The Veronese square of the dendriform operad (2512.11703v1)

Abstract: Veronese powers of operads were introduced in 2020 By Dotsenko, Markl, and Remm. 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 defining the Veronese square.

• The paper introduces the Veronese square of the dendriform operad, generated by five ternary operations and governed by 33 quadratic relations.
• It employs computational linear algebra and symbolic operad rewriting to translate dendriform identities into a Rota-Baxter setting.
• The explicit quadratic presentation paves the way for further classification and construction of higher-arity dendriform and triple systems.

The Veronese Square of the Dendriform Operad: Structure and Quadratic Relations

Introduction and Motivation

The concept of Veronese powers of operads establishes a systematic framework for transitioning from binary to higher-arity algebraic structures within the field of operad theory. The second Veronese power (Veronese square) $\mathcal{P}^{[2]}$ of a weight-graded operad $\mathcal{P}$ is the suboperad generated by all operations of weight $2$ and, in classical settings, yields a canonical notion of ternary algebras associated to binary operad-governed varieties. The present work systematically studies the Veronese square of the dendriform operad $\mathcal{D}$, which occupies a central position across nonassociative algebra, combinatorics, and the algebraic underpinning of Rota-Baxter operators.

Preliminaries: Operadic and Algebraic Context

The dendriform operad $\mathcal{D}$ is defined as the nonsymmetric operad generated by two binary operations, $\prec$ and $\succ$, subject to well-known dendriform relations. The free nonsymmetric operad $\mathcal{BB}$ generated by $\prec$ and $\succ$, modulo the ideal $\mathcal{I}$ generated by the dendriform identities, yields $\mathcal{D} = \mathcal{BB}/\mathcal{I}$. A key result is the embedding of $\mathcal{D}$ into the noncommutative Rota-Baxter operad $\mathcal{RB}$, which is defined as the quotient of the free operad $\mathcal{UB}$ (generated by a unary operator $U$ and an associative binary product $B$) by the ideal generated by the Rota-Baxter relation: $U(x)U(y) - U(U(x)y) - U(xU(y)) = 0.$ This embedding facilitates the translation of algebraic identities for $\mathcal{D}$ into consequences in the computationally amenable setting of $\mathcal{RB}$.

The categorical significance of the Veronese square is that for any variety $\mathbf{V}$ governed by a binary operad $\mathcal{V}$, algebras over $\mathcal{V}^{[2]}$ correspond precisely to ternary $\mathbf{V}$-algebras (triple systems).

Generating Operations for the Veronese Square

The study focuses on $\mathcal{D}^{[2]}$, the Veronese square of the dendriform operad, generated by the degree-three component $\mathcal{D}(3)$, known to be five-dimensional (with dimension coinciding with the third Catalan number). A convenient basis is given by the cosets of the five non-leading monomials in the dendriform relations: $$x \succ (y \prec z), \quad x \prec (y \prec z), \quad x \prec (y \succ z), \quad (x \succ y) \succ z, \quad (x \prec y) \succ z.$$ These elements are chosen for their symmetry properties and linear independence modulo $\mathcal{I}(3)$.

The free nonsymmetric operad $FT$ generated by five ternary operations $\omega_1, \ldots, \omega_5$ models the universal ternary framework and serves as the domain for a rewriting morphism that encodes quadratic relations.

Rota-Baxter Consequences and Computational Infrastructure

Consequences of the Rota-Baxter relation in higher arity and multiplicity are systematically generated by iterative partial compositions with $B$ and $U$, following a combinatorial enumeration closely linked to Dyck words and Narayana numbers. The core computational strategy involves constructing the "matrix of consequences" whose row space encodes all consequences of the Rota-Baxter identity in a given arity/multiplicity, hence modeling the ideal $\mathcal{J}(p,q)$. The dimensions and enumeration facilitate passage to quotient monomials forming a basis for $\mathcal{RB}$.

The Rewriting Morphism and Description of Quadratic Relations

A central construct is the rewriting morphism $r: FT \rightarrow \mathcal{RB}$, which maps the five ternary generators to explicit operator expressions in $\mathcal{RB}$ corresponding to the dendriform generators under the Rota-Baxter embedding. The arity-5 monomials (compositions of two ternary operations) in $FT(5)$ are enumerated (75 in total) and mapped explicitly into $\mathcal{UB}(5,4)$. The quadratic relations for the operad structure of $\mathcal{D}^{[2]}$ are identified as the kernel of the restriction of $r$ to arity-5, with computational linear algebra (RCF, LLL basis reduction) used to extract a minimal generating set.

Main Theorem: Explicit Structure of the Veronese Square

The main result is that the Veronese square of the dendriform operad is generated by five ternary operations subject to 33 linearly independent quadratic relations. These are explicitly presented as relations in the generators, compacted into expressions involving at most nine terms per relation, and with most involving only 2--5 terms thanks to lattice basis reduction.

All aspects of the construction are explicitly functorial and characteristic-free (for fields of characteristic zero), and the computations make essential use of combinatorial generation, operad rewriting, and symbolic linear algebra.

Implications and Prospects

This structural result realizes the Veronese square $\mathcal{D}^{[2]}$ as an explicit quadratic operad, paving the way for classification and construction of higher-arity dendriform algebras, and deeper investigation into the operadic and combinatorial underpinnings of Rota-Baxter-type structures. Through explicit computational scaffolding, the work bridges the gap between abstract operadic devices and concrete algebraic models, providing a platform for further studies in universal algebra, deformation theory, and higher algebraic structures linked to dendriform and Rota-Baxter operads.

From a theoretical viewpoint, these explicit presentations may inform homological inquiries (e.g., Koszulness, resolutions), representation theory of triple systems, and the design of computational tools to handle a wider class of ternary and higher-arity operads. The connections to Dyck paths, Catalan and Narayana combinatorics, and symbolic computation underscore the interplay between combinatorial and algebraic approaches.

Conclusion

The paper provides an explicit and algorithmically effective description of the quadratic relations for the Veronese square of the dendriform operad, realized via the Rota-Baxter embedding and computational linear algebra. This work not only clarifies the algebraic structure of ternary dendriform systems but also sets a foundation for further explorations in higher operadic theory, explicit presentations of higher-arity analogues, and their computational and theoretical ramifications (2512.11703).