Linear Symplectic Algebra
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 t
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Linear Symplectic Algebra
In this chapter, we present linear symplectic algebra, starting with a discussion of the elementary properties of subspaces of a symplectic vector space and of the symplectic group. We also present linear symplectic reduction. In the second part of this chapter, we come to some more advanced topics, all related to the study of the space of Lagrangian subspaces of a given symplectic vector space. In particular, the Maslov index and the Kashiwara index will be presented in some detail. These topological invariants will play an essential role in Chap. 12, in the context of geometric asymptotics.
7.1 Symplectic Vector Spaces Let V be a finite-dimensional vector space over R and let ω ∈ form ω induces a linear mapping ω : V → V ∗ ,
ω (v), u := ω(v, u),
2
V ∗ . The bilinear
v, u ∈ V .
We define the kernel and the rank of ω to be the kernel and the rank of ω , respectively.1 The form ω is called non-degenerate if ω is an isomorphism. This is equivalent to rank ω = dim V or ker ω = 0, that is, vanishing of ω(u, v) for all v ∈ V implies u = 0. If ω is an isomorphism, we denote its inverse by ω . If there is no danger of confusion, we often write v ≡ ω (v) and ρ ≡ ω (ρ). Remark 7.1.1 Let {ei } be a basis in V and let {e∗i } be the dual basis in V ∗ . Then, 1 ω = ωij e∗i ∧ e∗j , 2 1 This
ω (ei ) = ωij e∗j ,
rank(ω) = rank(ωij ),
(7.1.1)
is consistent with the definition of the kernel of a multilinear form given in (4.2.12).
G. Rudolph, M. Schmidt, Differential Geometry and Mathematical Physics, Theoretical and Mathematical Physics, DOI 10.1007/978-94-007-5345-7_7, © Springer Science+Business Media Dordrecht 2013
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Linear Symplectic Algebra
where ωij := ω(ei , ej ). Let V˜ = V / ker ω and denote the induced bilinear form V˜ × V˜ → R by ω. ˜ Obviously, ω˜ is antisymmetric and non-degenerate. Let ˜ e˜i , e˜j ). Since {e˜1 , . . . , e˜r } be a basis in V˜ and let ω˜ ij = ω( det(ω˜ ij ) = det(−ω˜ j i ) = (−1)r det(ω˜ j i ) = (−1)r det(ω˜ ij ), we have (−1)r = 1, that is, r = dim V˜ = rank ω is even. For a subspace W ⊂ V , the ω-orthogonal subspace is defined by W ω := v ∈ V : ω(v, u) = 0 for all u ∈ W .
(7.1.2)
Proposition 7.1.2 For every antisymmetric bilinear form ω on a finite-dimensional real vector space V , there exists an ordered basis {ei } in V such that ω=
n
e∗i ∧ e∗(i+n) ,
(7.1.3)
i=1
that is, ωij has the form
ωij =
with
Jn =
Jn 0
0 0
1n . 0
0 −1n
(7.1.4)
Proof We carry out the following iterative procedure, starting with V0 := V . If ω = 0 on V0 , we can choose any basis in V0 and thus we are done. Otherwise, there exist v1 , u1 ∈ V0 such that ω(v1 , u1 ) = 1. By bilinearity, v1 and u1 are nonzero. By antisymmetry, they cannot be parallel. Hence, they are linearly independent. Let E1 be the subspace spanned by v1 and u1 . We show E1 ∩ E1ω = 0,
E1 + E1ω = V0 .
For the first equation, we decompose v ∈ E1 ∩ E1ω as v = αv1 + βu1 and calculate β = −ω(v, v1 ) = 0 and α = ω(v, u1 ) = 0. For the second equation, let v ∈ V0 .
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