Miyamoto involutions in axial algebras of Jordan type half

Nonassociative commutative algebras A, generated by idempotents e whose adjoint operators ad (e) : A -> A, given by x a dagger broken vertical bar xe, are diagonalizable and have few eigenvalues, are of recent interest. When certain fusion (multiplication) rules between the associated eigenspaces are imposed, the structure of these algebras remains rich yet rather rigid. For example, vertex operator algebras give rise to such algebras. The connection between the Monster algebra and Monster group extends to many axial algebras which then have interesting groups of automorphisms. Axial algebras of Jordan type eta are commutative algebras generated by idempotents whose adjoint operators have a minimal polynomial dividing (x-1)x(x-eta), where eta ae {0, 1} is fixed, with well-defined and restrictive fusion rules. The case of eta not equal 1/2 was thoroughly analyzed by Hall, Rehren and Shpectorov in a recent paper, in which axial algebras were introduced. Here we focus on the case where eta = 1/2, which is less understood and is of a different nature.