  We extend the Poisson bracket from a Lie bracket of phase space functions to
a Lie bracket of functions on the space of canonical histories and investigate
the resulting algebras. Typically, such extensions define corresponding Lie
algebras on the space of Lagrangian histories via pull back to a space of
partial solutions. These are the same spaces of histories studied with regard
to path integration and decoherence. Such spaces of histories are familiar from
path integration and some studies of decoherence. For gauge systems, we extend
both the canonical and reduced Poisson brackets to the full space of histories.
We then comment on the use of such algebras in time reparameterization
invariant systems and systems with a Gribov ambiguity, though our main goal is
to introduce concepts and techniques for use in a companion paper.
