Algebraic identities for linear operators on associative triple systems (long version)

TL;DR

This paper classifies algebraic identities for linear operators on associative triple systems, extending Rota's classification problem to n-ary operations and using computational linear algebra techniques.

Contribution

It provides the first classification of operator identities in associative triple systems, applying operadic methods and computational linear algebra to extend Rota's problem to higher arity.

Findings

Identified 6 families with 1 parameter for multiplicity 1 identities.

Found 6 families with 2 parameters and 27 with 1 parameter for multiplicity 2.

Discovered 9 isolated solutions for the operator identities.

Abstract

We present the first classification of algebraic identities in 3 variables for linear operators on associative structures. We work in the context of associative triple systems, but since any associative algebra with product $xy$ becomes an associative triple system with product $xyz$, our results apply to associative algebras as well. This is the first time that Rota's classification problem for linear operators has been extended to algebras with an $n$-ary operation for $n \ge 3$. Our work is an application of computational linear algebra to the classification problem for linear operators. We begin with a generic operator identity with indeterminate coefficients. From this we use operadic partial compositions to derive a large sparse matrix whose nonzero entries are the indeterminates. We follow the rank principle which states that significant operator identities correspond to coefficients which produce submaximal rank of the matrix. For operator identities of multiplicity 1 (each term contains the operator once) we obtain 6 families with 1 parameter, and 1 isolated solution. For multiplicity 2, we obtain 6 families with 2 parameters, 27 families with 1 parameter, and 9 isolated solutions.