The Veronese square of the dendriform operad

TL;DR

This paper investigates the Veronese square of the dendriform operad, identifying its generators and quadratic relations through combinatorial and computational methods, thus advancing the understanding of ternary algebraic structures.

Contribution

It explicitly describes the generators and quadratic relations of the Veronese square of the dendriform operad, a novel algebraic structure related to ternary algebras.

Findings

Identified five generators for the Veronese square of the dendriform operad.

Determined 33 quadratic relations among these generators.

Represented the operad as a suboperad of the Rota-Baxter operad.

Abstract

Veronese powers of operads were introduced in 2020 By Dotsenko, Markl, and Remm \cite{DMR}. The $m$-th Veronese power of a weight-graded operad $\mathcal{V}$ is the suboperad $\mathcal{V}^{[m]}$ generated by the operations of weight $m$. If $\mathcal{V}$ is generated by binary operations and governs the variety $\mathbf{V}$ of algebras, this gives a natural definition of the concept of $(m{+}1)$-ary $\mathbf{V}$-algebras. In particular, the Veronese square ($m=2$) corresponds to ternary algebras. We choose five generating operations for the Veronese square of the dendriform operad. We represent the dendriform operad as a suboperad of the Rota-Baxter operad, and express the quadratic relations satisfied by the generating operations as the kernel of a rewriting map. We use combinatorics of monomials and computational linear algebra to determine the kernel. We obtain 33 linearly independent quadratic relations satisfied by the Veronese square.