## 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 nonzero 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 ℤ_{2}-graded. In doing so, we exhibit an infinite family of ℤ_{2}× ℤ_{2}-graded axial algebras—these are the first known examples of axial algebras with a non-trivial grading other than a ℤ_{2}-grading.

Original language | English |
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Pages (from-to) | 401-438 |

Number of pages | 38 |

Journal | Israel Journal of Mathematics |

Volume | 233 |

Issue number | 1 |

Early online date | 26 Jul 2019 |

DOIs | |

Publication status | Published - 1 Aug 2019 |