Graded dihedral axial algebras

Axial algebras are non-associative algebras generated by semisimple idempotents, known as axes, whose eigenvectors obey a given fusion law. These algebras have been shown to occur in many areas of mathematics but are of particular interest thanks to their rich relationship to certain groups. Dihedral axial algebras are those generated by two axes and are fundamental to the study of axial algebras in general. We present a new result that classifies in high generality dihedral axial algebra whose fusion laws have three or fewer eigenvalues. This result represents a significant broadening of our understanding of axial algebras.