Structural and combinatorial theory of Poisson algebras

Wayne State University, Detroit MI

Investigators

Abstract

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.