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Uniqueness of left invariant symplectic structures on the affine Lie group. (English) Zbl 1021.53052

Summary: We show the uniqueness of left invariant symplectic structures on the affine Lie group \(A(n,{\mathbb{R}})\) under the adjoint action of \(A(n,{\mathbb{R}})\), by giving an explicit formula of the Pfaffian of the skew symmetric matrix naturally associated with \(A(n,{\mathbb{R}})\), and also by giving an unexpected identity on it which relates two left invariant symplectic structures. As an application of this result, we classify maximum rank left invariant Poisson structures on the simple Lie groups \(SL(n,{\mathbb{R}})\) and \(SL(n, {\mathbb{C}})\). This result is a generalization of Stolin’s classification of constant solutions of the classical Yang-Baxter equation for \(\mathfrak{sl}(2, {\mathbb{C}})\) and \(\mathfrak{sl}(3,{\mathbb{C}})\).

MSC:

53D05 Symplectic manifolds (general theory)
53D17 Poisson manifolds; Poisson groupoids and algebroids
53C30 Differential geometry of homogeneous manifolds
17B63 Poisson algebras
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