Fusion rules from root systems I: case ${\rm A}_n$
  Axial algebras are commutative algebras generated by idempotents; they generalise associative algebras by allowing the idempotents to have additional eigenvectors, controlled by fusion rules. If the fusion rules are $\mathbb{Z}/2$-graded, axial algebras afford representations of transposition groups. We consider axial representations of Weyl groups of simply-laced root systems, which are examples of regular $3$-transposition groups. We introduce coset axes, a special class of idempotents based on embeddings of transposition groups, and use them to study the propagation of fusion rules in axial algebras, for root system ${\rm A}_n$. This is related to the construction of lattice vertex operator algebras and we show it reflects on the fusion of modules for the Virasoro algebra when we specialise our construction. 
Felix Rehren
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