Abstract:
Poisson algebras play a key role in the Hamiltonian mechanics, symplectic geometry and also are
central in
the study of quantum groups. At present, Poisson algebras are investigated by the many
mathematicians of Russia, France, the USA, Brazil, Argentina, Bulgaria etc. The purpose of the
present paper is to describe the automorphism groups of polynomial algebras endowed with additional
structure, namely, with Poisson
brackets. For any f ∈ k [x, y] one can transform associative-commutative algebra k [x, y] into a
Poisson
algebra Pf by defining a Poisson bracket by the rule {x, y} = f . Obviously, a structure of the
automorphism
group Gf of Poisson algebra Pf depends on the element f . A complete description of group Gf is
given for
the polynomial f of rank less or equals to 1. In present paper all algebras are considered over any
field k
of characteristic 0.
