Rota-Baxter operators of nilpotent evolution algebras with maximal nilindex

TL;DR

This paper classifies Rota-Baxter operators of weights zero and one on nilpotent evolution algebras with maximal nilindex, revealing their structural properties and providing a complete description of such operators.

Contribution

It provides the first comprehensive classification of Rota-Baxter operators on these specific nilpotent evolution algebras, highlighting their upper triangular form and recurrence relations.

Findings

Operators are upper triangular in a natural basis.

Weight zero operators are diagonal up to last basis vector perturbations.

Weight one operators include both triangular and non-triangular forms.

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.