# zbMATH — the first resource for mathematics

Poisson polynomial rings. (English) Zbl 1135.17012
Many quantum groups have been constructed from Poisson algebras which are polynomial rings with certain Poisson brackets. In this article, one finds that many Poisson brackets on polynomial rings are given by derivations with certain conditions, which may be considered as a Poisson version of a skew-polynomial ring constructed by an endomorphism $$\alpha$$ and an $$\alpha$$-derivation. Moreover a Poisson structure of a class of Poisson algebras including the coordinate rings of Poisson $$2\times 2$$-matrices and Poisson symplectic 4-space is investigated. More precisely, given a Poisson algebra $$(A,\{\;,\;\}_A)$$ and linear maps $$\alpha, \delta :A\rightarrow A$$, one gives a necessary and sufficient condition for the pair $$(\alpha, \delta)$$ such that the polynomial ring $$A[x]$$ has the Poisson bracket $\{a, b\}=\{a,b\}_A,\qquad \{a,x\}=\alpha (a) x+\delta(a)$ for all $$a, b\in A$$ and constructs a class of Poisson algebras including the coordinate rings of Poisson $$2\times 2$$-matrices and Poisson symplectic 4-space.

##### MSC:
 17B63 Poisson algebras 16W25 Derivations, actions of Lie algebras
##### Keywords:
derivation; Poisson algebra
Full Text:
##### References:
 [1] Brown K. A., Lectures on Algebraic Quantum Groups (2002) · Zbl 1027.17010 [2] Chari V., A Guide to Quantum Groups (1994) · Zbl 0839.17009 [3] Dixmier J., Enveloping Algebras 11 (1996) [4] Korogodski L. I., Algebras of Functions on Quantum Groups 56 (1998) · Zbl 0923.17017 [5] McConnell J. C., Noncommutative Noetherian Rings (1987) · Zbl 0644.16008 [6] DOI: 10.1080/00927879908826555 · Zbl 0936.16041 · doi:10.1080/00927879908826555 [7] DOI: 10.1023/A:1009914728281 · Zbl 0939.16018 · doi:10.1023/A:1009914728281
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.