We discuss the geometry of the Marsden-Ratiu (MR) reduction theorem for a bihamiltonian manifold. We consider the case of the manifolds associated with the Gel'fand-Dickey theory, i.e., loop algebras over SIn. We provide an explicit identification, tailored on the MR reduction, of the Adler-Gel'fand-Dickey brackets (AGD) with the Poisson brackets on the reduced bihamiltonian manifold N. Such an identification relies on a suitable immersion of T*N into the algebra of pseudodifferential operators connected to geometrical features of the theory of (classical) W-n-algebras
Casati, P., Falqui, G., Magri, F., Pedroni, M. (1998). Bihamiltonian reductions and Wn-algebras. JOURNAL OF GEOMETRY AND PHYSICS, 26(3-4), 291-310 [10.1016/S0393-0440(97)00060-0].
Bihamiltonian reductions and Wn-algebras
FALQUI, GREGORIO;MAGRI, FRANCO;
1998
Abstract
We discuss the geometry of the Marsden-Ratiu (MR) reduction theorem for a bihamiltonian manifold. We consider the case of the manifolds associated with the Gel'fand-Dickey theory, i.e., loop algebras over SIn. We provide an explicit identification, tailored on the MR reduction, of the Adler-Gel'fand-Dickey brackets (AGD) with the Poisson brackets on the reduced bihamiltonian manifold N. Such an identification relies on a suitable immersion of T*N into the algebra of pseudodifferential operators connected to geometrical features of the theory of (classical) W-n-algebras
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