Papers › An expansion algorithm for constructing axial algebras
An expansion algorithm for constructing axial algebras
Justin McInroy, Sergey Shpectorov
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An axial algebra A is a commutative non-associative algebra generated by primitive idempotents, called axes, whose adjoint action on A is semisimple and multiplication of eigenvectors is controlled by a certain fusion law. Different fusion laws define different classes of axial algebras. Axial algebras are inherently related to groups. Namely, when the fusion law is graded by an abelian group T, every axis a leads to a subgroup of automorphisms Tₐ of A. The group generated by all Tₐ is called the Miyamoto group of the algebra. We describe a new algorithm for constructing axial algebras with a given Miyamoto group. A key feature of the algorithm is the expansion step, which allows us to overcome the $2$-closeness restriction of Seress's algorithm computing Majorana algebras. At the end we provide a list of examples for the Monster fusion law, computed using a MAGMA implementation of our algorithm.
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