Killing metrized commutative nonassociative algebras associated with Steiner triple systems

TL;DR

This paper introduces a family of commutative, nonassociative algebras linked to Steiner triple systems, showing they are Killing metrized and simple for most parameters, with special cases related to axial algebras.

Contribution

It constructs and analyzes a new class of algebras associated with Steiner triple systems, demonstrating their Killing metrized and simple properties, and distinguishing them from Matsuo algebras.

Findings

Algebras are exact, Killing metrized, and simple for generic parameters.

For Hall triple systems, the algebra is a primitive axial algebra with a $rac{1}{2}$-graded fusion law.

The algebras differ from Matsuo algebras despite similar definitions.

Abstract

With each Steiner triple system there is associated a one-parameter family of commutative, nonassociative, nonunital algebras that are by construction exact, meaning that the trace of every multiplication operator vanishes, and these algebras are shown to be Killing metrized, meaning the Killing type trace-form is nondegenerate and invariant (Frobenius), and simple, except for certain parameter values. The definition of these algebras resembles that of the Matsuo algebra of the Steiner triple system, but they are different. For a Hall triple system, the associated algebra is a primitive axial algebra for a $\mathbb{Z}/2\mathbb{Z}$-graded fusion law.