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arXiv · 2512.11703

The Veronese square of the dendriform operad

Murray R. Bremner
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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.

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Murray R. Bremner. 2025-12-12. The Veronese square of the dendriform operad. https://arxiv.org/abs/2512.11703

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