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Free Poisson fields and their automorphisms

Leonid Makar-Limanov

The Weizmann Institute of Science, Rehovot, Israel

University of Michigan, Ann Arbor, USA

Wayne State University, Detroit, MI 48202, USA

E-mail Address: lml@math.wayne.edu

Corresponding author.

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and

Ualbai Umirbaev

Wayne State University, Detroit, MI 48202, USA

Eurasian National University, Astana, Kazakhstan

E-mail Address: umirbaev@math.wayne.edu

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https://doi.org/10.1142/S0219498816501966Cited by:5 (Source: Crossref)

Abstract

Let $k$ $k$ be an arbitrary field of characteristic $0$ . We prove that the group of automorphisms of a free Poisson field $P (x, y)$ in two variables $x, y$ over $k$ is isomorphic to the Cremona group ${Cr}_{2} (k)$ . We also prove that the universal enveloping algebra $P {(x_{1}, \dots, x_{n})}^{e}$ of a free Poisson field $P (x_{1}, \dots, x_{n})$ is a free ideal ring and give a characterization of the Poisson dependence of two elements of $P (x_{1}, \dots, x_{n})$ via universal derivatives.

Communicated by Efim Zelmanov

Keywords:

AMSC: 17B63, 17B40, 17A36, 16W20.

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