The structure of axial algebras

Joint work with Yoav Segev.
``Fusion rules'' are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to axial algebras, introduced recently by Hall, Rehren and Shpectorov, defined as nonassociative algebras generated by semisimple idempotents of degree 3, satisfying fusion rules based on a natural 2-grading . Axial algebras, in turn, are closely related to 3-transposition groups and vertex operator algebras.

We introduce a noncommutative generalization of primitive axial algebras (PAJ for short), and show that they all have Jordan type. Extending the known theory, we bring in the fundamental notion of Miyamoto involutions, and the ensuing topology on the set
of primitive axes.

Accompanying this is the ``axial graph'' on a generating set of axes X, where two axes
are neighbors if and only if their product is nonzero. The axial graph aids us in
decomposing a PAJ into connected components. The PAJ's which are not commutative are easily described, implying that all PAJ's are flexible, and any PAJ is a direct product of noncommutative PAJ 's and a commutative PAJ .

We obtain a Frobenius form for any PAJ which is not quite unique, and prove some properties which previously had been axioms. We give a complete description of all axes of 2-generated PAJ's, thereby enabling a solution of the question of whether primitive axes are conjugate.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062