  We show that the low-lying excitations of the one-dimensional Bose gas are
described, at all orders in a 1/N expansion and at the first order in the
inverse of the coupling constant, by an effective hamiltonian written in terms
of an extended conformal algebra, namely the Cartan subalgebra of the
$W_{1+\infty}\times \bar{W}_{1+\infty}$ algebra. This enables us to construct
the first interaction term which corrects the hamiltonian of free fermions
equivalent to a hard-core boson system.
