Decomposition algebras and axial algebras

Tom De Medts, Simon F. Peacock, Sergey Shpectorov, Michiel Van Couwenberghe

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We introduce decomposition algebras as a natural generalization of axial algebras, Majorana algebras and the Griess algebra. They remedy three limitations of axial algebras: (1) They separate fusion laws from specific values in a field, thereby allowing repetition of eigenvalues; (2) They allow for decompositions that do not arise from multiplication by idempotents; (3) They admit a natural notion of homomorphisms, making them into a nice category. We exploit these facts to strengthen the connection between axial algebras and groups. In particular, we provide a definition of a universal Miyamoto group which makes this connection functorial under some mild assumptions. We illustrate our theory by explaining how representation theory and association schemes can help to build a decomposition algebra for a given (permutation) group. This construction leads to a large number of examples. We also take the opportunity to fix some terminology in this rapidly expanding subject.
Original languageEnglish
Pages (from-to)287-314
Number of pages28
JournalJournal of Algebra
Early online date31 Mar 2020
Publication statusPublished - 15 Aug 2020


  • Association schemes
  • Axial algebras
  • Decomposition algebras
  • Fusion laws
  • Griess algebra
  • Majorana algebras
  • Norton algebras
  • Representation theory


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