Yamaguti algebras and noncrossing partitions

TL;DR

This work identifies a simple combinatorial model for the Yamaguti operad by proving an isomorphism with a noncrossing-partitions operad without singleton blocks. It constructs $\mathscr{B}$ with basis $B(n)$ and explicit composition rules, and shows that the Yamaguti operad $\mathrm{Yam}$ maps onto and bijects with $\mathscr{B}$ via $\psi$, with dimensions governed by Riordan numbers. The approach yields a concrete, purely combinatorial presentation of Yamaguti algebras and establishes a Gröbner-basis framework for the relations, while outlining rich open questions on Koszulity, identities, and connections to the Lie–Yamaguti operad and root-system combinatorics. The results bridge Yamaguti algebra theory with noncrossing-partition combinatorics and cluster-structure geometry, providing new tools for deformation theory and operadic analysis.

Abstract

Recently, Das defined a new type of algebras, the Yamaguti algebras, which are supposed to serve as envelopes of Lie-Yamaguti algebras appearing naturally in differential geometry. We show that the nonsymmetric operad of Yamaguti algebras admit a simple combinatorial description via noncrossing partitions without singleton blocks.

Figures (1)

• Figure 1: The noncrossing partition with blocks $\{0,1,4,5\}$ and $\{2,3\}$.

Theorems & Definitions (10)

• Definition 1
• Remark 2
• Lemma 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Theorem 6
• proof