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Generalised dihedral subalgebras from the Monster

Author: Felix Rehren
Journal: Trans. Amer. Math. Soc. 369 (2017), 6953-6986
MSC (2010): Primary 20D08, 17C27, 17Dxx, 20Bxx, 13F20
Published electronically: March 1, 2017
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Abstract: 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 $ \sfrac {1}{4},\sfrac {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; $ \sfrac {1}{4},\sfrac {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

Received by editor(s): October 10, 2014
Received by editor(s) in revised form: October 2, 2015
Published electronically: March 1, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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