Rota–Baxter operators of nilpotent evolution algebras with maximal nilindex

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.

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