# 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

generate.

This is joint work with J. Hall, S. Shpectorov.

- Last modified: 8/05/2016