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In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space ''V''. Its dimension is ''n(n+1)/2'' (where the dimension of ''V'' is ''2n''). It may be identified with the homogeneous space :''U''(''n'')/''O''(''n''), where ''U''(''n'') is the unitary group and ''O''(''n'') the orthogonal group. Following Vladimir Arnold it is denoted by Λ(''n''). The Lagrangian Grassmannian is a submanifold of the ordinary Grassmannian of V. A complex Lagrangian Grassmannian is the complex homogeneous manifold of Lagrangian subspaces of a complex symplectic vector space ''V'' of dimension 2''n''. It may be identified with the homogeneous space of complex dimension ''n(n+1)/2'' :''Sp''(''n'')/''U''(''n''), where ''Sp''(''n'') is the compact symplectic group. ==Topology== The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the Bott periodicity theorem: , and – they are thus exactly the homotopy groups of the stable orthogonal group, up to a shift in indexing (dimension). In particular, the fundamental group of is infinite cyclic, with a distinguished generator given by the square of the determinant of a unitary matrix, as a mapping to the unit circle. Its first homology group is therefore also infinite cyclic, as is its first cohomology group. Arnold showed that this leads to a description of the Maslov index, introduced by V. P. Maslov. For a Lagrangian submanifold ''M'' of ''V'', in fact, there is a mapping :''M'' → Λ(''n'') which classifies its tangent space at each point (cf. Gauss map). The Maslov index is the pullback via this mapping, in :''H''1(''M'', Z) of the distinguished generator of :''H''1(Λ(''n''), Z). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lagrangian Grassmannian」の詳細全文を読む スポンサード リンク
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