A Poisson algebra is a commutative associative algebra 
over a field
with a bilinear Poisson bracket 
. The bracket makes 
a Lie algebra and obeys the
product rule with respect to multiplication,
 |
Thus for fixed 
the map 
is a derivation
of the commutative algebra (Laurent-Gengoux et al.
2013, p. 4).
For example, the polynomial ring in 
and 
over a field of characteristic zero is a Poisson algebra with
bracket
 |
In particular, 
. The Poisson
conjecture concerns endomorphisms preserving this bracket in the generalization
to 
pairs of variables (Adjamagbo and van den Essen 2007).
The opposite sign convention for the Poisson bracket
gives an equivalent definition.
