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The general Sakuma Theorem

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If you have a question about this talk, please contact Simon Goodwin.

The original Sakuma Theorem classifies vertex operator algebras (VOAs) generated by two Ising vectors. The properties it relies on were turned by Ivanov into the axioms of a new class of non-associative algebras called Majorana algebras and thus axial algebras were born.

The first axial version of the Sakuma theorem (for Majorana algebras) was published by Ivanov, Pasechnik, Seress and Shpectorov in 2010 and it was followed in 2015 by a more general version due to Hall, Rehren and Shpectorov, where many Majorana-specific assumptions were removed. The same year, Rehren attempted an even more general version, completely parting with Majorana restrictions and allowing arbitrary parameters a and b in the fusion rules to substitute the Majorana-specific values a=1/4 and b=1/32. He did not manage to obtain a complete classification, but he did show that the dimension of a 2-generated algebra is bounded by eight except when a=2b or a=4b.

In a joint project with Franchi and Mainardis, we reprove and generalise Rehren’s theorem to cover also the exceptional cases. In the case a=2b, we obtain the same bound, 8, on the dimension of a 2-generated algebra, although for a different spanning set. Even more interesting is the other exceptional case, where a=4b. Here we also have the upper bound eight, except when a=2 and b=1/2. In this final case, we found an unexpected example of an infinite dimensional 2-generated algebra.

This talk is part of the Algebra Seminar series.

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