Code algebras, axial algebras and VOAs

Alonso Castillo-Ramirez*, Justin McInroy, Felix Rehren

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

3 Citations (Scopus)
117 Downloads (Pure)


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 AC 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 H8 where the code algebra AH8 is an axial algebra and embeds in the code VOA VH8 .

Original languageEnglish
Pages (from-to)146-176
Number of pages31
JournalJournal of Algebra
Early online date22 Oct 2018
Publication statusPublished - 15 Jan 2019


  • Axial algebras
  • Binary linear codes
  • Nonassociative algebras
  • Vertex operator algebras


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