Algebraic bosonisation: the case of the Dirac fermion

On the lattice fermions paired and then composed with precise phases are the lattice bosons. We construct rules for deforming the underlying algebra that keep unchanged the boson interpretation. These deformations alter the phases of the fermion pairs. The solutions of the deformed algebra are the pair-phase excitations of the lattice bosons. We find these eigenstates of the Dirac Hamiltonian of fermions with mass. The gauge interactions are shown to excite the k-deformed modes.