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.
As a homogeneous space To see that the Lagrangian Grassmannian Λ(n) can be identified with U(n)/O(n), note that C n {\displaystyle \mathbb {C} ^{n}} is a 2n-dimensional real vector space, with the imaginary part of its usual inner product making it into a symplectic vector space. The Lagrangian subspaces of C n {\displaystyle \mathbb {C} ^{n}} are then the real subspaces L ⊆ C n {\displaystyle L\subseteq \mathbb {C} ^{n}} of real dimension n on which the imaginary part of the inner product vanishes. An example is R n ⊆ C n {\displaystyle \mathbb {R} ^{n}\subseteq \mathbb {C} ^{n}} . The unitary group U(n) acts transitively on the set of these subspaces, and the stabilizer of R n {\displaystyle \mathbb {R} ^{n}} is the orthogonal group O ( n ) ⊆ U ( n ) {\displaystyle \mathrm {O} (n)\subseteq \mathrm {U} (n)} . It follows from the theory of homogeneous spaces that Λ(n) is isomorphic to U(n)/O(n) as a homogeneous space of U(n). It is a compact manifold of dimension n ( n + 1 ) / 2 {\displaystyle n(n+1)/2} . It is a (real, nonsingular) projective algebraic variety. Given a Lagrangian subspace A, the set of Lagranigian subspaces complementary to A is affine. Given an arbitrary complementary subspace B, this affine space consists of the graphs of symmetric linear operators u : B → A {\displaystyle u:B\to A} , G ( u ) = { b + u ( b ) | b ∈ B } {\displaystyle G(u)=\{b+u(b)|b\in B\}} . This is an affine space of dimension n ( n + 1 ) / 2 {\displaystyle n(n+1)/2} since the dimensions of A and B are both n. Symmetry here means that the form ω ( b , u ( b ′ ) ) {\displaystyle \omega (b,u(b'))} is a symmetric form on B. Likewise, the tangent space at a lagrangian subspace A is the space of symmetric opeators A → A ∗ {\displaystyle A\to A^{*}} . From the fibration
1 → O ( n ) → U ( n ) → Λ ( n ) {\displaystyle 1\to O(n)\to U(n)\to \Lambda (n)}
the fundamental group may be inferred from the long exact homotopy sequence:
π 1 ( Λ ( n ) ) = Z . {\displaystyle \pi _{1}(\Lambda (n))=\mathbb {Z} .}
Topology 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 (\mathrm {Sp} /\mathrm {U} )\simeq \mathrm {U} /\mathrm {O} } , and Ω ( U / O ) ≃ Z × B O {\displaystyle \Omega (\mathrm {U} /\mathrm {O} )\simeq \mathbb {Z} \times \mathrm {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. Its first homology group is therefore also infinite cyclic, as is its first cohomology group, with a distinguished generator given by the square of the determinant of a unitary matrix, as a mapping to the unit circle. 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 ) {\displaystyle M\to \Lambda (n)}
which classifies its tangent space at each point (cf. Gauss map). The Maslov index is the pullback via this mapping, in
… excerpt ends here. Continue reading the full article.
