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Rota-Baxter operators of nilpotent evolution algebras with maximal nilindex

Izzat Qaralleh, Farrukh Mukhamedov, Otabek Khakimov

TL;DR

This work classifies Rota-Baxter operators of weights $0$ and $1$ on nilpotent evolution algebras of maximal nilindex. By reducing to upper-triangular operators in a natural basis, the authors obtain a complete description: weight $0$ operators are rigid and essentially diagonal up to last-column perturbations, while weight $1$ operators split into triangular and non-triangular regimes with the diagonal entries governed by the rational map $f(x)=\frac{x^2}{2x+1}$ and periodicity phenomena. The triangular RB$_1$ regime yields explicit normal forms and period constraints, whereas the non-triangular RB$_1$ regime forces diagonals to be either $0$ or $-1$ with precise coefficient recursions. Overall, the paper provides a comprehensive, dynamical-systems–influenced classification of RB operators on this extremal class of evolution algebras, highlighting a sharp dichotomy between weights $0$ and $1$.

Abstract

Nilpotent evolution algebras of maximal nilindex admit a natural basis in which the structure matrix is strictly upper triangular. In this paper we classify Rota{Baxter operators of weights zero and one on such algebras. We prove that every Rota{Baxter operator is upper triangular with respect to a natural basis. For weight zero, a strong rigidity phenomenon occurs: the operators are diagonal up to possible perturbations supported in the last basis vector. For weight one, a richer structure appears, including both triangular and non-triangular families, with the diagonal entries governed by a rational recurrence relation. Our results provide a complete description of Rota{Baxter operators on nilpotent evolution algebras of maximal nilindex.

Rota-Baxter operators of nilpotent evolution algebras with maximal nilindex

TL;DR

This work classifies Rota-Baxter operators of weights and on nilpotent evolution algebras of maximal nilindex. By reducing to upper-triangular operators in a natural basis, the authors obtain a complete description: weight operators are rigid and essentially diagonal up to last-column perturbations, while weight operators split into triangular and non-triangular regimes with the diagonal entries governed by the rational map and periodicity phenomena. The triangular RB regime yields explicit normal forms and period constraints, whereas the non-triangular RB regime forces diagonals to be either or with precise coefficient recursions. Overall, the paper provides a comprehensive, dynamical-systems–influenced classification of RB operators on this extremal class of evolution algebras, highlighting a sharp dichotomy between weights and .

Abstract

Nilpotent evolution algebras of maximal nilindex admit a natural basis in which the structure matrix is strictly upper triangular. In this paper we classify Rota{Baxter operators of weights zero and one on such algebras. We prove that every Rota{Baxter operator is upper triangular with respect to a natural basis. For weight zero, a strong rigidity phenomenon occurs: the operators are diagonal up to possible perturbations supported in the last basis vector. For weight one, a richer structure appears, including both triangular and non-triangular families, with the diagonal entries governed by a rational recurrence relation. Our results provide a complete description of Rota{Baxter operators on nilpotent evolution algebras of maximal nilindex.
Paper Structure (7 sections, 10 theorems, 93 equations)

This paper contains 7 sections, 10 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Evolution algebras
  3. Rota-Baxter operators
  4. Description of $\mathcal{RB}_0({\bf E})$
  5. On description of $\mathcal{RB}_1({\bf E})$
  6. Case $\mathcal{RB}_1^\vartriangle({\bf E})$
  7. Case $\mathcal{RB}_1^\blacktriangle({\bf E})$

Key Result

Theorem 2.2

rozomir An $n$-dimensional evolution algebra $\bf E$ is nilpotent iff it admits a natural basis such that the matrix of the structural constants corresponding to $\bf E$ in this basis is represented as follows

Theorems & Definitions (21)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • ...and 11 more