The universality of one half in commutative nonassociative algebras with identities

TL;DR

This paper demonstrates that in finite-dimensional commutative nonassociative algebras satisfying identities, the number one-half always appears in the Peirce spectrum, and explores the structure of the associated Peirce modules.

Contribution

It introduces the concept of the Peirce symbol, providing explicit methods to analyze the Peirce spectrum and fusion laws in such algebras, with applications to genetic and Hsiang algebras.

Findings

Half is always in the Peirce spectrum of these algebras

The half-Peirce module satisfies Jordan type fusion laws

Explicit determination of the Peirce symbol for various algebra classes

Abstract

In this paper we will explain an interesting phenomenon which occurs in general nonassociative algebras. More precisely, we establish that any finite-dimensional commutative nonassociative algebra over a field satisfying an identity always contains $\frac12$ in its Peirce spectrum. We also show that the corresponding $\frac12$-Peirce module satisfies the Jordan type fusion laws. The present approach is based on an explicit representation of the Peirce polynomial for an arbitrary algebra identity. To work with fusion rules, we develop the concept of the Peirce symbol and show that it can be explicitly determined for a wide class of algebras. We also illustrate our approach by further applications to genetic algebras and algebra of minimal cones (the so-called Hsiang algebras).