The automorphism group of Poisson algebras on k[x; y]

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 P_f by defining a Poisson bracket by the rule {x, y} = f. Obviously, a structure of the automorphism group G_f of Poisson algebra P_f depends on the element f. A complete description of group G_f 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.