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 that its
exact solutions can be easily and naturally obtained within this formalism.
The Hamiltonian matrix of each $j$ multiplet can be split into two
submatrices associated to two distinct irreps of the deformed algebra.
Their invariant subspaces correspond to even and odd numbers of
particle-hole excitations.