The universality of one half in commutative nonassociative algebras with identities

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).