Representation of an algebra associated with the Dynkin graph $\tilde E_7$

Abstract

We describe the structure of pairs of self-adjoint operators $A$ and $B$ whose spectra belong to the set $\{±1/2, ±3/2\}$ and for which $(A+B)^2 = I$. Such pairs of operators determine a representation of a *-algebra $A_{\tilde E_7 }$ associated with the extended Dynkin graph $\tilde E_7$.