Code algebras, axial algebras and VOAs

TL;DR

This paper introduces code algebras, a new class of commutative non-associative algebras derived from binary codes, and explores their structure, simplicity, and relation to axial algebras and VOAs.

Contribution

It defines code algebras, analyzes their properties, and establishes connections to axial algebras and vertex operator algebras, including classification and examples.

Findings

Code algebras are almost always simple.

They admit an associating bilinear form under mild conditions.

Certain code algebras are proven to be axial algebras.

Abstract

Inspired by code vertex operator algebras (VOAs) and their representation theory, we define code algebras, a new class of commutative non-associative algebras constructed from binary linear codes. Let $C$ be a binary linear code of length $n$. A basis for the code algebra $A_C$ consists of $n$ idempotents and a vector for each non-constant codeword of $C$. We show that code algebras are almost always simple and, under mild conditions on their structure constants, admit an associating bilinear form. We determine the Peirce decomposition and the fusion law for the idempotents in the basis, and we give a construction to find additional idempotents, called the $s$-map, which comes from the code structure. For a general code algebra, we classify the eigenvalues and eigenvectors of the smallest examples of the $s$-map construction, and hence show that certain code algebras are axial algebras. We give some examples, including that for a Hamming code $H_8$ where the code algebra $A_{H_8}$ is an axial algebra and embeds in the code VOA $V_{H_8}$.