Lagrangian Grassmannian

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 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 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 2n. It may be identified with the homogeneous space of complex dimension n(n+1)/2 where Sp(n) is the compact symplectic group. The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the Bott periodicity theorem: Ω ( S p / U ) ≃ U / O {displaystyle Omega (Sp/U)simeq U/O} , and Ω ( U / O ) ≃ Z × B O {displaystyle Omega (U/O)simeq Z imes BO} – 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 U / O {displaystyle U/O} 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 which classifies its tangent space at each point (cf. Gauss map). The Maslov index is the pullback via this mapping, in