Axial algebras are a class of non-associative algebras with a strong natural link to groups and have recently received much attention. They are generated by axes which are semisimple idempotents whose eigenvectors multiply according to a so-called fusion law. Of primary interest are the axial algebras with the Monster type (α,β) fusion law, of which the Griess algebra (with the Monster as its automorphism group) is an important motivating example. By previous work of Yabe, and Franchi and Mainardis, any symmetric 2-generated axial algebra of Monster type (α,β) is either in one of several explicitly known families, or is a quotient of the infinite-dimensional Highwater algebra H, or its characteristic 5 cover Ĥ. We complete this classification by explicitly describing the infinitely many ideals and thus quotients of the Highwater algebra (and its cover). As a consequence, we find that there exist 2-generated algebras of Monster type (α,β) with any number of axes (rather than just 1,2,3,4,5,6,∞ as we knew before) and of arbitrarily large finite dimension. In this talk, we will begin with a reminder of axial algebras which were introduced last week.
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