Jordan algebras of rank 4 and minimal representations (Q1576603)

Let \(J\) be a simple Euclidean Jordan algebra of rank 4 and \(V=J^\mathbb C\) the complexification of \(J\). Denote by \(\mathfrak k=V \oplus V\square V\oplus V\) the associated Kantor-Koecher-Tits algebra, by \(\mathcal P(V)\) the space of holomorphic polynomials on \(V\), \(\Delta\) the determinant of \(V\) and \(\mathfrak p\subset {\mathcal P}(V)\) the subspace generated by the polynomials \(v\mapsto \Delta(v-a)\), \(a\in V\). The author defines a Lie algebra structure on \(\mathfrak g=\mathfrak k\oplus\mathfrak p\) which turns out to be of type \(E_6\), \(E_7\) or \(E_8\) and constructs an explicit realization of the representation of \(\mathfrak g\) associated to the minimal nilpotent orbit.