The graded modules for the Poisson algebras K are inverstigated and their compositionfactors are determined. Also, a revision on the realizations of the irreducible PTG H(n,m)-modules V(n, m) due to Shen is made.
Non-commutative Poisson algebras are the algebras having both an associa- tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures...Non-commutative Poisson algebras are the algebras having both an associa- tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on the Lie algebras sln(Cq^-) are determined.展开更多
Noncommutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this article,the noncommutative Poisson algebra structures on...Noncommutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this article,the noncommutative Poisson algebra structures on sp2l(^~CQ) are determined.展开更多
We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equ...We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.展开更多
The Hopf dual H~? of any Poisson Hopf algebra H is proved to be a co-Poisson Hopf algebra provided H is noetherian. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. It is...The Hopf dual H~? of any Poisson Hopf algebra H is proved to be a co-Poisson Hopf algebra provided H is noetherian. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. It is proved that there is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. So the polynomial Hopf algebra, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, is considered. The Poisson Hopf structures on polynomial Hopf algebras are exactly linear Poisson structures. The co-Poisson structures on polynomial Hopf algebras are characterized.Some correspondences between co-Poisson and Poisson structures are also established.展开更多
We introduce the notions of differential graded(DG) Poisson algebra and DG Poisson module. Let A be any DG Poisson algebra. We construct the universal enveloping algebra of A explicitly, which is denoted by A^(ue). We...We introduce the notions of differential graded(DG) Poisson algebra and DG Poisson module. Let A be any DG Poisson algebra. We construct the universal enveloping algebra of A explicitly, which is denoted by A^(ue). We show that A^(ue) has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over A is isomorphic to the category of DG modules over A^(ue). Furthermore, we prove that the notion of universal enveloping algebra A^(ue) is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.展开更多
Let A = F [x, y] be the polynomial algebra on two variables x, y over an algebraically closed field F of characteristic zero. Under the Poisson bracket, A is equipped with a natural Lie algebra structure. It is proven...Let A = F [x, y] be the polynomial algebra on two variables x, y over an algebraically closed field F of characteristic zero. Under the Poisson bracket, A is equipped with a natural Lie algebra structure. It is proven that the maximal good subspace of A* induced from the multiplication of the associative commutative algebra A coincides with the maximal good subspace of A* induced from the Poisson bracket of the Poisson Lie algebra A. Based on this, structures of dual Lie bialgebras of the Poisson type are investigated. As by-products,five classes of new infinite-dimensional Lie algebras are obtained.展开更多
For a Poisson algebra, the category of Poisson modules is equivalent to the module category of its Poisson enveloping algebra, where the Poisson enveloping algebra is an associative one. In this article, for a Poisson...For a Poisson algebra, the category of Poisson modules is equivalent to the module category of its Poisson enveloping algebra, where the Poisson enveloping algebra is an associative one. In this article, for a Poisson structure on a polynomial algebra S, we first construct a Poisson algebra R, then prove that the Poisson enveloping algebra of S is isomorphic to the specialization of the quantized universal enveloping algebra of R, and therefore, is a deformation quantization of R.展开更多
For a Poisson algebra,we prove that the Poisson cohomology theory introduced by Flato et al.(1995)is given by a certain derived functor.We show that the(generalized)deformation quantization is equivalent to the formal...For a Poisson algebra,we prove that the Poisson cohomology theory introduced by Flato et al.(1995)is given by a certain derived functor.We show that the(generalized)deformation quantization is equivalent to the formal deformation for Poisson algebras under certain mild conditions.Finally we construct a long exact sequence,and use it to calculate the Poisson cohomology groups via the Yoneda-extension groups of certain quasi-Poisson modules and the Lie algebra cohomology groups.展开更多
The Poisson structures on a basic cycle are determined completely via quiver techniques. As a consequence, all Poisson structures on basic cycles are inner.
We present a variety of superintegrable systems in polar coordinates possessing a cubic and a quadratic integral of motion, where Noether integrals of kinetic energy are used to build the integrals. In addition, the a...We present a variety of superintegrable systems in polar coordinates possessing a cubic and a quadratic integral of motion, where Noether integrals of kinetic energy are used to build the integrals. In addition, the associated polynomial Poisson algebras with their algebraic dependence relations are exhibited.展开更多
基金Acknowledgements Part of the work was done during the author's visit to SCMS (Shanghai Center for Mathematical Sciences), and the author would like to thank for the hospitality. The author also thank the referees for their careful reading, helpful suggestions and comments. This work was supported by the National Natural Science Foundation of China (Grant No. 11301180).
文摘We compute explicitly the modular derivations for Poisson-Ore extensions and tensor products of Poisson algebras.
文摘The graded modules for the Poisson algebras K are inverstigated and their compositionfactors are determined. Also, a revision on the realizations of the irreducible PTG H(n,m)-modules V(n, m) due to Shen is made.
基金supported by NSF of China(11071187)Innovation Program of Shanghai Municipal Education Commission(09YZ336)
文摘Non-commutative Poisson algebras are the algebras having both an associa- tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on the Lie algebras sln(Cq^-) are determined.
基金This work is partially supported by NNSF of China(10671124)the Specialized Research Fund for the Doctoral Program of Higher Education(20040247024).
文摘Noncommutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this article,the noncommutative Poisson algebra structures on sp2l(^~CQ) are determined.
基金Supported by National Natural Science Foundation of China(Grant No.10971206)
文摘We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.
基金supported by National Natural Science Foundation of China (Grant Nos. 11331006 and 11171067)
文摘The Hopf dual H~? of any Poisson Hopf algebra H is proved to be a co-Poisson Hopf algebra provided H is noetherian. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. It is proved that there is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. So the polynomial Hopf algebra, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, is considered. The Poisson Hopf structures on polynomial Hopf algebras are exactly linear Poisson structures. The co-Poisson structures on polynomial Hopf algebras are characterized.Some correspondences between co-Poisson and Poisson structures are also established.
基金supported by National Natural Science Foundation of China(Grant Nos.11571316 and 11001245)Natural Science Foundation of Zhejiang Province(Grant No.LY16A010003)
文摘We introduce the notions of differential graded(DG) Poisson algebra and DG Poisson module. Let A be any DG Poisson algebra. We construct the universal enveloping algebra of A explicitly, which is denoted by A^(ue). We show that A^(ue) has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over A is isomorphic to the category of DG modules over A^(ue). Furthermore, we prove that the notion of universal enveloping algebra A^(ue) is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.
基金supported by National Natural Science Foundation of China(Grant Nos.11071147,11431010 and 11371278)Natural Science Foundation of Shandong Province(Grant Nos.ZR2010AM003and ZR2013AL013)+1 种基金Shanghai Municipal Science and Technology Commission(Grant No.12XD1405000)Fundamental Research Funds for the Central Universities
文摘Let A = F [x, y] be the polynomial algebra on two variables x, y over an algebraically closed field F of characteristic zero. Under the Poisson bracket, A is equipped with a natural Lie algebra structure. It is proven that the maximal good subspace of A* induced from the multiplication of the associative commutative algebra A coincides with the maximal good subspace of A* induced from the Poisson bracket of the Poisson Lie algebra A. Based on this, structures of dual Lie bialgebras of the Poisson type are investigated. As by-products,five classes of new infinite-dimensional Lie algebras are obtained.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 11771085).
文摘For a Poisson algebra, the category of Poisson modules is equivalent to the module category of its Poisson enveloping algebra, where the Poisson enveloping algebra is an associative one. In this article, for a Poisson structure on a polynomial algebra S, we first construct a Poisson algebra R, then prove that the Poisson enveloping algebra of S is isomorphic to the specialization of the quantized universal enveloping algebra of R, and therefore, is a deformation quantization of R.
基金supported by National Natural Science Foundation of China(Grant Nos.11401001,11871071,11431010 and 11571329)。
文摘For a Poisson algebra,we prove that the Poisson cohomology theory introduced by Flato et al.(1995)is given by a certain derived functor.We show that the(generalized)deformation quantization is equivalent to the formal deformation for Poisson algebras under certain mild conditions.Finally we construct a long exact sequence,and use it to calculate the Poisson cohomology groups via the Yoneda-extension groups of certain quasi-Poisson modules and the Lie algebra cohomology groups.
文摘The Poisson structures on a basic cycle are determined completely via quiver techniques. As a consequence, all Poisson structures on basic cycles are inner.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11771352 and 11371293
文摘We present a variety of superintegrable systems in polar coordinates possessing a cubic and a quadratic integral of motion, where Noether integrals of kinetic energy are used to build the integrals. In addition, the associated polynomial Poisson algebras with their algebraic dependence relations are exhibited.