Code algebras which are axial algebras and their ${\mathbb{Z}}_2$-gradings

Abstract

A code algebra $A_C$ is a non-associative commutative algebra defined via a binary linear code $C$. We study certain idempotents in code algebras, which we call small idempotents, that are determined by a single non-zero codeword. For a general code $C$, we show that small idempotents are primitive and semisimple and we calculate their fusion law. If $C$ is a projective code generated by a conjugacy class of codewords, we show that $A_C$ is generated by small idempotents and so is, in fact, an axial algebra. Furthermore, we classify when the fusion law is ${\mathbb{Z}}_2$-graded. In doing so, we exhibit an infinite family of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded axial algebras - these are the first known examples of axial algebras with a non-trivial grading other than a ${\mathbb{Z}}_2$-grading.

29 pages