Modules over Axial Algebras
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We introduce axial representations and modules over axial algebras as new tools to study axial algebras. All known interesting examples of axial algebras fall into this setting, in particular the Griess algebra whose automorphism group is the Monster group. Our results become especially interesting for Matsuo algebras. We vitalize the connection between Matsuo algebras and 3-transposition groups by relating modules over Matsuo algebras with representations of 3-transposition groups. As a by-product, we define, given a Fischer space, a group that can fulfill the role of a universal 3-transposition group.
KeywordsAxial algebras Modules Axial representations Matsuo algebras 3-transposition groups Fischer spaces
Mathematics Subject Classification (2010)Primary 17A99 20B25 20F29 20C05 Secondary 17B69 20C34
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We have benefited from fruitful discussions between Jonathan Hall and the second author during a visit at Michigan State University. We thank the two anonymous referees of earlier versions of this paper for their suggestions, which improved the exposition of the results at many places.
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