Abstract
Chapters 7 and 8 introduce the reader to symplectic geometry, which plays a tremendous role both in pure mathematics and in physics. In particular, symplectic geometry provides the natural mathematical framework for the study of Hamiltonian systems. In this chapter, we present linear symplectic algebra. We start with a discussion of the elementary properties of symplectic vector spaces, the various types of their subspaces and linear symplectic reduction. Next, we study the symplectic group and its Lie algebra and prove the Symplectic Eigenvalue Theorem. In the second part of this chapter, we investigate the Graßmann manifold of Lagrangian subspaces of a symplectic vector space. In particular, we discuss the Maslov index and the Kashiwara index. These are homotopy invariants which contain information on how the members of a 1-parameter family of Lagrangian subspaces intersect a given Lagrangian subspace. They will play an essential role in the study of geometric asymptotics in Chap. 12.
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- 1.
This is consistent with the definition of the kernel of a multilinear form given in (4.2.12).
- 2.
That is, a positive definite Hermitian form.
- 3.
By means of a local section of the submersion
(right cosets). - 4.
Recall that S2 L ∗ denotes the vector space of symmetric bilinear forms on L.
- 5.
Cf. Remark 4.3.6/4. Both S 1=ℝ/2πℤ and S 1⊂ℂ are endowed with the natural orientations.
- 6.
It decomposes into the isomorphisms
provided by, respectively, the de Rham Theorem, the Universal Coefficient Theorem and the Hurewicz Theorem. Here, H 1(⋅,ℝ) and H 1(⋅) denote the first integer-valued singular cohomology group and the first singular homology group, respectively.
- 7.
This is the Killing vector field generated by J under the action of Sp(V,ω) on
. - 8.
That is, an orientation of the normal bundle of the submanifold
, cf. Remark 2.7.18/2. - 9.
The number of positive minus the number of negative eigenvalues.
- 10.
Some authors call it Wall-Kashiwara index, others Hörmander-Kashiwara index. We refer to the textbooks by de Gosson [72], Libermann and Marle [181], as well as to the Lecture Notes of Meinrenken [208], where the reader can find an exhaustive treatment including historical remarks. An axiomatic approach can be found in [58].
- 11.
This manifold is also called the universal Maslov bundle, see [72].
(right cosets).
.
, cf. Remark 2.7.18/2.References
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Rudolph, G., Schmidt, M. (2013). Linear Symplectic Algebra. In: Differential Geometry and Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5345-7_7
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