Structural and combinatorial theory of Poisson algebras

交换代数、结合代数和李代数的结构与组合理论是现代数学的一个重要分支。尽管泊松代数与这些代数有着密切的联系，但目前还没有系统的关于泊松代数的代数理论。如果考虑到泊松代数在数学和物理的许多分支中是多么重要（和流行），这是相当令人惊讶的。从代数的角度研究泊松代数是很重要的。泊松代数的代数理论将有助于理解泊松结构的几何。泊松代数的纯代数研究也将为交换代数、李代数和结合代数的许多问题提供新的途径。同样清楚的是，泊松结构的研究将使我们更好地理解它们的变形量子化。西蒙-丹尼斯·泊松(1781 ？（1840年）是有史以来最多产的数学家之一。在他的众多贡献中，他介绍了泊松括号作为经典动力学的工具。卡尔·古斯塔夫·雅各布·雅可比(1804 ？1851年)意识到这些括号的重要性，并发现了它们的代数性质。Marius Sophus Lie（1842年12月17日- 1899年2月18日）开始研究它们的几何学。在过去的40年里，泊松几何已经成为一个活跃的研究领域，与许多领域的联系，包括非交换，几何，李群的调和分析，无限维李代数，粒子力学和连续体，奇点理论，和完全可积系统。对泊松结构进行系统的代数研究，并发展适当的代数理论和工具，对于研究这些结构的科学家（主要是物理学家和几何学家）具有潜在的价值，就像交换代数对代数几何学家一样。这种方法应该使主题更加清晰，并允许获得更详细和可理解的结果。

The structural and combinatorial theory of commutative algebras, associative algebras, and Lie algebras is one of the most important branches of modern mathematics. Though Poisson algebras are very closely interconnected with these algebras, at present there is no systematic algebraic theory of Poisson algebras. This is rather surprising if one takes in account how important (and popular) Poisson algebras are in many branches of mathematics and physics. It is important to study Poisson algebras from an algebraic point of view. An algebraic theory of Poisson algebras will be useful for understanding the geometry of Poisson structures. Purely algebraic study of Poisson algebras should also give new approaches to many problems of commutative algebras, Lie algebras, and associative algebras. It is clear as well that the study of Poisson structures will bring a better understanding of their deformation quantization.Siméon-Denis Poisson (1781 ?1840) was one of the most prolific mathematicians of all times. Among his numerous contributions, he introduced Poisson brackets as a tool for classical dynamics. Carl Gustav Jacob Jacobi (1804 ? 1851) realized the importance of these brackets and discovered their algebraic properties. Marius Sophus Lie (17 December 1842 - 18 February 1899) began the study of their geometry. During the past 40 years Poisson geometry has become an active field of research stimulated by connections with a number of areas, including non-commutative, geometry, harmonic analysis on Lie groups, infinite dimensional Lie algebras, mechanics of particles and continua, singularity theory, and completely integrable systems.Systematic algebraic approach to the Poisson structures and development of appropriate algebraic theory and tools has potentially the same value for the scientist working with these structures (primarily physicists and geometers) as commutative algebra has for algebraic geometers. This approach should bring better clarity to the subject and allow obtaining more detailed and understandable results.