On the exact solutions of the Lipkin-Meshkov-Glick model

We present the many-particle Hamiltonian model of Lipkin, Meshkov and Glick in the context of deformed polynomial algebras and show chat its exact sol utions can be easily and naturally obtained within this formalism. The Hami ltonian matrix of each j multiplet can be split into two submatrices associ ated with two distinct irreps of the deformed algebra. Their invariant subs paces correspond to even and odd numbers of particle-hole excitations.