Algebraic Combinatorics· 2026Q1
Combinatorics of pre-Lie products sharing the same Lie bracket
- 3citations
- Q1SCImago
- 2026year
Short summary
A new operad controlling multiple pre-Lie products with the same Lie bracket is described by the species of rooted Greg trees, where black vertices have at least two children.
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Key points
- A novel operad is introduced for pre-Lie products with identical Lie brackets.
- The operad's combinatorial structure is identified as the species of rooted Greg trees.
- Rooted Greg trees are defined by having black vertices with a minimum of two children.
- The new operad shares key algebraic properties with the established pre-Lie operad.
AI-generated from the title and abstract; the full text is not read.
Abstract
We study in detail the operad controlling several pre-Lie products sharing the same Lie bracket. We show that this operad admits a combinatorial description similar to that of Chapoton and Livernet for the pre-Lie operad, namely the underlying species of the operad controlling two pre-Lie products sharing the same Lie bracket is the species of rooted Greg trees, which are rooted trees with black and white vertices such that the black vertices have at least two children, and the composition is the insertion. Moreover, we show that it has many of the remarkable algebraic properties of the pre-Lie operad.
The authors' abstract, as published at the source. Algebraic Combinatorics, 2026 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics