Double axes and subalgebras of Monster type in Matsuo algebras

Alexey Galt, Vijay Joshi, Andrey Mamontov, Sergey Shpectorov*, Alexey Staroletov

*Corresponding author for this work

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Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov (in 2015) as a broad generalization of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster sporadic simple group. The class of axial algebras of Monster type includes Majorana algebras for the Monster and many other sporadic simple groups, Jordan algebras for classical and some exceptional simple groups, and Matsuo algebras corresponding to 3-transposition groups. Thus, axial algebras of Monster type unify several strands in the theory of finite simple groups. It is shown here that double axes, i.e., sums of two orthogonal axes in a Matsuo algebra, satisfy the fusion law of Monster type (Formula presented.) Primitive subalgebras generated by two single or double axes are completely classified and 3-generated primitive subalgebras are classified in one of the three cases. These classifications further lead to the general flip construction outputting a rich variety of axial algebras of Monster type. An application of the flip construction to the case of Matsuo algebras related to the symmetric groups results in three new explicit infinite series of such algebras.

Original languageEnglish
Pages (from-to)4208-4248
Number of pages41
JournalCommunications in Algebra
Issue number10
Publication statusPublished - 25 Jun 2021

Bibliographical note

Funding Information:
This work was partly supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation. Mathematical Center in Akademgorodok, Novosibirsk, Russia.

Publisher Copyright:
© 2021 Taylor & Francis Group, LLC.


  • 3-transposition group
  • Axial algebra
  • non-associative algebra

ASJC Scopus subject areas

  • Algebra and Number Theory


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