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The purpose of this introductory Chapter is to both establish basic notation and also to give a reasonably complete account of symplectic reduction theory. The first section is a basic introduction, the second provides proofs and the third gives a history of the subject. This chapter focuses on reduction theory in the general setting of symplectic manifolds. The next chapter deals with, amongst other things, the important case of cotangent bundle reduction. Both of these cases are fundamental ingredients in the reduction by stages program.

This chapter gives some additional background on symplectic reduction theory, the main topic being one of the most important cases, namely the symplectic reduction of cotangent bundles. The main results concerning cotangent bundle reduction make use of the theory of principal connections and so we provide the necessary background on this theory in the first section. The chapter closes with a description of the setting for the major topic of the book: reduction by stages.

In this chapter we develop two of the basic results on reduction by stages, namely the case of commuting reduction and semidirect product reduction. While one could view these as special cases of more general theorems to follow in the next chapter, it is worthwhile to see them on their own as more structured preludes to more general cases. In addition, these cases are important in applications as well as for the historical development of the subject.

In this chapter we formulate the first of several reduction by stages theorems in the regular, that is, free actions, case. We state a sufficient condition, called the stages hypothesis under which the two step reduced space is symplectically diffeomorphic to the space obtained by reducing all at once by the original group. In Chapters 11 and 12 we will come up with alternative conditions for reduction by stages based on the use of distribution theory.

As was discussed in the general setting of reduction by stages, we consider a Lie group M with a normal subgroup N ; recall that the goal is to reduce the action of M in two stages, the first stage being reduction by N . The goal of this chapter is to introduce hypotheses under which reduction by stages works—that is, the stages hypothesis (see Definition 5.2.8) is automatically satisfied. The actual reduction by stages procedure for these examples will be carried out in Chapters 8, 9, and 10.

The introductory chapters have discussed the history and the theory of cotangent bundle reduction in some detail. In particular, §2.2 and §2.3 reviewed the basic standard theory of the embedding version of cotangent bundle reduction.

This chapter addresses the basic theory of symplectic reduction by stages for central extensions. Examples are given in the following Chapter. The main feature of this theory is that already after the first reduction, one encounters curvature, or magnetic terms and this complicates the subsequent reductions. To deal with this situation, we use the theory developed in the preceding chapter. The same sort of phenomenon also occurs in Lagrangian reduction by stages, as presented in Cendra, Marsden, and Ratiu [2001a].

We begin this chapter by revisiting the example of the Heisenberg group from §5.1 to illustrate the general theory developed in Chapter 8. As will become evident, now that the general theory is at hand, when it is applied to this example, the results done by hand in §5.1 can be obtained very quickly. This example is interesting because it is a simple, yet nontrivial, example of a group extension that is not a semidirect product.

This chapter is concerned with two major themes. The first theme, which is presented in §10.1 and §10.2, deals with the semidirect product M of a group G with an Abelian group A , where the construction of the semidirect product itself involves an A -valued cocycle of G . In this context, A (which is N in the general theory) is still a normal subgroup and the reduction by stages program is fully carried out. In particular, the stages hypothesis holds and so the reduction by stages program for the action of M on a symplectic manifold can be implemented.We focus on the case of the action of M on T * M (by the lift of right translation) so that the final reduced spaces will be the coadjoint orbits of M .