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C. Give a short summary of the text



 

Text 6. Some History of Poisson Structures

Following from the work of Lagrange and Poisson discussed at the end of §8.1, the general concept of a Poisson manifold should be credited to Sophus Lie in his treatise on trans­formation groups written around 1880 in the chapter on "function groups." Lie uses the word "group" for both "group" and "algebra." For example, a "function group" should really be translated as "function algebra."

Lie defines what today is called a Poisson structure. The title of Chapter 19 is The Coadjoint Group, which is explicitly identified on page 334. Chapter 17, pages 294-298, defines a linear Poisson structure on the dual of a Lie algebra, today called the Lie-Poisson structure, and "Lie's third theorem" is proved for the set of regular elements. On page 349, together with a remark on page 367, it is shown that the Lie-Poisson structure naturally induces a symplectic structure on each coadjoint orbit. As we shall point out in §11.2, Lie also had many of the ideas of momentum maps. For many years this work appears to have been forgotten.

Because of the above history. Marsden and Weinstein [1983] coined the phrase "Lie-Poisson bracket" for this object, and this terminology is now in common use. However, it is not clear that Lie understood the fact that the Lie-Poisson bracket is obtained by a simple reduction process, namely, that it is induced from the canonical cotangent Poisson bracket on T*G by passing to g* regarded as the quotient T*G/G, as will be explained in Chapter 13. The link between the closedness of the symplectic form and the Jacobi identity is a little harder to trace explicitly; some comments in this direction are given in Souriau [1970], who gives credit to Maxwell.

Lie's work starts by taking functions f1, ... ,Fr on a symplectic manifold M, with the property that there exist functions of r variables such that

{Fi,Fj} = Gij(F1,...,Fr).

In Lie's time, all functions in sight are implicitly assumed to be analytic. The collection of all functions of f1, ... ,Fr is the "function group"; it is provided with the bracket

where

and

Considering F = (F1,... ,Fr) as a map from M to an r-dimensional space P, and and as functions on P, one may formulate this as saying that [ , ] is a Poisson structure on P, with the property that

.





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