New Operated Polynomial Identities and Gröbner-Shirshov Bases

Abstract

Twenty years ago, Rota posed the problem of finding all possible algebraic identities that can be satisfied by a linear operator on an algebra, named Rota’s Classification Problem later. Rota’s Classification Problem has proceeded two steps to understand it and has been studied actively recently. In particular, the method of Gröbner-Shirshov bases has been used successfully in the study of Rota’s Classification Problem. Quite recently, a new approach introduced to Rota’s Classification Problem and classified some (new) operated polynomial identities. In this paper, we prove that all operated polynomial identities classified via this new approach are Gröbner-Shirshov. This gives a partial answer of Rota’s Classification Problem.