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- W890486392 abstract "Dirac structures are characterized in terms of their characteristic pairs dened in this note and then Poisson reductions are discussed from the point of view of Dirac structures. 1. Introduction. Dirac structures on manifolds include closed 2-forms, Poisson structures, and foliations which were introduced by Courant and Weinstein and thor- oughly investigated in (2). Their Lie algebroid version was developed by Weinstein, Xu and the author in (6) and (7). In this note, we characterize a Dirac structure in terms of its characteristic pair (D; ), which consists of a subbundle and a bivector eld. One can see that it becomes more convenient to check the integrability and to study the reduction of a Dirac structure described this way. In the Lie bialgebra case, such a description is given by Diatta and Menida in (3). As natural generalizations of symplectic reductions, Poisson reductions admit many applications. There are already several versions and dieren t approaches for doing it, e.g., Marsden and Ratiu's theorem in (10) and that in (13) given by Weinstein by means of the coisotropic calculus. In fact, it is also an original purpose of the construction of Dirac structures to describe Poisson reductions. Of course, the same conclusion can be obtained from all of other approaches. But, anyway, we wish to illuminate more geometric pictures and algebraic relations behind the Poisson reduction in order to construct a unied framework for its various generalizations (e.g., (1), (4), (8), (11) and (12)) by use of the theory of Dirac structure, which has been shown to be a powerful tool as well as a beautiful theory." @default.
- W890486392 created "2016-06-24" @default.
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- W890486392 date "2000-01-01" @default.
- W890486392 modified "2023-10-17" @default.
- W890486392 title "Some Remarks on Dirac Structures and Poisson Reductions" @default.
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