Rota-type operators on 2-dimensional dendriform algebras

TL;DR

This work classifies Rota-type operators on 2-dimensional dendriform algebras over $\mathbb{C}$, articulating explicit conditions and matrices for Rota–Baxter, Reynolds, Nijenhuis, and Averaging operators within the low-dimensional family $Dend_2^{i}$ (including $i=1$ and $i=2$–$12$). It provides complete weight-0 and weight-1 classifications for Rota–Baxter operators and similarly enumerates operator families for the other three operator types, with detailed parameter restrictions. The resulting catalogs illustrate how the dendriform identities constrain operator equations in 2D, enabling concrete constructions and potential deformations of these algebras. The findings contribute to the broader understanding of operadic and algebraic deformation frameworks in low dimensions, with potential implications for non-commutative geometry and combinatorics.

Abstract

We describe Rota-Baxter operators, Reynolds operators, Nijenhuis operators, and Averaging operators on 2-dimensional dendriform algebras over $\mathbb{C}$.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5