Gröbner-Shirshov bases of Rota-Baxter algebra of weight $λ$ with spectrum lying in $\{0,-λ\}$
H. Alhussein
TL;DR
The paper addresses the problem of deriving a Gröbner–Shirshov basis for the ideal generated by the Rota–Baxter relation $R(x)R(y)=R(xR(y))+R(R(x)y)+\\lambda R(xy)$ together with the nilpotence constraint $R^k(R+\\lambda \text{id})^l=0$ in the free RB-algebra. Building on the noncommutative Gröbner–Shirshov framework for operated algebras and Guo’s monomial order, it provides an explicit finite basis of operator-relations $(R1)$–$(R10)$ whose all compositions vanish modulo the basis, thereby yielding a confluent rewriting system and computable normal forms. Special cases are treated: when $\\lambda=0$ a reduced basis $(R1)$–$(R6)$ appears, and for $k+l=3$ with $\\lambda\\neq0$ a compact basis $(R1)$–$(R5)$ is given, generalizing the previously solved $k=l=1$ case. Overall, the work enables algorithmic ideal membership testing and normal-form computation for nilpotent RB-algebras, with potential impact on combinatorics, operad theory, and quantum algebra.
Abstract
It is known that if $A$ is a finite-dimensional unital algebra equipped with a Rota-Baxter operator $R$ of weight $λ$, then spectrum of $R$ is a subset of $\{0,-λ\}$. We are interested on finding all consequences of the Rota-Baxter relation and the relation of the form $R^k(R+λ{\rm id})^l = 0$. In 2024, H.~Qiu, S. Zheng, Y. Dan solved this problem for $k = l = 1$ and $λ\neq0$. We find a Gröbner-Shirshov basis of the ideal generated by these two relations in general case.