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arXiv · 1802.03342

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

Alonso Castillo-Ramirez·Justin McInroy
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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.

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Alonso Castillo-Ramirez, Justin McInroy. 2018-12-03. Code algebras which are axial algebras and their $\mathbb{Z}_2$-gradings. https://doi.org/10.1007/s11856-019-1911-5

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