Abstract:
One of the important directions in modern mathematics is applications of Poisson structures and to
various problems of mathematics and theoretical mechanics. These problems arise in dynamics of a rigid
body, the celestial mechanics, the theory of curls, cosmological models. Poisson algebras play a
key role in the Hamiltonian mechanics, symplectic geometry and also are central in the study of
quantum groups. Note that a development of the theory of Poisson structures in many respects was
stimulated by the dynamics of many-dimensional tops since the latter allows to make the abstract
statements of many theorems more vivid and substantial. Note also that some important examples of
the Lie-Poisson brackets were already known to Jacobi. In his examples the Poisson brackets
appeared on a space of the first integrals of the Hamilton equations. Until recently, an algebraic
theory of Poisson structures was scarcely studied. At present, Poisson algebras are investigated by
the many mathematicians of Russia, France, the USA, Brazil, Argentina, Bulgaria etc. This paper is
devoted to the description of the automorphism group of Poisson algebra P on polynomial algebra k [x, y, z], such that {x, y} = z2, {y, z} = x2, {z, x} = y2. One interesting Poisson relation between the homogeneous algebraically dependent elements is established and is proved that the group of automorphisms Autk P of algebra P is generated by automorphisms ϕα = (αx, αy, αz), α ∈ k∗, τ = (y, z, x) and δ = (x, εy, ε2z), where ε – a solution of an equation x2 + x + 1 = 0.
