A Gröbner--Shirshov Basis for Nilpotent Rota--Baxter Algebras of Weight Zero

Abstract

We construct an explicit Gröbner--Shirshov basis for free associative Rota--Baxter algebras of weight zero with nilpotent operator $R^n=0$, where $n\ge 2$. First, we define a monomial order on the standard linear basis $RS(X)$ of the free algebra $R\mathrm{As}⟨X⟩$ and establish fundamental identities for Rota--Baxter operators. For the case $n=2$, the basis consists of the Rota--Baxter relation $R(u)R(v)\to R(uR(v))+R(R(u)v)$ and the nilpotency relation $R(R(w))\to 0$. For general $n\ge 3$, we prove that the Gröbner--Shirshov basis is finite and consists of six families of relations $(R1)$--$(R6)$ derived from resolving all composition ambiguities. Using the Composition-Diamond Lemma, we describe the corresponding irreducible basis $\mathrm{Irr}(S)$, which provides normal forms for elements in the quotient algebra. This result gives a complete solution to the word problem for nilpotent Rota--Baxter algebras and establishes their operadic Gröbner--Shirshov basis.

This paper is a special case of the more general work "A Gröbner--Shirshov Basis for Nilpotent Rota--Baxter Algebras of {0, -λ}" (arXiv:2507.01614). The results presented here are fully contained in that article. Therefore, the authors wish to withdraw this version to avoid duplication. Readers should refer to the more comprehensive article for the complete results