  Let H_T=C[T,T^{-1}] be the Hopf algebra of symmetries of a lattice of rank 1,
or equivalently, H_T is the group algebra of a free Abelian group with one
generator T. We construct conformal algebras, vertex Poisson algebras and
vertex algebras with H_T as symmetry. For example, the Hamiltonian structure
for the infinite Toda lattice gives rise to an H_T-vertex Poisson structure on
a free difference algebra. Examples of H_T-vertex algebras are constructed from
representations of a class of infinite dimensional Lie algebras related to H_T
in the same way loop algebras are related to the Hopf algebra H_D=C[D] of
infinitesimal translations used in the usual vertex algebras.
