A new classification of algebraic identities for linear operators on associative algebras

Abstract

We introduce a new approach to the classification of operator identities, based on basic concepts from the theory of algebraic operads together with computational commutative algebra applied to determinantal ideals of matrices over polynomial rings. We consider operator identities of degree 2 (the number of variables in each term) and multiplicity 1 or 2 (the number of operators in each term), but our methods apply more generally. Given an operator identity with indeterminate coefficients, we use partial compositions to construct a matrix of consequences, and then use computer algebra to determine the values of the indeterminates for which this matrix has submaximal rank. For multiplicity 1 we obtain six identities, including the derivation identity. For multiplicity 2 we obtain eighteen identities and two parametrized families, including the left and right averaging identities, the Rota-Baxter identity, the Nijenhuis identity, and some new identities which deserve further study.

New operated polynomial identities and Gröbner-Shirshov bases

“…In the present paper, we prove that OPIs classified in [7] are Gröbner-Shirshov in the framework of [11], via the method of Gröbner-Shirshov bases. In other words, we show that OPIs classified in [7] are "good" OPIs searched in Rota's Classification Problem. Our study is a partial answer of Rota's Classification Problem.…”

“…In this section, we prove that all OPIs classified in [7] are Gröbner-Shirshov. In the rest of the paper, in order to be consistent with the notations in [7], we use L to denote the operator ⌊ ⌋.…”

“…Quite recently, Bremner et al introduced a new approach to Rota's Classification Problem, based on the rank of matrixes from OPIs [7]. They obtained six OPIs with degree 2 and multiplicity 1, and eighteen OPIs and two parametrized families with degree 2 and multiplicity 2.…”

“…We then review the necessary concepts of Gröbner-Shirshov bases and the notation of Gröbner-Shirshov OPIs. Section 3 is devoted to prove that all OPIs in [7] are Gröbner-Shirshov. Three monomial orders ≤ dt , ≤ db and ≤ Dl are recalled.…”

“…Three monomial orders ≤ dt , ≤ db and ≤ Dl are recalled. Based on these three monomial orders, we first prove that all OPIs classified in [7] of degree 2 and multiplicity 1 are Gröbner-Shirshov (Theorem 3.2). Then we show that all OPIs in [7] of degree 2 and multiplicity 2 are Gröbner-Shirshov (Theorem 3.11).…”