Local Rota--Baxter operators of weight zero on nilpotent evolution algebras with maximal nilindex

Abstract

Let $E$ be an $n$-dimensional nilpotent evolution algebra of maximal nilindex over a field of characteristic zero. The Rota--Baxter operators of weights zero and one on such algebras were recently classified. In this paper we investigate the local analogue of the weight-zero case. We prove that every local Rota--Baxter operator of weight zero has a rigid diagonal part: the nonzero diagonal coefficients, when they occur, begin after the obstruction index determined by the structural matrix of $E$ and form a final geometric string governed by one scalar. In contrast, the last-column coefficients are arbitrary. This gives an explicit description of the class $\LRB_0(E)$ and shows that it is generally strictly larger than $\RB_0(\E)$. We also prove that the corresponding $s$-local notion produces no new operators. The classification is further interpreted as a finite union of quasi-affine strata, yielding a dimension comparison between $\LRB_0(E)$ and $\RB_0(E)$. Finally, we study ordinary weight-zero Rota--Baxter operators commuting with derivations and automorphisms and describe the resulting conditions in terms of the directed graph associated with $E$.