  In many Lagrangian field theories, there is a Poisson bracket on the space of
local functionals. One may identify the fields of such theories as sections of
a vector bundle. It is known that the Poisson bracket induces an sh-Lie
structure on the graded space of horizontal forms on the jet bundle of the
relevant vector bundle. We consider those automorphisms of the vector bundle
which induce mappings on the space of functionals preserving the Poisson
bracket and refer to such automorphisms as canonical automorphisms.
  We determine how such automorphisms relate to the corresponding sh-Lie
structure. If a Lie group acts on the bundle via canonical automorphisms, there
are induced actions on the space of local functionals and consequently on the
corresponding sh-Lie algebra. We determine conditions under which the sh-Lie
structure induces an sh-Lie structure on a corresponding reduced space where
the reduction is determined by the action of the group. These results are not
directly a consequence of the corresponding theorems on Poisson manifolds as
none of the algebraic structures are Poisson algebras.
