 
Summary: POISSON STRUCTURES
TUAN M. NGUYEN
Abstract. A Poisson algbera is a Lie algebra on which there is
defined a multiplication which is required to satisfy certain con
ditions. Let M be an ndimensional manifold; if there is a Pois
son algebra on the space C
(M) of all smooth functions on M,
then this Poisson algebra is called a Poisson structure on C
(M),
and the manifold M is called a Poisson manifold. Poisson struc
tures on the space of all smooth functions are important examples
of infinitedimensional Lie algebras. Denote by {} (the Poisson
bracket) the commutator of the underlying Lie algebra of the Pois
son structure on C
(M), and let M be a domain of Rn
, f and g
are realanalytic functions of x1
, . . . , xn
in M, then
{f, g} =
