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SUMMARY:Classifying quotients of the Highwater algebra - Justin McInroy (U
 niversity of Chester)
DTSTART:20220727T090000Z
DTEND:20220727T093000Z
UID:TALK176555@talks.cam.ac.uk
DESCRIPTION:Axial algebras are a class of non-associative algebras with a 
 strong natural link to groups and have recently received much attention.&n
 bsp\; They are generated by axes which are semisimple idempotents whose ei
 genvectors multiply according to a so-called fusion law.&nbsp\; Of primary
  interest are the axial algebras with the Monster type $(\\alpha\, \\beta)
 $ fusion law\, of which the Griess algebra (with the Monster as its automo
 rphism group) is an important motivating example.\nBy previous work of Yab
 e\, and Franchi and Mainardis\, any symmetric 2-generated axial algebra of
  Monster type $(\\alpha\, \\beta)$ is either in one of several explicitly 
 known families\, or is a quotient of the infinite-dimensional Highwater al
 gebra $\\mathcal{H}$\, or its characteristic 5 cover $\\hat{\\mathcal{H}}$
 .&nbsp\; We complete this classification by explicitly describing the infi
 nitely many ideals and thus quotients of the Highwater algebra (and its co
 ver).&nbsp\; As a consequence\, we find that there exist 2-generated algeb
 ras of Monster type $\\mathcal{M}(\\alpha\, \\beta)$ with any number of ax
 es (rather than just $1\,2\,3\,4\,5\,6\, \\infty$ as we knew before) and o
 f arbitrarily large finite dimension.\nIn this talk\, we do not assume any
  knowledge of axial algebras.\nThis is joint work with:Clara Franchi\, Cat
 holic University of the Sacred Heart\, MilanMario Mainardis\, University o
 f Udine
LOCATION:Seminar Room 1\, Newton Institute
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