Mathematical quantization | Topology of homogeneous spaces | Symplectic geometry

Lagrangian Grassmannian

In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space V. Its dimension is 1/2n(n + 1) (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 2n. It may be identified with the homogeneous space of complex dimension 1/2n(n + 1) Sp(n)/U(n), where Sp(n) is the compact symplectic group. (Wikipedia).

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Vladimir Arnold | Tangent space | Unitary group | Hamiltonian vector field | Symplectomorphism | Grassmannian | Gauss map | Symplectic vector space | Determinant | Mathematics | Cauchy–Riemann equations | Unit circle | Reeb vector field | Floer homology | Symplectic geometry | Unitary matrix | Quantum chaos | Fundamental group | Orthogonal group | Symplectic group | Symplectic manifold | Homogeneous space | WKB approximation | Bott periodicity theorem