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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
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References:
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[7] DOI: 10.1023/A:1009914728281 · Zbl 0939.16018 · doi:10.1023/A:1009914728281
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