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A Guide to Symplectic Geometry

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A Guide to Symplectic Geometry OSU — SYMPLECTIC GEOMETRY CRASH COURSE IVO TEREK A GUIDE TO SYMPLECTIC GEOMETRY IVO TEREK* These are lecture notes for the SYMPLECTIC GEOMETRY CRASH COURSE held at The Ohio State University during the summer term of 2021, as our first attempt for a series of mini-courses run by graduate students for graduate students. Due to time and space constraints, many things will have to be omitted, but this should serve as a quick introduction to the subject, as courses on Symplectic Geometry are not currently offered at OSU. There will be many exercises scattered throughout these notes, most of them routine ones or just really remarks, not only useful to give the reader a working knowledge about the basic definitions and results, but also to serve as a self-study guide. And as far as references go, arXiv.org links as well as links for authors’ webpages were provided whenever possible. Columbus, May 2021 *[email protected] Page i OSU — SYMPLECTIC GEOMETRY CRASH COURSE IVO TEREK Contents 1 Symplectic Linear Algebra1 1.1 Symplectic spaces and their subspaces....................1 1.2 Symplectomorphisms..............................6 1.3 Local linear forms................................ 11 2 Symplectic Manifolds 13 2.1 Definitions and examples........................... 13 2.2 Symplectomorphisms (redux)......................... 17 2.3 Hamiltonian fields............................... 21 2.4 Submanifolds and local forms......................... 30 3 Hamiltonian Actions 39 3.1 Poisson Manifolds................................ 39 3.2 Group actions on manifolds.......................... 46 3.3 Moment maps and Noether’s Theorem................... 53 3.4 Marsden-Weinstein reduction......................... 63 Where to go from here? 74 References 78 Index 82 Page ii OSU — SYMPLECTIC GEOMETRY CRASH COURSE IVO TEREK 1 Symplectic Linear Algebra 1.1 Symplectic spaces and their subspaces There is nothing more natural than starting a text on Symplecic Geometry1 with the definition of a symplectic vector space. All the vector spaces considered will be real and finite-dimensional unless otherwise specified. Definition 1 A symplectic vector space is a pair (V, W), where: • V is a vector space, and; • W : V × V ! R is a non-degeneratea skew-symmetric bilinear form. We say that W is a linear symplectic form. aThat is, if W(v, w) = 0 for all w 2 V, then v = 0. Before anything else, a quick observation: every symplectic vector space (V, W) is even-dimensional. To wit, let m = dim V and pick any basis (v1,..., vm) for V. m Identify W with the matrix (W(vi, vj))i,j=1. Non-degeneracy of W says that det W 6= 0, while skew-symmetry leads to det W = (−1)m det W, and thus (−1)m = 1. Thus m is even. Our first example will be manifestly even-dimensional: Example 1 The mother of all examples (for precise reasons we’ll see later in the chapter) is 2n 2n 2n (R , W2n), where W2n : R × R ! R is given by 0 0 . 0 0 W2n((x, y), (x , y )) = hx, y i − hx , yi. Here, we write R2n = Rn ⊕ Rn and h·, ·i stands for the standard Euclidean inner product in Rn. Here’s a concrete instance (up to perhaps relabeling some coordi- 4 nates): in R , the symplectic form W4 is given by 0 0 0 0 0 0 0 0 W4((x, y, z, w), (x , y , z , w )) = xy − x y + zw − z w. 2n The matrix representing W2n relative to the standard basis of R , in block form, is simply 0 1 . 0 Idn −J2n = @ A . −Idn 0 2n We call (R , W2n) the canonical symplectic prototype. 1By the way, the name “symplectic” is due to Hermann Weyl, changing the Latin com/plex to the Greek sym/plectic. Page 1 OSU — SYMPLECTIC GEOMETRY CRASH COURSE IVO TEREK Here’s another example, more abstract in spirit. Example 2 . Let L be a vector space, and consider V = L ⊕ L∗, where L∗ is the dual space of L. We define WL : V × V ! R by . WL((x, f ), (y, g)) = g(x) − f (y). ∗ This WL is called the canonical symplectic structure of L ⊕ L . Exercise 1 (a) Check that the symplectic forms given in the previous two examples are in- deed non-degenerate. (b) In the previous example, let (e1,..., en) be a basis for L, take the dual ba- sis (e1,..., en) in L∗, and compute the matrix of W relative to the joint basis 1 n ∗ ((e1, 0),..., (en, 0), (0, e ),..., (0, e )) of L ⊕ L . Does the result surprise you? When dealing with spaces equipped with inner products, we have orthogonal com- plements. There’s a natural analogue here: Definition 2 Let (V, W) be a symplectic vector space and S ⊆ V be a subspace. The symplectic . complement of S is defined to be SW = fx 2 V j W(x, y) = 0 for all y 2 Sg. However, the behavior of symplectic complements is not exactly the same as the one for orthogonal complements. The most striking difference is that the sum of S and SW does not need to be direct, nor equal to V, in general. But there’s a saving grace: Proposition 1 Let (V, W) be a symplectic vector space, and S ⊆ V a subspace. Then: (i) dim S + dim SW = dim V. (ii) (SW)W = S. W W (iii) S ⊕ S = V () S \ S = f0g () (S, WjS×S) is a symplectic vector space. Proof: ∗ W (i) The map V 3 x 7! W(x, ·)jS 2 S is surjective with kernel S , so the formula follows from the rank-nullity theorem, noting that dim S = dim S∗. (ii) Clearly S ⊆ (SW)W, so equality follows since dim S = dim(SW)W by the previous item applied twice. Page 2 OSU — SYMPLECTIC GEOMETRY CRASH COURSE IVO TEREK (iii) Since dim(S + SW) = dim S + dim SW − dim(S \ SW), item (i) now says that dim(S + S?) = dim V if and only if dim(S \ SW) = 0. In particular, this proposition shows that it is not true that restricting W to sub- spaces of V gives symplectic spaces. This is a very special condition. There are other types of subspaces worthy of attention. Definition 3 Let (V, W) be a symplectic vector space and S ⊆ V be a subspace. We say that S is: (i) symplectic if S \ SW = f0g. (ii) isotropic if S ⊆ SW. (iii) coisotropic if SW ⊆ S. (iv) Lagrangian if S = SW. As immediate examples, all lines passing through the origin are isotropic, and all hyperplanes passing through the origin are coisotropic. And Lagrangian subspaces are the isotropic subspaces which are maximal (relative to inclusion) — it is not hard to see that Lagrangian subspaces are always mid-dimensional. Here are more concrete examples: Example 3 (1) Let S ⊆ Rn be any subspace. Then S × f0g and f0g × S are isotropic subspaces 2n n ? of (R , W2n), and they’re Lagrangian if and only if S = R . Moreover, S × S is always Lagrangian. (2) Let L be a vector space, and consider the canonical symplectic structure WL on V = L ⊕ L∗. If S ⊆ L and X ⊆ L∗ are vector subspaces, then S ⊕ f0g and f0g ⊕ X are both isotropic, and Lagrangian if and only if S = L and X = L∗, respectively. If Ann(S) ⊆ L∗ is the annihilator of S, then S ⊕ Ann(S) is Lagrangian. (3) If (V1, W1) and (V2, W2) are symplectic vector spaces, so is their (external) di- rect sum (V1 ⊕ V2, W1 ⊕ W2), where 0 0 . 0 0 (W1 ⊕ W2)((v1, v2), (v1, v2)) = W1(v1, v1) + W2(v2, v2). W1⊕W2 W1 W2 If S1 ⊆ V1 and S2 ⊆ V2 are subspaces, then (S1 × S2) = S1 ⊕ S2 , so that if S1 and S2 are both of the same “type” (i.e., symplectic, isotropic, coisotropic, Lagrangian), then S1 × S2 will also have the same type. Page 3 OSU — SYMPLECTIC GEOMETRY CRASH COURSE IVO TEREK Exercise 2 n Explore the above example further: if S1, S2 ⊆ R are two subspaces, consider 2n W2n ? ? (R , W2n) and show that (S1 × S2) = S2 × S1 . Conclude that S1 × S2 is ? ? isotropic if and only if S1 ⊆ S2 , coisotropic if and only if S1 ⊇ S2 , and Lagrangian ? 2n if and only if S1 = S2 . State and prove a similar result replacing (R , W2n) with ∗ (L ⊕ L , WL) and orthogonal complements with annihilators. Check the claims made in item (3). The following exercise is also useful to practice using the definitions: Exercise 3 Let (V, W) be a symplectic vector space, and S1, S2 ⊆ V be two subspaces. Show that: W W W (a) (S1 + S2) = S1 \ S2 ; W W W (b) (S1 \ S2) = S1 + S2 . Conclude that if S is any subspace of V, then S \ SW is always isotropic and S + SW is always coisotropic. Hint: how does (b) follow from (a)? Let’s make sure you’re comfortable with everything seen so far. Exercise 4 (Challenge #1) Let V and W be two vector spaces. A map B : V × W ! R is called a perfect pairing if both maps V ! W∗ and W ! V∗ induced by B are isomorphisms. 0 0 0 0 Define WB : (V × W) × (V × W) ! R by WB((v, w), (v , w )) = B(v, w ) − B(v , w). (a) Show that (V × W, WB) is a symplectic vector space and explain how the two main examples we have been dealing with so far fit here. Namely, what are the perfect pairings? (b) Let T : V ! W be a linear map. Show that the graph gr(T) is a Lagrangian subspace of V × W if and only if the diagram V T W =∼ =∼ W∗ V∗ T∗ commutes.
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