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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Generalised dihedral subalgebras from the Monster
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by Felix Rehren PDF
Trans. Amer. Math. Soc. 369 (2017), 6953-6986 Request permission


The conjugacy classes of the Monster which occur in the McKay observation correspond to the isomorphism types of certain $2$-generated subalgebras of the Griess algebra. Sakuma, Ivanov and others showed that these subalgebras match the classification of vertex algebras generated by two Ising conformal vectors, or of Majorana algebras generated by two axes. In both cases, the eigenvalues $\alpha ,\beta$ parametrising the theory are fixed to $\frac {1}{4}$, $\frac {1}{32}$. We generalise these parameters and the algebras which depend on them, in particular finding the largest possible (nonassociative) axial algebras which satisfy the same key features, by working extensively with the underlying rings. The resulting algebras admit an associating symmetric bilinear form and satisfy the same $6$-transposition property as the Monster; $\frac {1}{4}$, $\frac {1}{32}$ turns out to be distinguished.
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Additional Information
  • Felix Rehren
  • Affiliation: School of Mathematics, University of Birmingham, B15 2TT, United Kingdom
  • MR Author ID: 1084481
  • Email:
  • Received by editor(s): October 10, 2014
  • Received by editor(s) in revised form: October 2, 2015
  • Published electronically: March 1, 2017
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 6953-6986
  • MSC (2010): Primary 20D08, 17C27, 17Dxx, 20Bxx, 13F20
  • DOI:
  • MathSciNet review: 3683099