# Universal enveloping algebras and universal derivations of Poisson algebras

```@article{Umirbaev2011UniversalEA,
title={Universal enveloping algebras and universal derivations of Poisson algebras},
author={U. Umirbaev},
journal={arXiv: Rings and Algebras},
year={2011}
}```
• U. Umirbaev
• Published 2011
• Mathematics
• arXiv: Rings and Algebras
Let \$k\$ be an arbitrary field of characteristic \$0\$. It is shown that for any \$n\geq 1\$ the universal enveloping algebras of the Poisson symplectic algebra \$P_n(k)\$ and the Weyl algebra \$A_n(k)\$ are isomorphic and the canonical isomorphism between them easily leads to the Moyal product. A basis of the universal enveloping algebra \$P^e\$ of a free Poisson algebra \$P=k\{x_1,...,x_n\}\$ is constructed and proved that the left dependency of a finite number of elements of \$P^e\$ over \$P^e\$ is… Expand
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