Quadratic Poisson algebras on k[x, y, z]and their automorphisms

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 Aut_kP of algebra P is generated by automorphisms Aut_kP of algebra P is generated by automorphisms ϕ_α = (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.