LECTURE1:LINEARSYMPLECTICGEOMETRYCONTENTS31.Linear symplectic structure52.Distinguished subspaces73.Linear complex structure104.Thesymplecticgroup**************Information:CourseName:SymplecticGeometryInstructor: Zuoqin WangTime/Ro0m:Wed.2:00pm-6:00pm@1318Reference books:.Lectures on Symplectic Geometry by A. Canas de Silver?Symplectic Techniques in Physics by V.Guillemin and S.Sternberg. Lectures on Symplectic Manifolds by A. Weinstein.Introduction to Symplectic Topology by D.McDuff and D.Salamon.Foundations of MechanicsbyR.Abrahamand J.Marsden.GeometricQuantizationbyWoodhouseCourse webpage:http://staff.ustc.edu.cn/~wangzuoq/Symp15/SympGeom.html*******************************?Introduction:The word symplectic was invented by Hermann Weyl in 1939:he replaced theLatin roots in the word complex, com-plexus, by the corresponding Greek rootssym-plektikos.What is symplectic geometry?.Geometry=background space (smoothmanifold)+extra structure (tensor).-Riemannian geometry = smooth manifold + metric structure*metricstructure=positive-definitesymmetric2-tensor-Complexgeometry=smoothmanifold+complex structure.* complex structure = involutive endomorphism (1,1)-tensor)Symplecticgeometry=smoothmanifold+symplecticstructure
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY Contents 1. Linear symplectic structure 3 2. Distinguished subspaces 5 3. Linear complex structure 7 4. The symplectic group 10 ********************************************************************************* Information: Course Name: Symplectic Geometry Instructor: Zuoqin Wang Time/Room: Wed. 2:00pm-6:00pm @ 1318 Reference books: • Lectures on Symplectic Geometry by A. Canas de Silver • Symplectic Techniques in Physics by V. Guillemin and S. Sternberg • Lectures on Symplectic Manifolds by A. Weinstein • Introduction to Symplectic Topology by D. McDuff and D. Salamon • Foundations of Mechanics by R. Abraham and J. Marsden • Geometric Quantization by Woodhouse Course webpage: http://staff.ustc.edu.cn/˜ wangzuoq/Symp15/SympGeom.html ********************************************************************************* Introduction: The word symplectic was invented by Hermann Weyl in 1939: he replaced the Latin roots in the word complex, com-plexus, by the corresponding Greek roots, sym-plektikos. What is symplectic geometry? • Geometry = background space (smooth manifold) + extra structure (tensor). – Riemannian geometry = smooth manifold + metric structure. ∗ metric structure = positive-definite symmetric 2-tensor – Complex geometry = smooth manifold + complex structure. ∗ complex structure = involutive endomorphism ((1,1)-tensor) – Symplectic geometry = smooth manifold + symplectic structure 1
2LECTURE1:LINEARSYMPLECTICGEOMETRY* Symplectic structure = closed non-degenerate 2-form:2-form=anti-symmetric 2-tensor-Contactgeometry=smooth manifold+contact structure*contactstructure=localcontact1-form. Symplectic geometry v.s. Riemannian geometryVery different (although the definitions look similar)* All smooth manifolds admit a Riemannian structure, but onlysome of them admit symplectic structures.* Riemannian geometry is very rigid (isometry group is small), whilesymplectic geometry is quite soft (the group of symplectomor-phisms is large)* Riemannian manifolds have rich local geometry (curvature etc).whilesymplecticmanifoldshavenolocalgeometry(Darbouxthe-orem)- Still closely related* Each cotangent bundle is a symplectic manifold.* Many Riemannian geometry objects have their symplectic inter-pretations, e.g. geodesics on Riemannian manifolds lifts to geo-desic flow on their cotangent bundles.Symplecticgeometryv.s.complexgeometry-Many similarities. For example, in complex geometry one combine pairsof real coordinates (r,y) into complex coordinates z= r + iy. In sym-plectic geometry one has Darboux coordinates that play a similar role.. Symplectic geometry v.s. contact geometrycontact geometry = the odd-dim analogue of symplectic geometry.Symplectic geometry v.s. analysis-Symplecticgeometryis a languagewhichcanfacilitatecommunicationbetween geometry and analysis (Alan Weinstein).(LAST SEMESTER) Quantization: one can construct analytic objects(e.g. FIOs) from symplectic ones (e.g. Lagrangians)..Symplectic geometry v.s.algebraThe orbit method (Kostant,Kirillov etc)in constructing Liegroup rep-resentations uses symplectic geometry in an essential way: coadjointorbits are naturally symplectic manifolds.-→ geometric quantization.Symplectic geometry v.s.physicsmathematics is created tosolvespecificproblems inphysicsandprovidesthe very language in which the laws of physics are formulated. (VictorGuillemin and Shlomo Sternberg)+general relativity* Riemannian geometry←*Symplecticgeometry←classical mechanics (and quantum me-chanics via quantization),geometrical optics etc.-Symplecticgeometryhas its origin in physics
2 LECTURE 1: LINEAR SYMPLECTIC GEOMETRY ∗ Symplectic structure = closed non-degenerate 2-form · 2-form = anti-symmetric 2-tensor – Contact geometry = smooth manifold + contact structure ∗ contact structure = local contact 1-form • Symplectic geometry v.s. Riemannian geometry – Very different (although the definitions look similar) ∗ All smooth manifolds admit a Riemannian structure, but only some of them admit symplectic structures. ∗ Riemannian geometry is very rigid (isometry group is small), while symplectic geometry is quite soft (the group of symplectomorphisms is large) ∗ Riemannian manifolds have rich local geometry (curvature etc), while symplectic manifolds have no local geometry (Darboux theorem) – Still closely related ∗ Each cotangent bundle is a symplectic manifold. ∗ Many Riemannian geometry objects have their symplectic interpretations, e.g. geodesics on Riemannian manifolds lifts to geodesic flow on their cotangent bundles • Symplectic geometry v.s. complex geometry – Many similarities. For example, in complex geometry one combine pairs of real coordinates (x, y) into complex coordinates z = x + iy. In symplectic geometry one has Darboux coordinates that play a similar role. • Symplectic geometry v.s. contact geometry – contact geometry = the odd-dim analogue of symplectic geometry • Symplectic geometry v.s. analysis – Symplectic geometry is a language which can facilitate communication between geometry and analysis (Alan Weinstein). – (LAST SEMESTER) Quantization: one can construct analytic objects (e.g. FIOs) from symplectic ones (e.g. Lagrangians). • Symplectic geometry v.s. algebra – The orbit method (Kostant, Kirillov etc) in constructing Lie group representations uses symplectic geometry in an essential way: coadjoint orbits are naturally symplectic manifolds. −→ geometric quantization • Symplectic geometry v.s. physics – mathematics is created to solve specific problems in physics and provides the very language in which the laws of physics are formulated. (Victor Guillemin and Shlomo Sternberg) ∗ Riemannian geometry ←→ general relativity ∗ Symplectic geometry ←→ classical mechanics (and quantum mechanics via quantization), geometrical optics etc. – Symplectic geometry has its origin in physics
3LECTURE1:LINEARSYMPLECTICGEOMETRY* Lagrange's work (1808)on celestial mechanics, Hamilton, Jacobi,Liouville,Poisson,Poincare.Arnold etc.-An old name of symplectic geometry: the theory of canonical transformationsIn this course, we plan to cover.Basic symplectic geometry-Linear symplectic geometry Symplectic manifolds- Local normal forms- Lagrangian submanifolds v.s. symplectomorphisms-Related geometric structures-Hamiltonian geometry.Symplectic group actions (= symmetry in classical mechanics)-Themoment mapSymplectic reduction-The convexity theorem-Toricmanifolds.Geometric quantization-Prequantization-Polarization一Geometric quantization***************************************************1.LINEAR SYMPLECTICSTRUCTURE Definitions and examples.Let V be a (finite dimensional) real vector space and 2: V × V-→R a bilinearmap. 2 is called anti-symmetric if for all u, v e V,(1)2(u, v) = -2(v, u).It is called non-degenerate if the associated map: V→V*, 2(u)(u) =2(u, v)(2)is bijective. Obviously the non-degeneracy is equivalent to the condition2(u,)=0,VE2u=0.Note that one can regard as a linear 2-form 2 E A2(V*) via2(u,0)=tutu2Definition 1.1. A symplectic vector space is a pair (V,2), where V is a real vectorspace, and 2 a non-degenerate anti-symmetric bilinear map. 2 is called a linearsymplectic structure or a linear symplectic form on V
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY 3 ∗ Lagrange’s work (1808) on celestial mechanics, Hamilton, Jacobi, Liouville, Poisson, Poincare, Arnold etc. – An old name of symplectic geometry: the theory of canonical transformations In this course, we plan to cover • Basic symplectic geometry – Linear symplectic geometry – Symplectic manifolds – Local normal forms – Lagrangian submanifolds v.s. symplectomorphisms – Related geometric structures – Hamiltonian geometry • Symplectic group actions (= symmetry in classical mechanics) – The moment map – Symplectic reduction – The convexity theorem – Toric manifolds • Geometric quantization – Prequantization – Polarization – Geometric quantization ********************************************************************************* 1. Linear symplectic structure ¶ Definitions and examples. Let V be a (finite dimensional) real vector space and Ω : V × V → R a bilinear map. Ω is called anti-symmetric if for all u, v ∈ V , (1) Ω(u, v) = −Ω(v, u). It is called non-degenerate if the associated map (2) Ω : e V → V ∗ , Ω( e u)(v) = Ω(u, v) is bijective. Obviously the non-degeneracy is equivalent to the condition Ω(u, v) = 0, ∀v ∈ Ω =⇒ u = 0. Note that one can regard Ω as a linear 2-form Ω ∈ Λ 2 (V ∗ ) via Ω(u, v) = ιvιuΩ. Definition 1.1. A symplectic vector space is a pair (V, Ω), where V is a real vector space, and Ω a non-degenerate anti-symmetric bilinear map. Ω is called a linear symplectic structure or a linear symplectic form on V
4LECTURE1:LINEARSYMPLECTICGEOMETRYErample.Let V=R2n=Rn×Rn and define2o((r,s), (y,n)) := (r,n) -(s,y))then (V,2o)is a symplectic vector space.Let [ei,..,en,fi,..*,fn) bethe stan-dard basis of Rn × Rn, then 2 is determined by the relationsVi,j.2o(ei,e,)=o(fi,fi)=0,2o(ei,f)=dij,Denote by [ei,... ,en, ft, ... , f) the dual basis of (Rn)* × (R")*, then as a linear2-form onehas2o=e A ft.Erample. More generally, for any finitely dimensional vector space U, the vectorspace V = U @ U* admits a canonical symplectic structure2((u, α), (v, β)) = β(u) - α(v).Erample. For any nondegenerate skew-symmetric 2n × 2n matrix, the 2-form ZA onIR2ndefinedby2A(X,Y) = (X,AY)isasymplecticformonR2n. Linear Darboux theorem.Definition 1.2. Let (Vi,21) and (V2,2) be symplectic vector spaces. A linearmap F : Vi -→ V2 is called a linear symplectomorphism (or a linear canonical trans-formation)if it is a linear isomorphism and satisfies(3)F*22=21.Erample.AnylinearisomorphismL:U→U2liftstoalinearsymplectomorphismF : Ui④ U* → U2 由U2, F(u, α)) = (L(u), (L*)-1(α),It is not hard to check that F is a linear symplectomorphism.Theorem1.3(LinearDarboux theorem).For any linear symplectic wector space(V,2), there erists a basis [ei,..-,en, fi,.., fn] of V so that(4)Vi,j.2(ei,ej)=2(fi, fi)= 0,2(es,fi)=dij,The basis is called a Darboux basis of (V,2)Remark. The theorem is equivalent to saying that given any symplectic vector space(V,), there exists a dual basis (ei, ..,en, fi,..-, fn] of V* so that as a linear 2-form,(5)n-eifti=1
4 LECTURE 1: LINEAR SYMPLECTIC GEOMETRY Example. Let V = R 2n = R n × R n and define Ω0((x, ξ),(y, η)) := hx, ηi − hξ, yi, then (V, Ω0) is a symplectic vector space. Let {e1, · · · , en, f1, · · · , fn} be the standard basis of R n × R n , then Ω is determined by the relations Ω0(ei , ej ) = Ω0(fi , fj ) = 0, Ω0(ei , fj ) = δij , ∀i, j. Denote by {e ∗ 1 , · · · , e∗ n , f ∗ 1 , · · · , f ∗ n} the dual basis of (R n ) ∗ × (R n ) ∗ , then as a linear 2-form one has Ω0 = Xn i=1 e ∗ i ∧ f ∗ i . Example. More generally, for any finitely dimensional vector space U, the vector space V = U ⊕ U ∗ admits a canonical symplectic structure Ω((u, α),(v, β)) = β(u) − α(v). Example. For any nondegenerate skew-symmetric 2n×2n matrix, the 2-form ΩA on R 2n defined by ΩA(X, Y ) = hX, AY i is a symplectic form on R 2n . ¶ Linear Darboux theorem. Definition 1.2. Let (V1, Ω1) and (V2, Ω2) be symplectic vector spaces. A linear map F : V1 → V2 is called a linear symplectomorphism (or a linear canonical transformation) if it is a linear isomorphism and satisfies (3) F ∗Ω2 = Ω1. Example. Any linear isomorphism L : U1 → U2 lifts to a linear symplectomorphism F : U1 ⊕ U ∗ 1 → U2 ⊕ U ∗ 2 , F((u, α)) = (L(u),(L ∗ ) −1 (α)). It is not hard to check that F is a linear symplectomorphism. Theorem 1.3 (Linear Darboux theorem). For any linear symplectic vector space (V, Ω), there exists a basis {e1, · · · , en, f1, · · · , fn} of V so that (4) Ω(ei , ej ) = Ω(fi , fj ) = 0, Ω(ei , fj ) = δij , ∀i, j. The basis is called a Darboux basis of (V, Ω). Remark. The theorem is equivalent to saying that given any symplectic vector space (V, Ω), there exists a dual basis {e ∗ 1 , · · · , e∗ n , f ∗ 1 , · · · , f ∗ n} of V ∗ so that as a linear 2- form, (5) Ω = Xn i=1 e ∗ i ∧ f ∗ i
LECTURE1:LINEARSYMPLECTICGEOMETRY5This is also equivalent to saying that there exists a linear symplectomorphismF : (V,2) → (R2n,20).Inparticular,. Any symplectic vector space is even-dimensional. Any even dimensional vector space admits a linear symplectic form..Up to linear symplectomorphisms, there is a unique linear symplecticformon each even dimensional vector space.Proof of the linear Darbouc theorem. Apply the Gram-Schmidt process with respect口to the linear symplectic form 2. Details left as an exercise. Symplectic volume form.Since a linear symplectic form is a linear 2-form, a natural question is: which2-form in A?(V*) is a linear symplectic form on V?Proposition 1.4.Let V be a 2n dimensional vector space.A linear 2-form EA2(V*) is a linear symplectic form on V if and only if as a 2n-form,(6)2" =...20EA2n(V*).[We will call a symplectic volume form or a Liouville volume form on V.]Proof. If is symplectic, then according to the linear Darboux theorem, one canchoose a dual basis of V* so that 2 is given by (5). It follows"=nleift..enfn0.Conversely, if is degenerate, then there exists u e V so that (u, u) = o forall u e V.Extend u into a basis {ui,...,u2n) of Vwith ui = u.Then sincedim A2n(V) = 1, ui A.... Au2n is a basis of A2n(V). But (2n(ui A ..- A u2n) = 0. So口2n = 0.2. DISTINGUISHED SUBSPACESSymplectic ortho-complement.Nowwe turn to study interesting vector subspaces of a symplectic vector space(V,2). A vector subspace W of V is called a symplectic subspace if 2/wxw is a linearsymplectic form on W. Symplectic subspaces are of course important.However,insymplectic vector spaces there are many other types of vector subspaces that areeven more important.Definition 2.1. The symplectic ortho-complement of a vector subspace W V is(7)w? = (e V I 2(u, w) = o for all w E W).Erample.If (V,2)=(R2n,2o)and W= span[ei,e2,fi,f3],thenW?= spanfe2, f3,es,...,en, fa,..., fn]
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY 5 This is also equivalent to saying that there exists a linear symplectomorphism F : (V, Ω) → (R 2n , Ω0). In particular, • Any symplectic vector space is even-dimensional. • Any even dimensional vector space admits a linear symplectic form. • Up to linear symplectomorphisms, there is a unique linear symplectic form on each even dimensional vector space. Proof of the linear Darboux theorem. Apply the Gram-Schmidt process with respect to the linear symplectic form Ω. Details left as an exercise. ¶ Symplectic volume form. Since a linear symplectic form is a linear 2-form, a natural question is: which 2-form in Λ2 (V ∗ ) is a linear symplectic form on V ? Proposition 1.4. Let V be a 2n dimensional vector space. A linear 2-form Ω ∈ Λ 2 (V ∗ ) is a linear symplectic form on V if and only if as a 2n-form, (6) Ωn = Ω ∧ · · · ∧ Ω 6= 0 ∈ Λ 2n (V ∗ ). [We will call Ωn n! a symplectic volume form or a Liouville volume form on V .] Proof. If Ω is symplectic, then according to the linear Darboux theorem, one can choose a dual basis of V ∗ so that Ω is given by (5). It follows Ω n = n!e ∗ 1 ∧ f ∗ 1 ∧ · · · ∧ e ∗ n ∧ f ∗ n 6= 0. Conversely, if Ω is degenerate, then there exists u ∈ V so that Ω(u, v) = 0 for all v ∈ V . Extend u into a basis {u1, · · · , u2n} of V with u1 = u. Then since dim Λ2n (V ) = 1, u1 ∧ · · · ∧ u2n is a basis of Λ2n (V ). But Ωn (u1 ∧ · · · ∧ u2n) = 0. So Ω n = 0. 2. Distinguished subspaces ¶ Symplectic ortho-complement. Now we turn to study interesting vector subspaces of a symplectic vector space (V, Ω). A vector subspace W of V is called a symplectic subspace if Ω|W×W is a linear symplectic form on W. Symplectic subspaces are of course important. However, in symplectic vector spaces there are many other types of vector subspaces that are even more important. Definition 2.1. The symplectic ortho-complement of a vector subspace W ⊂ V is (7) WΩ = {v ∈ V | Ω(v, w) = 0 for all w ∈ W}. Example. If (V, Ω) = (R 2n , Ω0) and W = span{e1, e2, f1, f3}, then WΩ = span{e2, f3, e4, · · · , en, f4, · · · , fn}.
6LECTURE1:LINEARSYMPLECTICGEOMETRYFrom the definition one immediately see that if Wi C W2, then W c W, and.asaconsequence,Lemma 2.2. Let Wi,W2 be subspaces of (V,2), then(1) (Wi +W2)? = wnwg.(2) (WinW2)"=w?+w?One can easily observe the difference the symplectic ortho-complement and thestandard ortho-complement Wl with respect to an inner product on V. For exam-ple, one always have wn wl = [0] while in most cases W n w + [0]. However,w and Ww+ do have the same dimensions:Proposition 2.3. dim W = 2n - dim W.Proof. Let W = Im(2/w) C V*. Then dim W = dim W since 2 is bijective. But wealsohavew = (w)° = (ue V : l(u) = o for all l e W).口So the conclusion follows.As an immediateconsequence,we getCorollary 2.4. (w?)? = W.Proof.Thisfollows from dimension counting and thefact W c (w)口Obviously Wn w is a subspace of W, so one can form the quotient spacew/wnw2.The symplectic form2isreduced toa2-form2'on W/wnw?, sinceifwi,w2ewandwi,wewnw?,then2(w1 +wi,w2 +w)=2(W1,W2)Moreover, ' is non-degenerate, since if w e W, and 2(u, w) = o for all w e W, thenby definition w eW?. So we getProposition 2.5. 2' is a symplectic form on W/W n W?.Using this proposition one can extend the linear Darboux theorem toTheorem 2.6 (The Linear “Relative Darboux Theorem"). Given any subspace W CV, we can choose a symplectic basis [ei,... ,en, fi,..., fn) of (v2n,) such thatW = span[ei, ...,ek+, fi,.., f], W? = span[ek+1,... ,en, fk++1,..., fn] andthus Wn w? - span[ek+1,... ,ek+t].口Proof. Exercise.In particular,Corollary 2.7. W is a symplectic subspace of (V,2) Wn w = [0) V -W@w?
6 LECTURE 1: LINEAR SYMPLECTIC GEOMETRY From the definition one immediately see that if W1 ⊂ W2, then WΩ 2 ⊂ WΩ 1 , and, as a consequence, Lemma 2.2. Let W1, W2 be subspaces of (V, Ω), then (1) (W1 + W2) Ω = WΩ 1 ∩ WΩ 2 . (2) (W1 ∩ W2) Ω = WΩ 1 + WΩ 2 . One can easily observe the difference the symplectic ortho-complement and the standard ortho-complement W⊥ with respect to an inner product on V . For example, one always have W ∩ W⊥ = {0} while in most cases W ∩ WΩ 6= {0}. However, WΩ and W⊥ do have the same dimensions: Proposition 2.3. dim WΩ = 2n − dim W. Proof. Let Wf = Im(Ωe|W ) ⊂ V ∗ . Then dim Wf = dim W since Ω is bijective. But we e also have WΩ = (Wf) 0 = {u ∈ V : l(u) = 0 for all l ∈ Wf}. So the conclusion follows. As an immediate consequence, we get Corollary 2.4. (WΩ) Ω = W. Proof. This follows from dimension counting and the fact W ⊂ (WΩ) Ω. Obviously W ∩ WΩ is a subspace of W, so one can form the quotient space W/W ∩WΩ. The symplectic form Ω is reduced to a 2-form Ω0 on W/W ∩WΩ, since if w1, w2 ∈ W and w 0 1 , w0 2 ∈ W ∩ WΩ, then Ω(w1 + w 0 1 , w2 + w 0 2 ) = Ω(w1, w2). Moreover, Ω0 is non-degenerate, since if w ∈ W, and Ω(v, w) = 0 for all v ∈ W, then by definition w ∈ WΩ. So we get Proposition 2.5. Ω 0 is a symplectic form on W/W ∩ WΩ. Using this proposition one can extend the linear Darboux theorem to Theorem 2.6 (The Linear “Relative Darboux Theorem”). Given any subspace W ⊂ V , we can choose a symplectic basis {e1, · · · , en, f1, · · · , fn} of (V 2n , Ω) such that W = span{e1, · · · , ek+l , f1, · · · , fk}, WΩ = span{ek+1, · · · , en, fk+l+1, · · · , fn} and thus W ∩ WΩ = span{ek+1, · · · , ek+l}. Proof. Exercise. In particular, Corollary 2.7. W is a symplectic subspace of (V, Ω) ⇐⇒ W ∩ WΩ = {0} ⇐⇒ V = W ⊕ WΩ.
LECTURE1:LINEARSYMPLECTICGEOMETRY7 Isotropic, coisotropic, and Lagrangian subspaces.Definition 2.8. A vector subspace W of a symplectic vector space (V,2) is called. isotropic if Wcwn.- Equivalently: 2|wxw = 0.- Equivalently: t* = 0 e A?(W*), where t : W → V is the inclusion.- In particular dim W ≤ dim V/2..coisotropicif Ww?-Equivalently: w is isotropic.- In particular dim W ≥ dim V/2..Lagrangian ifW-w?- Equivalently: W is isotropic and dimW= dimV/2- Equivalently: W is coisotropic and dim W= dim V/2-Equivalently:W is both isotropic and coisotropic.- In particular dim W = dim V/2.Erample. If [ei,... ,en, fi, .., fn) is a Darboux basis of (V, 2), then for any 0 <k ≤ n, W, = span(ei, .*., ek, fe+1,... , fn) is a Lagrangian subspace of (V, 2).Erample.Let F:(Vi,2)-→(V2,22)beany linear symplectomorphism.Note that2=2 @ (-22) is a symplectic structure on V=Vi @V2. It is easy to check thatthe graph of F,F= (u1, F(ui)) I U E V),is a Lagrangian subspace of (V, 2). Linear symplectic reduction.Theorem 2.9. Let W be a coisotropic subspace of (V,2), then(1) The induced 2-form 2' is symplectic on the quotient V'= W/n w?(2)IfACVisaLagrangian subspace,thenA'= (Anw)+W")/W?is a Lagrangian subspace of W/w?.Proof. (1) This is a special case of proposition 2.5(2) We first check that A = An W + W? is a Lagrangian subspace of V:X"=(Anw)"nW=(A+w")nw=AnW+W?=^.口It follows that A' is isotropic in V' and dimA'=, dim V'.3.LINEAR COMPLEX STRUCTURE Linear complex structure.Definition 3.1. A compler structure on a vector space V is an automorphismJ: V-→ V such that J? = -Id. Such a pair (V, J) is called a compler vector space
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY 7 ¶ Isotropic, coisotropic, and Lagrangian subspaces. Definition 2.8. A vector subspace W of a symplectic vector space (V, Ω) is called • isotropic if W ⊂ WΩ. – Equivalently: Ω|W×W = 0. – Equivalently: ι ∗Ω = 0 ∈ Λ 2 (W∗ ), where ι : W ,→ V is the inclusion. – In particular dim W ≤ dim V/2. • coisotropic if W ⊃ WΩ. – Equivalently: WΩ is isotropic. – In particular dim W ≥ dim V/2. • Lagrangian if W = WΩ. – Equivalently: W is isotropic and dim W = dim V/2. – Equivalently: W is coisotropic and dim W = dim V/2. – Equivalently: W is both isotropic and coisotropic. – In particular dim W = dim V/2. Example. If {e1, · · · , en, f1, · · · , fn} is a Darboux basis of (V, Ω), then for any 0 ≤ k ≤ n, Wk = span{e1, · · · , ek, fk+1, · · · , fn} is a Lagrangian subspace of (V, Ω). Example. Let F : (V1, Ω1) → (V2, Ω2) be any linear symplectomorphism. Note that Ω = Ω1 ⊕ (−Ω2) is a symplectic structure on V = V1 ⊕ V2. It is easy to check that the graph of F, Γ = {(v1, F(v1)) | v1 ∈ V1}, is a Lagrangian subspace of (V, Ω). ¶ Linear symplectic reduction. Theorem 2.9. Let W be a coisotropic subspace of (V, Ω), then (1) The induced 2-form Ω 0 is symplectic on the quotient V 0 = W/ ∩ WΩ. (2) If Λ ⊂ V is a Lagrangian subspace, then Λ 0 = ((Λ ∩ W) + WΩ )/WΩ is a Lagrangian subspace of W/WΩ. Proof. (1) This is a special case of proposition 2.5. (2) We first check that Λ = Λ ˜ ∩ W + WΩ is a Lagrangian subspace of V : Λ˜Ω = (Λ ∩ W) Ω ∩ W = (Λ + WΩ ) ∩ W = Λ ∩ W + WΩ = Λ˜. It follows that Λ0 is isotropic in V 0 and dim Λ0 = 1 2 dim V 0 . 3. Linear complex structure ¶ Linear complex structure. Definition 3.1. A complex structure on a vector space V is an automorphism J : V → V such that J 2 = −Id. Such a pair (V, J) is called a complex vector space.
8LECTURE1:LINEARSYMPLECTICGEOMETRYThe basic example is of course Cn =R2n, with standard complex structure Jocorresponding to the map“multiplication by i=V-1:Jori=yo,Joyi=-ri.Remarks. Complex structure is very similar to symplectic structure:(1) Since det J? ≥ 0, dim V must be even.(2) For any 2n dimensional vector space V with basis i,-.., n, i, ..., yn, thelinearmapJ defined byJa =Yi, Jyi=-riis a complex structure on V.As in the symplectic case,(R2n, Jo)is essentiallythe only complex vector space of dimension 2n.Theorem 3.2. Let V be an 2n dimensional real vector space and let J be a comperstructure on V. Then there erists a vector space isomorphism : R2n → V suchthat JΦ = ΦJo.口Proof.Exercise. Compatible complex structure.Now suppose (V,2) is symplectic vector space which admits with a complexstructure J.Definition 3.3.Let (V,2) be a symplectic vector space, and J a complex structureon V.(1) We say that J is tamed by if the quadratic form 2(u, Ju) is positivedefinite.(2) We say that J is compatible with 2 if it is tamed by 2 and J is a symplec-tomorphism, i.e.2(Jv, Jw) = 2(, w).An equivalent condition forJ compatiblewith is thatG(v, w) = 2(v, Jw)defines a positive definite inner product on V.One can easily check that Jo iscompatiblewith2oonR2n.The space of 2 compatible complex structures is denoted by J(V,2). It is asubset of End(V). We will see later that it is in fact a smooth submanifold.Proposition 3.4.Every symplectic vector space admits a compatible compler struc-ture. Moreover, given any inner product g(,) on V, one can canonically constructsuch a J
8 LECTURE 1: LINEAR SYMPLECTIC GEOMETRY The basic example is of course C n = R 2n , with standard complex structure J0 corresponding to the map “multiplication by i = √ −1: J0xi = y0, J0yi = −xi . Remarks. Complex structure is very similar to symplectic structure: (1) Since det J 2 ≥ 0, dim V must be even. (2) For any 2n dimensional vector space V with basis x1, · · · , xn, y1, · · · , yn, the linear map J defined by Jxi = yi , Jyi = −xi is a complex structure on V . As in the symplectic case, (R 2n , J0) is essentially the only complex vector space of dimension 2n. Theorem 3.2. Let V be an 2n dimensional real vector space and let J be a compex structure on V . Then there exists a vector space isomorphism Φ : R 2n → V such that JΦ = ΦJ0. Proof. Exercise. ¶ Compatible complex structure. Now suppose (V, Ω) is symplectic vector space which admits with a complex structure J. Definition 3.3. Let (V, Ω) be a symplectic vector space, and J a complex structure on V . (1) We say that J is tamed by Ω if the quadratic form Ω(v, Jv) is positive definite. (2) We say that J is compatible with Ω if it is tamed by Ω and J is a symplectomorphism, i.e. Ω(Jv, Jw) = Ω(v, w). An equivalent condition for J compatible with Ω is that G(v, w) = Ω(v, Jw) defines a positive definite inner product on V . One can easily check that J0 is compatible with Ω0 on R 2n . The space of Ω compatible complex structures is denoted by J (V, Ω). It is a subset of End(V ). We will see later that it is in fact a smooth submanifold. Proposition 3.4. Every symplectic vector space admits a compatible complex structure. Moreover, given any inner product g(·, ·) on V , one can canonically construct such a J.
9LECTURE1:LINEARSYMPLECTICGEOMETRYProof.Take an inner product g on V.Since both g and 2 arenondegenerate,thereexists aAE End(V)suchthat2(u,w)=g(A,w)forall v,w EV.In other words.A is the transpose matrix of 2 in an orthogonal basis.Since is skew-symmetricand nondegenerate, we conclude that A is skew-symmetric and invertible. Moreover,AA*=-A? is symmetric and positive definite, which has a square root VAA*. Itis easy to see that A preserves the eigenspace of AA*, thus preserves the eigenspaceof VAA*.SoA commuteswith VAA*.DefineJ= (VAA)Then J, VAA* and A commutes with each other.But A is skew-symmetric andVAA*is symmetric, so J is skew-symmetric.Moreover, J is an orthogonal matrixJ*J = A*(VAA*)-1(VAA*)-1A= A*(AA*)-1A = Id.(This shows that thedecompositionA=AA*J is just thepolar decomposition ofA). As a corollary, J? - -JJ* = -Id, i.e. J is a almost complex structure. Now itis straightforward to check the compatibility:2(u, Ju) = g(Av, Ju) = g(-JAv, ) = g(VAA*v, v) > 0,2(Jv, Jw) = g(AJv, Jw) = g(JAv, Jw) = g(Av, w) = 2(v, w)口This completes the proof.Remark. 1. In general the given inner product g doesn't equal the inner product Gconstructed via 2 and J above. In fact, there are related to each other viaG(v, w) = 2(u, Jw) = g(VAA*v, w).However, if the inner product g was already compatible with 2, then AA* = Id andthus g coincides with G.2. If (Vt,2t)is a smooth family of symplectic vector spaces, then we can choosea smoothfamilyof innerproductsgt and get a smoothfamilyofcompatiblecomplexstructures Jt.Nowwe can proveTheorem 3.5. The set J(V,2) is contractible.Proof.Fix a 2-compatible complex structureJon V.Define the contractionmapf:[0.1l xJ(V.2)-→J(V.2)asfollows:ForanyJ'EJ(V.2).wehaveanaturallydefined inner product g'. Let gt = tg + (1 - t)g', then gt is an inner product on Vwhich gives us a canonically defined continuous family of complex structure Jt, seeremark2above.Moreover,byremark1weknowthatJo=J',Ji=J.Thusf is口continuous with f(o, J')= J' and f(1, J')= J
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY 9 Proof. Take an inner product g on V . Since both g and Ω are nondegenerate, there exists a A ∈ End(V ) such that Ω(v, w) = g(Av, w) for all v, w ∈ V . In other words, A is the transpose matrix of Ω in an orthogonal basis. Since Ω is skew-symmetric and nondegenerate, we conclude that A is skew-symmetric and invertible. Moreover, AA∗ = −A2 is symmetric and positive definite, which has a square root √ AA∗ . It is easy to see that A preserves the eigenspace of AA∗ , thus preserves the eigenspace of √ AA∗ . So A commutes with √ AA∗ . Define J = √ AA∗ −1 A. Then J, √ AA∗ √ and A commutes with each other. But A is skew-symmetric and AA∗ is symmetric, so J is skew-symmetric. Moreover, J is an orthogonal matrix J ∗ J = A ∗ ( √ AA∗ ) −1 ( √ AA∗ ) −1A = A ∗ (AA∗ ) −1A = Id. (This shows that the decomposition A = √ AA∗J is just the polar decomposition of A). As a corollary, J 2 = −JJ∗ = −Id, i.e. J is a almost complex structure. Now it is straightforward to check the compatibility: Ω(v, Jv) = g(Av, Jv) = g(−JAv, v) = g( √ AA∗v, v) > 0, Ω(Jv, Jw) = g(AJv, Jw) = g(JAv, Jw) = g(Av, w) = Ω(v, w). This completes the proof. Remark. 1. In general the given inner product g doesn’t equal the inner product G constructed via Ω and J above. In fact, there are related to each other via G(v, w) = Ω(v, Jw) = g( √ AA∗v, w). However, if the inner product g was already compatible with Ω, then AA∗ = Id and thus g coincides with G. 2. If (Vt , Ωt) is a smooth family of symplectic vector spaces, then we can choose a smooth family of inner products gt and get a smooth family of compatible complex structures Jt . Now we can prove Theorem 3.5. The set J (V, Ω) is contractible. Proof. Fix a Ω-compatible complex structure J on V . Define the contraction map f : [0, 1]×J (V, Ω) → J (V, Ω) as follows: For any J 0 ∈ J (V, Ω), we have a naturally defined inner product g 0 . Let gt = tg + (1 − t)g 0 , then gt is an inner product on V , which gives us a canonically defined continuous family of complex structure Jt , see remark 2 above. Moreover, by remark 1 we know that J0 = J 0 , J1 = J. Thus f is continuous with f(0, J0 ) = J 0 and f(1, J0 ) = J.
10LECTURE 1:LINEARSYMPLECTIC GEOMETRY4.THESYMPLECTICGROUPStudent presentation after lecture 2:ZHANG Pei.I will add more details later
10 LECTURE 1: LINEAR SYMPLECTIC GEOMETRY 4. The symplectic group Student presentation after lecture 2: ZHANG Pei. I will add more details later