Idempotents inducing a Z_2 grading on nonassociative algebras and their corresponding involutions.
Jordan algebras J of charateristic not 2 sometimes contain
a set of idempotents (e^2=e) that generate J such that their adjoint
map ad_e: u \mapsto ue (u\in J) has the minimal polynomial
x(x-1)(x-1/2), and with additional restrictions on products
of elements in the eigenspaces of ad_e (for each e).
Generalizing these properties (not only of such Jordan
algebras) Hall, Rehren, Shpectorov (HRS) introduced ``Axial algebras
of Jordan type''. In my talk I will present structural results
on Axial algebras of Jordan type 1/2 (a case which was not
dealt with in HRS), I will discuss their idempotents e, the corresponding
``Miyamoto involutions'' \tau(e) and the group that these involutions
This is joint work with J. Hall, S. Shpectorov.