Maslov index

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: ${\displaystyle \Omega (Sp/U)\simeq U/O}$, and ${\displaystyle \Omega (U/O)\simeq Z\times BO}$ – they are thus exactly the homotopy groups of the stable orthogonal group, up to a shift in indexing (dimension).

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