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Mastermath Course Symplectic Geometry Assignment 2 the Exercises Marked with a ∗ Are Inleveropdrachten

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Mastermath Course Symplectic Geometry Assignment 2 the Exercises Marked with a ∗ Are Inleveropdrachten Utrecht University, second semester, 2020/ 2021 Fabian Ziltener mastermath course Symplectic Geometry Assignment 2 The exercises marked with a ∗ are inleveropdrachten. The exercises marked with a + are particularly important. * Exercise 1 (canonical linear symplectic form) Let W be a vector space. We denote by W ∗ its ∗ dual space. We define the canonical linear symplectic form ΩW to be the bilinear form on W × W given by w w0 Ω ; := α0(w) − α(w0): W α α0 Show that this is a symplectic bilinear form. The following exercise will be used in Exercise 3 below. Exercise 2 (nondegeneracy and flat map) Let (V; Ω) be a presymplectic vector space. Show that Ω is nondegenerate iff the flat map [Ω is bijective. * Exercise 3 (dimension of symplectic complement) Let (V; Ω) be a symplectic vector space and W a linear subspace of V . We define the Ω-complement (or symplectic complement) of W to be the linear subspace Ω W := v 2 V Ω(v; w) = 0; 8w 2 W ⊆ V: Prove that dim W + dim W Ω = dim V: Ω Hint: Express W in terms of [Ω and the dual of the canonical inclusion ι : W ! V . Use Exercise 2. Remarks: This means that the dimension of the symplectic complement equals the dimension of the orthogonal complement w.r.t. an inner product. (However, in contrast with this, the symplectic complement of W may intersect W nontrivially.) In the lecture we will generalize the above dimension formula to the presymplectic situation. Exercise 4 (rank of direct sum) Show that the rank of the direct sum of two presymplectic vector spaces is the sum of the ranks of the spaces. The following exercise is used in the proof of the classification of presymplectic vector spaces. + Exercise 5 (ranks of isomorphic presymplectic spaces) Show that the ranks of two isomor- phic presymplectic vector spaces agree. Exercise 6 (automorphism group of a tuple of tensors) Let V be a (real finite-dimensional) vector space, m 2 N, and α1; : : : ; αm be tensors on V . Prove that the set Aut α1; : : : ; αm := T 2 Aut(V ) T preserves α1; : : : ; αm is a subgroup of Aut(V ). 1 + Exercise 7 (characterization of symplectic linear automorphism of standard symplectic space) We define the standard linear complex structure on R2n to be the map J st : R2n ! R2n;J st(v) := − v2; v1;:::; −v2n; v2n−1: We identify J st with its matrix w.r.t. the standard basis, i.e., 0 0 −1 1 B 1 0 C B C st B .. C J = B . C : B C @ 0 −1 A 1 0 Prove that a linear map Φ: R2n ! R2n is symplectic if and only if it satisfies ΦT J stΦ = J st; where we view Φ as an 2n × 2n-matrix. Exercise 8 (linear symplectic map) Let (V; Ω) and (V 0; Ω0) be symplectic vector spaces and Φ: V ! V 0 a linear symplectic map. Show the following: (i) Φ is injective. (ii) Assume that V and V 0 have the same dimension. Then Φ is surjective. Remark: Hence every linear symplectic map between symplectic vector spaces of the same dimension is a linear symplectic isomorphism. Exercise 9 (presymplectic basis) Let (V; Ω) be a presymplectic vector space. We denote by n its n dimension. Show that there exist a nonnegative integer m ≤ 2 and a basis 1 m u1; : : : ; um; v ; : : : ; v ; w1; : : : ; wn−2m of V , such that i i Ω(ui; v ) = 1 = −Ω(v ; ui); 8i 2 f1; : : : ; mg; and Ω vanishes on every other ordered pair of basis vectors. Hint: Use the proof of a theorem about presymplectic vector spaces from the lecture. Remark: A basis as above is called a presymplectic basis for (V; Ω). The following exercise will be used in the lecture in the proof of the formula for the dimension of the presymplectic complement. Exercise 10 (flat maps) Let V; W be vector spaces and B : V × W ! R a bilinear map. We define the left and right flat maps for B to be the maps ∗ ∗ [B : V ! W ;[B(v) := B(v; ·); B[ : W ! V ; B[(w) := B(·; w): Show that dim ker [B + dim W = dim V + dim ker B[: Hint: Define a canonical map ∗ V= ker [B ! W= ker B[ and show that it is an isomorphism. Use that a linear map is surjective if its dual map is injective. Canonically identify a finite dimensional vector space with its bidual (:= dual space of the dual space). 2.
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