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Dirac structures and dynamical \(r\)-matrices. (English) Zbl 1029.53088

Summary: The purpose of this paper is to establish a connection between various objects such as dynamical \(r\)-matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures and Courant algebroids. In particular, we give a new method of classifying dynamical \(r\)-matrices of simple Lie algebras \(\mathfrak g\), and prove that dynamical \(r\)-matrices are in one-one correspondence with certain Lagrangian subalgebras of \({\mathfrak g}\oplus{\mathfrak g}\).

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
17B62 Lie bialgebras; Lie coalgebras
58H05 Pseudogroups and differentiable groupoids
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
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