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arXiv · 1402.2007

Universal enveloping algebras of Poisson Hopf algebras

Jiafeng Lü·Xingting Wang·Guangbin Zhuang
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Abstract

For a Poisson algebra $A$, by exploring its relation with Lie-Rinehart algebras, we prove a Poincar\'e-Birkoff-Witt theorem for its universal enveloping algebra $A^e$. Some general properties of the universal enveloping algebras of Poisson Hopf algebras are studied. Given a Poisson Hopf algebra $B$, we give the necessary and sufficient conditions for a Poisson polynomial algebra $B[x; \alpha, \delta]_p$ to be a Poisson Hopf algebra. We also prove a structure theorem for $B^e$ when $B$ is a pointed Poisson Hopf algebra. Namely, $B^e$ is isomorphic to $B#_\sigma \mathcal{H}(B)$, the crossed product of $B$ and $\mathcal{H}(B)$, where $\mathcal{H}(B)$ is the quotient Hopf algebra $B^e/B^eB^+$.

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Jiafeng Lü, Xingting Wang, Guangbin Zhuang. 2014-02-09. Universal enveloping algebras of Poisson Hopf algebras. https://arxiv.org/abs/1402.2007

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