Free generic Poisson fields and algebras

ABSTRACT

The free generic Poisson algebras (GP-algebras) over a field k of characteristic 0 are studied. We prove that certain properties of free Poisson algebras are true for free GP-algebras as well. In particular, the universal multiplicative enveloping algebra $U=U\left(GP\right(x_1,\dots ,x_n\left)\right)$ of a free GP-field $GP(x_1,\dots ,x_n)$ is a free ideal ring. Besides, the Poisson and polynomial dependence of two elements are equivalent in $GP(x_1,\dots ,x_n)$. As a corollary, all automorphisms of the free GP-algebra GP{x,y} are tame and we have the isomorphisms of groups of automorphisms Aut GP{x,y}≅Aut P{x,y}≅Aut k[x,y].

KEYWORDS:

• Automorphism
• free algebra
• generic Poisson algebra
• generic Poisson module

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B63
• 16S40
• 17A50
• 17A36

Acknowledgments

The main part of this research was performed during the visits of the first and the third authors to the University of San Paulo. The hospitality of this university is gratefully acknowledged.