced in [Menous et al., Adv. Applied Math. 51 (2013), 177-22] on the PBW basis of the Lie module. It is found that the coefficient of a tree depends only on its number of left and right internal edges. In particular, the Catalan idempotents belong to a preLie algebra based on naked binary trees, of which we identify several Lie and preLie subalgebras.