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

TL;DR

This paper extends Rota's operator-identity classification to algebras with a ternary operation by studying linear operators on associative triple systems. It develops a computational framework using operator monomials, partial compositions, and the matrix of consequences, then applies a rank-principle to extract significant identities. For degree 3, multiplicity 1, it yields six 1-parameter families and one isolated identity, including the ternary derivation identity; for degree 3, multiplicity 2, it identifies 387 submaximal-rank solutions organized into 9 isolated points, 27 one-parameter families, and 6 two-parameter families, with explicit examples and a complete appendix. The work demonstrates the power of linear-algebraic methods in classifying higher-arity operator identities and provides a detailed catalog for future exploration of n-ary 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.

Figures (2)

• Figure 1: Algorithm for generating $n$-ary operator monomials
• Figure 2: Matrix of consequences for the generic operator identity of degree 3 and multiplicity 1 (with dot for zero)

Theorems & Definitions (14)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Example 2.7
• Definition 3.1
• Theorem 4.1
• Conjecture 4.2