Abstract
An axial algebra A is a commutative nonassociative 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 T_{a} of A. The group generated by all T_{a} 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 2closedness 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.
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 T_{a} of A. The group generated by all T_{a} 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 2closedness 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.
Original language  English 

Pages (fromto)  379409 
Number of pages  34 
Journal  Journal of Algebra 
Volume  550 
Early online date  21 Jan 2020 
DOIs  
Publication status  Published  15 May 2020 
Keywords
 axial algebra
 nonassociative algebra
 Majorana algebra
 finite groups
 algorithm
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Dr Justin F McInroy
 School of Mathematics  Heilbronn Research Fellow
 Heilbronn Institute for Mathematical Research
 Pure Mathematics
Person: Academic , Member