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

TL;DR

This paper develops a novel classification framework for algebraic operator identities using algebraic operads and computational algebra, identifying key identities and families in associative algebras.

Contribution

It introduces a new method combining operad theory and computational algebra to classify operator identities, including new identities.

Findings

Identified six identities of degree 2 and multiplicity 1.

Found eighteen identities and two families of identities of degree 2 and multiplicity 2.

Included classical identities like derivation, Rota-Baxter, and Nijenhuis, plus new identities.

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.