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The reduced spaces of a symplectic Lie group action 5 0 0 Juan-Pablo Ortega1 and Tudor S. Ratiu2 2 n a J Abstract 7 There exist three main approaches to reduction associated to canonical Lie group actions on ] a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein re- G duction, and optimal reduction, introduced by the authors. When the action is free, proper, and S admits a momentum map these three approaches coincide. The goal of this paper is to study the . general case of a symplectic action that does not admit a momentum map and one needs to use h its natural generalization, a cylinder valued momentum map introduced by Condevaux, Dazord, t a and Molino [CDM88]. In this case it will be shown that the threereduced spaces mentioned above m do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not sym- [ plectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reductionproducesasymplecticreducedspacewhosePoisson quotientbyacertain Liegroupasso- 1 ciatedtothegroupofsymmetriesoftheproblemequalstheMarsden-Weinsteinreducedspace. We v illustratetheseconstructionswithconcreteexamples,specialemphasisbeinggiventothereduction 8 of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures 9 the non-existence of the momentum map for the lifted action. The precise relation of the cylinder 0 valued momentum map with group valued momentum maps for Abelian Lie groups is also given. 1 0 5 Contents 0 / h Abstract 1 t a m 1 Introduction 2 : v 2 The cylinder valued momentum map 3 i X 3 The equivariance properties of the cylinder valued map 7 r a 4 Poisson structures on g / 10 ∗ H 5 The reduction theorems 17 6 Example: Magnetic cotangent bundles of Lie groups 23 7 Appendix: The relation between Lie group and cylinder valued momentum maps 32 Bibliography 35 1Centre National de la Recherche Scientifique, Département de Mathématiques de Besançon, Université de Franche-Comté, UFR des Sciences et Techniques. 16, route de Gray. F-25030 Besançon cedex. France. Juan- [email protected] 2Section de Mathématiques and Centre Bernoulli. Ećole Polytechnique Fédérale de Lausanne. CH-1015 Lausanne. Switzerland. Tudor.Ratiu@epfl.ch 1 Ortega and Ratiu: The reduced spaces of a symplectic Lie group action 2 1 Introduction Let (M,ω) be a connected paracompactsymplectic manifold acted upon properly and canonically by a Lie group G. In this paper it is assumed that the G-action is free; the non-free case is the subject of [OR05]. Letgbe the Lie algebraofGandg itsdual. Assume forthe momentthatthe actionadmitsa ∗ standard equivariant momentum map J: M g . There are three main approaches to the symmetry ∗ → reduction of (M,ω) by G that yield, up to connected components, the same spaces: Foliation reduction [C22]: consider the fiber J 1(µ) and the characteristic distribution D = − • kerTJ (kerTJ)ω onit;theupperindexω onavectorsubbundleofTM denotestheω-orthogonal ∩ complement. The symplectic structure of (M,ω) drops naturally to the leaf space J 1(µ)/D. − Marsden-Weinsteinreduction[MW74]: letG betheisotropysubgroupoftheelementµ g µ ∗ • ∈ with respect to the coadjoint action of G on g . The orbit manifold J 1(µ)/G inherits from ∗ − µ (M,ω) a natural symplectic form ω uniquely characterized by the expression i ω = π ω , with µ ∗µ µ∗ µ i :J 1(µ)֒ M the inclusion and π :J 1(µ) J 1(µ)/G the projection. µ − µ − − µ → → Optimal reduction [OR02, O02]: let A be the distribution on M defined by A := X • ′G ′G { f | f C (M)G . The distribution A is smooth and integrable in the sense of Stefan and Suss- ∈ ∞ } ′G mann [St74a, St74b, Su73]. The optimal momentum map : M M/A is defined as the J −→ ′G canonical projection onto the leaf space of A which is, in most cases, not even a Hausdorff ′G topological space, let alone a manifold. For any g G, the map Ψ (ρ) = (g m) M/A de- ∈ g J · ∈ ′G fines a continuous G-action on M/A with respect to which is G-equivariant. The orbit space ′G J M := 1(ρ)/G isasmoothsymplecticregularquotientmanifoldwithsymplecticformω char- ρ − ρ ρ J acterizedbyπ ω =i ω,whereπ : 1(ρ) 1(ρ)/G istheprojectionandi : 1(ρ)֒ M ρ∗ ρ ∗ρ ρ J− →J− ρ ρ J− → the inclusion. These reductiontheoremsare importantforsymmetric Hamiltonian dynamics since the flow associated to aG-invariantHamiltonianfunctionprojectstoaHamiltonianflowonthesymplectic reducedspaces. Ourgoalinthispaperistocarryouttheregularreductionprocedureforany symplecticaction,even whenamomentummapdoesnotexist. Aswillbeshown,thethreeapproachestoreductionyieldspaces that are, in general, distinct but that are non-trivially related to each other in very interesting ways. Our results are based on a key construction of Condevaux, Dazord, and Molino [CDM88] naturally generalizingthe standardmomentummaptoa cylinder valued momentum mapK:M Ra Tb, a,b N, that always exists for any symplectic Lie group action. The cylinder Ra Tb is o→btain×ed as ∈ × the quotient g / , with a discrete subgroup of (g ,+) which is the holonomy of a flat connection ∗ ∗ H H on the trivial principal fiber bundle π : M g M with (g ,+) as Abelian structure group. This ∗ ∗ × → flatconnectionis constructedusingexclusivelythe canonicalG-actionandthe symplectic formω onM thereby justifying the name Hamiltonian holonomy for . H The main result. Let (M,ω) be a connected paracompact symplectic manifold and G a Lie group acting freely and properly on it by symplectic diffeomorphisms. Let K:M g / be a cylinder valued ∗ → H momentum map for this action. Then g / carries a natural Poisson structure and there exists a ∗ H smooth G-action on it with respect to which K is equivariant and Poisson. Moreover: (i) The Marsden-Weinstein reduced space M[µ] := K 1([µ])/G , [µ] g / , has a natural Poisson − [µ] ∗ ∈ H structure inherited from the symplectic structure (M,ω) that is, in general, degenerate. M[µ] will be referred to as the Poisson reduced space. (ii) The optimal reduced spaces can be naturally identified with the symplectic leaves of M[µ]. (iii) The reduced spaces obtained by foliation reduction equal the orbit spaces M := K 1([µ])/N , [µ] − [µ] where N is anormalconnectedLie subgroupofGwhose Liealgebra is theannihilator n:= Lie ◦ H ⊂ g of Lie g in g. The manifolds M will be referred to as the symplectic reduced(cid:0)spa(cid:0)ce(cid:1)s(cid:1). ∗ [µ] H ⊂ (iv)The(cid:0)qu(cid:1)otient group H := G /N acts canonically on M and the quotient Poisson manifold [µ] [µ] [µ] [µ] M /H is Poisson diffeomorphic to M[µ]. [µ] [µ] Ortega and Ratiu: The reduced spaces of a symplectic Lie group action 3 As will be shown in the course of this paper, one of the reasons behind the existence of the three distinct reducedmanifolds is the non-closednessofthe discrete Hamiltonianholonomy (asthe holon- H omy group of a flat connection). In fact, measures in some sense the degree of degeneracy of the H Poissonstructure ofthe Marsden-WeinsteinreducedspaceM[µ]. Moreover,when is closed,the three H reduction approaches yield (up to connected components) the same symplectic space. The present paper deals only with free actions. In our forthcoming paper [OR05] we will study the situation in which this hypothesis has been dropped. The contents of the paper are as follows. Section 2 introduces and presents in detail the properties of the cylinder valued momentum map. Section 3 studies the invariance properties of the Hamiltonian holonomy andconstructsa naturalactiononthe targetspaceofthe cylindervaluedmomentummap H with respect to which the cylinder valued momentum map is equivariant. This action is an essential ingredient for reduction. Section 4 defines a Poisson structure on the target space of the cylinder valued momentum map with respect to which this map is Poisson. It also provides a careful study of this Poisson structure and explicitly characterizes its symplectic leaves. This section also contains a general discussion on central extensions of Lie algebras and groups, their actions, and their role in the characterization of the symplectic leaves of affine and projected affine Lie-Poisson structures on duals of Lie algebras. Apart from its intrinsic interest, this information on central extensions will be heavily used in the example of Section 6. Section 5 contains a detailed statement and proof of the reduction results announced above. Section 6 contains an in-depth study of an example that illustrates some of the mainresultsinthe paper. The cotangentbundle ofaLiegroupis considered,butwithasymplectic structure that is the sum of the canonical one and of an invariant magnetic term, whose value at the identity does not integrateto a grouptwo-cocycle. This modification destroys,in general,the existence of a standardmomentum map for the lift of left translations and forces the use of all the developments in the paper. This section contains an interesting generalization of the classical result that states that the coadjoint orbits endowed with their canonical Kostant-Kirillov-Souriau symplectic structure are symplecticreducedspacesofthecotangentbundleofthecorrespondingLiegroup. Thepaperconcludes with anappendix thatspecifiesthe relation,inthe contextofAbelianLiegroupactions,ofthe cylinder valued momentum map and the so called Lie group valued momentum maps. Notations andgeneral assumptions. Manifolds: Inthispaperallmanifoldsarefinitedimensional. Group actions: The image of a point m in a manifold M under a group action Φ : G M M × → is denoted interchangeably by Φ(g,m) = Φ (m) = g m, for any g G. The symbol L : G G g g · ∈ → (respectively R : G G) denotes left (respectively right) translation on G by the group element g → g G. The group orbit containing m M is denoted by G m and its tangent space by T (G m) m ∈ ∈ · · or g m. The Lie algebra of the group G is usually denoted by g. Given any ξ g, the symbol ξ M · ∈ denotes the infinitesimal generator vector field associated to ξ defined by ξ (m) = d exp tξ m, M dt t=0 · for any m M. A right (left) Lie algebra action of g on M is a Lie algebra (anti)h(cid:12)omomorphism ∈ (cid:12) ξ g ξ X(M) such that the mapping (m,ξ) M g ξ (m) TM is smooth. If g acts M M ∈ 7−→ ∈ ∈ × 7−→ ∈ on a symplectic manifold (M,ω) we say that the g-action is canonical when £ ω =0, for any ξ g. ξM ∈ The Chu map: Given a symplectic manifold (M,ω) acted canonically upon by a Lie algebra g, the Chu map Ψ : M Z2(g) is defined by the expression Ψ(m)(ξ,η) := ω(m)(ξ (m),η (m)), for any M M → m M, ξ,η, g. ∈ ∈ 2 The cylinder valued momentum map In this section we define carefully the cylinder valued momentum map and study its elementary properties. This construction, first introduced by Condevaux, Dazord, and Molino in [CDM88] under the name of “reduced momentum map”, is the key stone of the main results in this paper. Thefollowingnotationswillbeusedthroughoutthiswork. If , :W W Risanondegenerate ∗ h· ·i × → duality pairing and V W, define the annihilator subspace V := α W α,v = 0 for all v ◦ ∗ ⊂ { ∈ | h i ∈ V W and similarly for a subset of W . If (S,ω) is a symplectic vectorspace and U S, define the ∗ ∗ }⊂ ⊂ ω-orthogonal subspace Uω := s S ω(s,u)=0for allu U . { ∈ | ∈ } Let (M,ω) be a connected and paracompact symplectic manifold and let g be a Lie algebra that Ortega and Ratiu: The reduced spaces of a symplectic Lie group action 4 acts canonically on M. Take the Cartesian product M g and let π :M g M be the projection ∗ ∗ × × → onto M. Consider π as the bundle map of the trivial principal fiber bundle (M g ,M,π,g ) that has ∗ ∗ × (g ,+) as Abelian structure group. The group (g ,+) acts on M g by ν (m,µ):=(m,µ ν), with ∗ ∗ ∗ × · − m M and µ,ν g . Let α Ω1(M g ;g ) be the connection one-form defined by ∗ ∗ ∗ ∈ ∈ ∈ × α(m,µ)(v ,ν),ξ :=(i ω)(m)(v ) ν,ξ , (2.1) h m i ξM m −h i where (m,µ) M g , (v ,ν) T M g , , denotes the natural pairing between g and g, and ∗ m m ∗ ∗ ∈ × ∈ × h· ·i ξ is the infinitesimal generator vector field associated to ξ g defined by ξ (m)= d exp tξ m, M ∈ M dt t=0 · m M. It is easy to check that α is a flat connection. For (z,µ) M g∗, let (M g(cid:12)∗)(z,µ) be the ∈ ∈ × × (cid:12) holonomy bundle through (z,µ) and let (z,µ) be the holonomy groupof α with reference point (z,µ) H (which is an Abelian discrete subgroup of g by the flatness of α). The Reduction Theorem [KN63, ∗ Theorem 7.1, page 83] guarantees that the principal bundle ((M g∗)(z,µ),M,π (M g∗)(z,µ), (z,µ)) is a reduction of the principal bundle (M g ,M,π,g ); it is her×e that we used t|he×paracompHactness ∗ ∗ × of M since it is a technical hypothesis in the Reduction Theorem. To simplify notation, we will write (M,M,p, ) instead of ((M g∗)(z,µ),M,π (M g∗)(z,µ), (z,µ)). Let K : M M g∗ g∗ be the projectionHinto the g -factor.× | × H ⊂ × → fLet e be the clos∗ure of in g . Since is a closed subgroup of (ge,+)f, the quotient C := g / ∗ ∗ ∗ is a cylHinder (that is, it isHisomorphic to Hthe Abelian Lie group Ra Tb for some a,b N). LHet × ∈ π : g g / = C be the projection. Define K : M C to be the map that makes the following C ∗ ∗ → H → diagram commutative: K M g ∗ −−−e−→ pf πC (2.2)   e K  My g∗/y . −−−−→ H Inotherwords,KisdefinedbyK(m)=π (ν),whereν g isanyelementsuchthat(m,ν) M. This C ∗ ∈ ∈ is agooddefinitionbecauseifwehavetwopoints (m,ν),(m,ν ) M,this implies that(m,ν),(m,ν ) ′ ∈ f ′ ∈ p 1(m) and, as is the structure group of the principal fiber bundle p : M M, there exists an − H f → element ρ such that ν =ν+ρ. Consequently, π (ν)=π (ν ). e We wil∈l rHefer to K : M′ g / =: C as a cyClinder vCalu′ed mome enftum map associated to ∗ → H the canonical g-action on (M,ω). The cylinder valued momentum map is a strict generalization of the standard(Kostant-Souriau)momentummapsinceitiseasytoprove(seeforinstance[OR04,Proposition 5.2.10])thattheG-actionhasastandardmomentummapifandonlyiftheholonomygroup istrivial. H In such a case the cylinder valued momentum map is a standard momentum map. Notice that we refer to “a” and not to “the” cylinder valued momentum map since each choice of the holonomy bundle of the connection (2.1) defines such a map. In order to see how the definition of K depends on the choice of the holonomy bundle M take M and M two holonomy bundles of 1 2 (M g ,M,π,g ). We now notice three things. First, there exists τ g such that M = R (M ), × ∗ ∗ f f ∈f∗ 2 τ 1 where R (m,µ) := (m,µ + τ), for any (m,µ) M g . Second, since (g ,+) is Abelian all the τ ∈ × ∗ ∗ f f holonomy groups based at any point are the same and hence the projection π : g g / in (2.2) C ∗ ∗ → H does not depend on the choice of M; in view of this remark we will refer to as the Hamiltonian H holonomy of the G-action on (M,ω). Third, π is a group homomorphism. Let now p : M M, f C Mi i → KMi :Mi →g∗, and KMi :M →g∗ be the maps in the diagram (2.2) constructed usingefthe hfolonomy beufndlesfMi, i ∈ {1,2}.fLet m ∈ M. By definition KM2(m) = KM2(pM2(m,ν)), where (m,ν) ∈ M2. Since M2f= Rτ(M1) there exists an element ν′ ∈ g∗ sufch that (m,fν′)e∈fM1 and (m,ν) = (m,ν′+fτ). Hence, f f f K (m)=K (p (m,ν))=K (p (m,ν +τ)) M2 M2 M2 M2 M2 ′ f =πCf(KMef2(m,ν′+τ))=fπCe(fν′+τ)=πC(ν′)+πC(τ)=KM1(m)+πC(τ). ef f Ortega and Ratiu: The reduced spaces of a symplectic Lie group action 5 Since in the previous chain of equalities the point m M is arbitrary and τ g depends only on M ∗ 1 ∈ ∈ and M we have that 2 f K =K +π (τ). f M2 M1 C Thefollowingpropositionsummarizestheeflementafrypropertiesofthecylindervaluedmomentummap. Proposition 2.1 Let (M,ω) be a connected and paracompact symplectic manifold and g a Lie algebra actingcanonically onit. Then anycylinder valuedmomentummapK:M C associated tothis action → has the following properties: (i) K is a smooth map that satisfies Noether’s Theorem, that is, for any g-invariant function h ∈ C (M)g := f C (M) dh(ξ ) = 0 for all ξ g , the flow F of its associated Hamiltonian ∞ ∞ M t { ∈ | ∈ } vector field X satisfies the identity h K F =K . ◦ t |Dom(Ft) (ii) For any v T M, m M, we have the relation m m ∈ ∈ T K(v )=T π (ν), m m µ C where µ g is any element such that K(m)=π (µ) and ν g is uniquely determined by: ∗ C ∗ ∈ ∈ ν,ξ =(i ω)(m)(v ), for any ξ g. (2.3) h i ξM m ∈ ω (iii) ker(TmK)= Lie( ) ◦ m . (cid:16)(cid:0) H (cid:1) · (cid:17) (iv) Bifurcation Lemma: range(T K)=T π ((g ) ), m µ C m ◦ where µ g is any element such that K(m)=π (µ). ∗ C ∈ Remark 2.2 Later on in Theorem 5.4 we will show that the cylinder valued momentum map remains constant along the flow of functions that are less invariant than those in part (i) of the previous proposition. Proof. Since g / is a homogeneous manifold, the canonical projection π : g g / is a ∗ C ∗ ∗ H → H surjectivesubmersion. Moreover,by (2.2),K p=π Kisasmoothmap. Thus,sincepisasurjective C ◦ ◦ submersion, it follows that the map K is necessarily smooth. We start by proving (ii). Let m M and (em,µ) pe 1(m). If v =T p(v ,ν) tehen (2.2) gives − m (m,µ) m ∈ ∈ e e T K(v )=T K T p(v ,ν) =T π T K(v ,ν) =T π (ν). m m m (m,µ) m µ C (m,µ) m µ C (cid:16) (cid:17) (cid:0) (cid:1) e e (i) We now check that K satisfies Noether’s condition. Let h C (M)g and let F be the flow of the ∞ t ∈ associated Hamiltonian vector field X . Using the expression for the derivative T K in (ii) it follows h m that T K(X (m)) = T π (ν), where µ g is any element such that K(m) = π (µ) and ν g is m h µ C ∗ C ∗ ∈ ∈ uniquely determined by ν,ξ =(i ω)(m)(X (m))= dh(m)(ξ (m))=ξ [h](m)=0, h i ξM h − M M for all ξ g, which proves that ν =0 and, consequently, T K(X (m))=0, for all m M. Finally, as m h ∈ ∈ d (K F )(m)=T K(X (F (m)))=0, dt ◦ t Ft(m) h t we have K F =K , as required. ◦ t |Dom(Ft) Ortega and Ratiu: The reduced spaces of a symplectic Lie group action 6 (iii) Due to the expression in (ii), a vector v kerT K if and only if the unique element ν g m m ∗ ∈ ∈ determined by (2.3) satisfies T π (ν) = 0, that is, ν Lie( ). Equivalently, we have that ν,ξ = 0, ν C ∈ H h i for any ξ (Lie( )) (g ) = g which, in terms of v , yields that (i ω)(m)(v ) = 0 for any ∈ H ◦ ⊂ ∗ ∗ m ξM ω m ξ (Lie( ))◦. This can obviously be rewritten by saying that vm Lie( ) ◦ m . ∈ H ∈(cid:16)(cid:0) H (cid:1) · (cid:17) (iv)Westartbycheckingthatrange(T K) T π ((g ) ). LetT K(v ) range(T K). Letν g m µ C m ◦ m m m ∗ ⊂ ∈ ∈ be the element determined by (2.3) which hence satisfies T K(v ) = T π (ν). Now, notice that for m m µ C any ξ g we have that m ∈ ν,ξ =ω(m)(ξ (m),v )=0 M m h i which implies that ν (g ) . This proves the inclusion range(T K) T π ((g ) ). Hence, the m ◦ m µ C m ◦ ∈ ⊂ equality will be proven if we show that rank(TmK)=dim(TµπC((gm)◦)). (2.4) On one hand we can use the equality in (iii) to obtain rank(TmK)=dimM dim(kerTmK)=dimM dimM +dim Lie( ) ◦ m − − (cid:16)(cid:0) H (cid:1) · (cid:17) =dim Lie( ) ◦ dim gm Lie( ) ◦ . (2.5) (cid:16)(cid:0) H (cid:1) (cid:17)− (cid:16) ∩(cid:0) H (cid:1) (cid:17) On the other hand, dim(TµπC((gm)◦))=dim(gm)◦−dim(kerTµπC|(gm)◦)=dimg−dimgm−dim(kerTµπC ∩(gm)◦) =dimg dimg dim(Lie( ) (g ) ) m m ◦ − − H ∩ =dimg dimgm dim(Lie( )) dim(gm)◦+dim(Lie( )+(gm)◦) − − H − H = dim(Lie( ))+dimg dim([Lie( )+(gm)◦]◦)=dim Lie( ))◦ dim gm Lie( ) ◦ , − H − H (cid:0) H (cid:1)− (cid:16) ∩(cid:0) H (cid:1) (cid:17) which coincides with (2.5), thereby establishing (2.4). (cid:4) Proposition 2.3 (The cylinder valued momentum map and restricted actions) Let(M,ω)be a connected and paracompact symplectic manifold, g a Lie algebra acting symplectically on it, and K :M g / an associated cylinder valued momentum map. Let h be a Lie subalgebra of g and let g ∗ g → H be the Hamiltonian holonomy of the h-action. Then h H (i) i ( ) , where i :g h is the dual of the inclusion i:h֒ g. Hence there is a unique Lie ∗ g h ∗ ∗ ∗ H ⊂H → → group epimorphism i∗ :g∗/Hg →h∗/Hh such that i∗◦πg∗ =πh∗ ◦i∗, with πh∗ :h∗ →h∗/Hh and πg∗ :g∗ g∗/ g the natural projections. → H (ii) Let i : M g M h be the map given by i (m,µ) := (m,i (µ)), (m,µ) M g , and M ∗ × ∗ → × ∗ ∗ ∗ ∈ × ∗ g the holonomy bundle used in the construction of K and that contains the point (m ,µ ). Let M b b g 0 0 fh be the holonomy bundle for the H-action containing the point (m,i (µ)). Then ∗ g i (M ) M . (2.6) ∗ g ⊂ h b f g (iii) Let K :M h / be the h-cylinder valued momentum map constructed using M . Then h ∗ h h → H K =i K . g (2.7) h ∗◦ g Proof. (i) Let µ and c(t) M be a loop in M such that c(0) = c(1) = m whose horizontal g 0 ∈ H ⊂ lift (c(t),µ(t)) is such that µ(0) = µ and µ(1) µ = µ. The horizontality of (c(t),µ(t)) means that 0 0 − for any ξ g the equality µ(t),ξ = ω(c(t))(ξ (c(t)),c (t)) holds. Consequently i µ(1) i µ = ′ M ′ ∗ ∗ 0 ∈ h i − i µ and i µ(t),η = ω(c(t))(η (c(t)),c(t)), for any η h which proves that (c(t),i µ(t)) is the h- ∗ ∗ ′ M ′ ∗ h i ∈ horizontal lift of c(t) passing through (m ,i µ ) and hence that i µ . The rest of the statement is 0 ∗ 0 ∗ h ∈H a straightforwardverification. Ortega and Ratiu: The reduced spaces of a symplectic Lie group action 7 (ii) Let (z,ν) M . By definition there exists a piecewise smooth g-horizontal curve (m(t),µ(t)) such g ∈ that (m(0),µ(0)) = (m ,µ ) and (m(1),µ(1)) = (z,ν). An argument similar to the one that we just f 0 0 used in the proof of (i) shows that the g-horizontality of (m(t),µ(t)) implies the h-horizontality of (m(t),i µ(t))=i ((m(t),µ(t))) and hence (m(1),i µ(1))=i (z,ν) M . ∗ ∗ ∗ ∗ ∈ h (iii) Let m Mbarbitrary. For some µ g∗ such that (m,µ)b Mg weghave that ∈ ∈ ∈ i∗◦Kg (m)= i∗◦πg∗ (µ)=(πhf∗ ◦i∗)(µ). (cid:0) (cid:1) (cid:0) (cid:1) Onthe otherhandKh(m)=πh∗(ν),forsomeν h∗ suchthat(m,ν) Mh. Sinceby(2.6)(m,i∗(µ)) ∈ ∈ ∈ M we have that K (m)=π (i (µ))= i K (m), as required. (cid:4) h h h ∗ ∗◦ g g (cid:0) (cid:1) g 3 The equivariance properties of the cylinder valued map Supposenowthattheg-Liealgebraactionon(M,ω)consideredintheprevioussectionisobtainedfrom acanonicalactionoftheLiegroupGon(M,ω)bytakingtheinfinitesimalgeneratorsofallelementsing. ThemaingoalofthissectionistheconstructionofaG-actiononthetargetspaceofthecylindervalued momentum map K : M g / with respect to which it is G-equivariant. The following paragraphs ∗ → H generalizeto the contextofthe cylindervaluedmomentummapthe strategyfollowedbySouriau[So69] for the standard momentum map. We start with an important fact about the Hamiltonian holonomy of a symplectic action. Proposition 3.1 Let (M,ω) be a connected and paracompact symplectic manifold and Φ:G M M × → a symplectic Lie group action. Then the Hamiltonian holonomy of the action is invariant under the H coadjoint action, that is, Ad∗g−1(H)⊂H, (3.1) for any g G. ∈ Proof. Letν arbitrary. By definition of the holonomygroupthere exists aloopc:[0,1] M at ∈H → a point m M, that is, c(0) = c(1) = m, whose horizontal lift c(t) = (c(t),µ(t)) satisfies the relations ∈ cT(a0k)e=th(eml,oµo)padn:d[0c,(11)] = (Mm,aµt+thνe)p,ofoinrtsgomme µde∈fing∗e.dWbyedn(otw):=eshoΦw(tch(ta)t).AWd∗ge−1wνil∈l pHro,vfeorthaencylagim∈ Gby. g e e → · showing that the horizontal lift d of d is given by e d(t)= Φg(c(t)),Ad∗g−1µ(t) . (3.2) (cid:0) (cid:1) e If this is the case thenAd∗g−1ν ∈H necessarilysinced(0)=(g·m,Ad∗g−1µ)andd(1)=(g·m,Ad∗g−1µ+ Ad∗g−1ν). In order to establish (3.2) it suffices to checek that d(t) is horizontal. Neotice first that e d′(t)= Tc(t)Φg(c′(t)),Ad∗g−1µ′(t) . (cid:0) (cid:1) Then, for any ξ g, we have e ∈ iξMω(Φg(c(t))) Tc(t)Φg(c′(t)) + Ad∗g−1µ′(t),ξ (cid:0) (cid:1) (cid:10) (cid:11) =ω(Φg(c(t))) Tc(t)Φg Adg−1ξ M(c(t)) ,Tc(t)Φg(c′(t)) + µ′(t),Adg−1ξ (cid:0) (cid:0)(cid:0) (cid:1) (cid:1) (cid:1) (cid:10) (cid:11) =ω(c(t)) Adg−1ξ M(c(t)),c′(t) + µ′(t),Adg−1ξ =0 (cid:0)(cid:0) (cid:1) (cid:1) (cid:10) (cid:11) because of the symplectic character of the action and the fact that the curve c(t) is horizontal. (cid:4) Remark 3.2 The connected component of the identity 0 of is a vector sepace. Consequently, H H Lie = . 0 H H ⊂H (cid:0) (cid:1) In order to prove this recall that any Abelian connected Lie group, like , is isomorphic to Ta Rb, 0 for some a,b N. Since is a closedLie subgroupof (g ,+) it cannotcHontain any compactsubg×roup 0 ∗ ∈ H and hence a=0 necessarily. Ortega and Ratiu: The reduced spaces of a symplectic Lie group action 8 Corollary 3.3 In the hypotheses of Proposition 3.1 the following statements hold (i) Ad∗g−1 H ⊂H, for any g ∈G. (cid:0) (cid:1) (ii) Ad∗g−1 Lie H ⊂Lie H , for any g ∈G. (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1) (iii) There is a unique group action d :G g / g / such that for any g G ∗ ∗ ∗ A × H→ H ∈ Ad∗g−1 ◦πC =πC ◦Ad∗g−1. (3.3) The map π : g g / is the projection. We will refer to d as the projected coadjoint C ∗ ∗ ∗ → H A action of G on g / . ∗ H Proof. (i) By (3.1) and the continuity of the coadjoint action we have Ad∗g−1 H ⊂Ad∗g−1(H)⊂H. (cid:0) (cid:1) (ii)Theinclusion(3.1)guaranteesthattherestrictedmapAd∗g−1 :H→HisaLiegrouphomomorphism so it induces a Lie algebra homomorphism (which is itself) Ad∗g−1 : Lie H → Lie H . In particular this implies the statement. (cid:0) (cid:1) (cid:0) (cid:1) (Aiisi)trTaihgehtmfoarpwAardd∗g−v1ergiifivceantiboynAshdo∗gw−1s(tµh+atHt)h:e=mAadp∗g−1dµ+dHefinisewsealnldaecfitinoend.by(cid:4)part(i)andsatisfies(3.3). ∗ A As we will see in the following paragraphs, the results that we just proved allow us to reproduce in the context of the cylinder valued momentum map the techniques introduced by Souriau [So69] to study the equivariance properties of the standard momentum map. Proposition 3.4 Let (M,ω) be a connected and paracompact symplectic manifold and Φ:G M M × → a symplectic Lie group action. Let K:M g / be a cylinder valued momentum map for this action. ∗ → H Define σ :G M g / by ∗ × → H σ(g,m):=K(Φg(m))−Ad∗g−1K(m). (3.4) Then: (i) The map σ does not depend on the points m M and hence it defines a map σ :G g / . ∗ ∈ → H (ii) The map σ : G g / is a group valued one-cocycle, that is, for any g,h G, it satisfies the ∗ → H ∈ equality σ(gh)=σ(g)+ d σ(h). A ∗g−1 (iii) The map Θ: G g / g / ∗ ∗ × H −→ H (g,π (µ)) d (π (µ))+σ(g), C 7−→ A ∗g−1 C defines a G-action on g / with respect to which the cylinder valued momentum map K is G- ∗ H equivariant, that is, for any g G, m M, ∈ ∈ K(Φ (m))=Θ (K(m)). g g (iv) The infinitesimal generators of the affine G-action on g / are given by the expression ∗ H ξ (π (µ))= T π (Ψ(m)(ξ, )), (3.5) g∗/ C − µ C · H for any ξ g, (m,µ) M, and where Ψ : M Z2(g) is the Chu map defined by Ψ(ξ,η) := ∈ ∈ → ω(ξ ,η ), for any ξ,η g. M M ∈f Wewillrefertoσ :G g / asthenon-equivarianceone-cocycleofthecylindervaluedmomentum ∗ → H map K:M g / and to Θ as the affine G-action on g / induced by σ. ∗ ∗ → H H Ortega and Ratiu: The reduced spaces of a symplectic Lie group action 9 Proof. (i) For any g G define the map τ : M g / by τ (m) := σ(g,m). We will prove the g ∗ g ∈ → H claim by showing that τ is a constant map. Indeed, for any point z M and any vector v T M g z z ∈ ∈ Tzτg(vz)=Tg·zK(TzΦg(vz))−TK(z)Ad∗g−1(TzK(vz)). (3.6) Recall now that by part (ii) of Proposition 2.1, T K(v ) = T π (ν), where µ g is any element z z µ C ∗ ∈ such that K(z) = π (µ) and ν g is uniquely determined by the equality ν,ξ = (i ω)(z)(v ), C ∈ ∗ h i ξM z for any ξ g. Equivalently, the relation between v and ν can be expressed by saying that the pair z ∈ (v ,ν) T M, where M is the holonomy bundle of the connection α in (2.1) used in the definition z (z,µ) ∈ of the cylinder valued momentum map K. Let now µ g be such that (g z,µ) M. We now show f f ′ ∗ ′ ∈ · ∈ tthhaetG(-vazc,tνio)n∈. TIn(zd,µee)Md, fiomrpalnieysξthatg(wTezΦhga(vvez),Ad∗g−1ν)∈T(g·z,µ′)M due to the symplfectic character of f ∈ f iξMω(g·z)(TzΦg(vz))=ω(g·z) TzΦg Adg−1ξ M(z),TzΦg(vz) =ω(z) Adg−1ξ M(z),vz (cid:0) (cid:0) (cid:1) (cid:1) (cid:0)(cid:0) (cid:1) (cid:1) =hν,Adg−1ξi=hAd∗g−1ν,ξi, whichprovesthat(TzΦg(vz),Ad∗g−1ν)∈T(g·z,µ′)M andhencethatTg·zK(TzΦg(vz))=Tµ′πC Ad∗g−1ν . If we use this fact to compute the derivatives in the right hand side of (3.6) we obtain (cid:0) (cid:1) f Tzτg(vz)=Tµ′πC Ad∗g−1ν −TK(z)Ad∗g−1(TµπC(ν))=Tµ′πC Ad∗g−1ν −Tµ πC ◦Ad∗g−1 (ν) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) =Tµ′πC Ad∗g−1ν −TµπC Ad∗g−1ν =0. (3.7) (cid:0) (cid:1) (cid:0) (cid:1) The lastequalityfollowsfromthe factthatπ isanAbelianLie grouphomomorphism. Indeed, forany C ρ,µ g ∗ ∈ d d d T π (ρ)= π (µ+tρ)= (π (µ)+π (tρ))= π (tρ)=T π (ρ), µ C dt(cid:12) C dt(cid:12) C C dt(cid:12) C 0 C (cid:12)t=0 (cid:12)t=0 (cid:12)t=0 (cid:12) (cid:12) (cid:12) and hence T π = T π(cid:12) , for any µ g . (cid:12)Finally, the expression (3.7(cid:12)) shows that Tτ = 0, for any µ C 0 C ∗ g ∈ g G. As M is, by hypothesis, connected this guarantees that σ(g,m)=σ(g,m) for any m,m M, ′ ′ ∈ ∈ which proves the claim. (ii) Using the definition (3.4) at the point h m we obtain σ(g) = K(gh m) d K(h m). If we · · −A ∗g−1 · now use the point m M we can write σ(h)=K(h m) d K(m). Consequently, ∈ · −A ∗h−1 σ(g)+ d σ(h)=K(gh m) d K(h m)+ d K(h m) d d K(m) A ∗g−1 · −A ∗g−1 · A ∗g−1 · −A ∗g−1A ∗h−1 =K(gh m) d K(m)=σ(gh), · −A ∗(gh)−1 as required. (iii) It is a straightforwardconsequence of the definition. (iv) By definition d ξ (π (µ))= Θ (π (µ)) g∗/ C dt(cid:12) exptξ C H (cid:12)t=0 (cid:12) d (cid:12) = dt(cid:12)(cid:12)t=0(cid:16)πC(Ad∗exp(−tξ)µ)+σ(exptξ)(cid:17)=−TµπC(ad∗ξµ)+Teσ(ξ). (3.8) (cid:12) (cid:12) Additionally, if K(m)=π (µ) then C d Teσ(ξ)= dt(cid:12)(cid:12)t=0(cid:16)K(exptξ·m)−Ad∗exp(−tξ)K(m)(cid:17)=TmK(ξM(m))+TµπC(cid:0)ad∗ξµ(cid:1). (cid:12) By (2.3), T K(ξ (m(cid:12) )) = T π (ν) with ν g uniquely determined by the expression ν,η = (i ω)(m)(ξm (m)M)=Ψ(m)(ηµ,ξ)C, for any η g∈. H∗mence (3.8) yields (3.5). (cid:4) h i ηM M ∈ Ortega and Ratiu: The reduced spaces of a symplectic Lie group action 10 4 Poisson structures on g / ∗ H In the following theorem we present a Poisson structure on the target space of the cylinder valued momentum map K:M g / with respect to which this mapping becomes a Poisson map. We also ∗ → H seethatthesymplecticleavesofthisPoissonstructurecanbedescribedastheorbitsoftheaffineaction introduced in the previous section with respect to a subgroup of G whose definition is related to the non-closedness of the Hamiltonian holonomy as a subspace of g . ∗ H Theorem 4.1 Let (M,ω) be a connected paracompact symplectic manifold acted symplectically upon by the Lie group G. Let K : M g / be a cylinder valued momentum map for this action with ∗ → H non-equivariance cocycle σ : M g / and defined using the holonomy bundle M M g . The ∗ ∗ bracket , :C (g / ) C→ (g /H ) R defined by ⊂ × {· ·}g∗/ ∞ ∗ H × ∞ ∗ H → f H δ(f π ) δ(g π ) C C f,g (π (µ))=Ψ(m) ◦ , ◦ , (4.1) { }g∗/ C (cid:18) δµ δµ (cid:19) H where f,g C (g / ), (m,µ) M, π :g g / is the projection, and Ψ:M Z2(g) is the Chu ∞ ∗ C ∗ ∗ ∈ H ∈ → H → map, defines a Poisson structure on g / such that ∗ f H (i) K:M g / is a Poisson map. ∗ → H (ii) Theannihilator Lie ◦ gofLie ing∗ isanideal ing. LetN Gbeaconnectednormal H ⊂ H ⊂ LarieesthuebgororbuiptsooffG(cid:0)thwehoa(cid:0)sffie(cid:1)nL(cid:1)eieNa-lgaecbtiroaniso(cid:0)nn:g=(cid:1)/(cid:0)Liein(cid:0)dHu(cid:1)c(cid:1)e◦d.bTyhσes:yGmplecgti/cle.avesof(cid:16)g∗/H,{·,·}g∗/H(cid:17) ∗ ∗ H → H (iii) For any [µ]:=π (µ) g / , the symplectic form ω+ on the affine orbit N [µ] induced by the C ∈ ∗ H N [µ] · · Poisson structure (4.1) is given by ω+ ([µ]) ξ ([µ]),η ([µ]) =ω+ ([µ])( T π (Ψ(m)(ξ, )), T π (Ψ(m)(η, ))) N·[µ] (cid:16) g∗/H g∗/H (cid:17) N·[µ] − µ C · − µ C · =Ψ(m)(ξ,η), for any ξ,η n, (m,µ) M. ∈ ∈ The proof of this theorem refquires several preliminary considerations. Lemma 4.2 Let be the Hamiltonian holonomy in the statement of the previous theorem. Then H [g,g] . (4.2) ◦ H⊂ Moreover [g,g] Lie ◦ (4.3) ⊂ H (cid:0) (cid:0) (cid:1)(cid:1) and hence Lie ◦ is an ideal of g. H (cid:0) (cid:0) (cid:1)(cid:1) Proof. Letµ be arbitrary. Bydefinition, there existsaloopc(t)inM suchthatc(0)=c(1)=m ∈H and a horizontal lift γ(t) := (c(t),µ(t)) M M g such that µ(1) µ(0) = µ. Since γ(t) ∗ ∈ ⊂ × − is horizontal we have i ω(c(t)) = µ(t),ξ , for any ξ g. Since the G-action is symplectic, the ξM ′ h ′ if ∈ infinitesimal generator vector fields ξ and η are locally Hamiltonian, for any ξ,η g, and hence M M ∈ [ξ,η] = [ξ ,η ] is globally Hamiltonian. Let f C (M) be such that X =[ξ,η] = [ξ ,η ]. M M M ∞ f M M M − ∈ − 1 The relation µ=µ(1) µ(0)= µ(t)dt implies thus − 0 ′ R 1 1 1 µ,[ξ,η] = µ(t),[ξ,η] dt= i ω(c(t))dt= ω(c(t))([ξ,η] (c(t)),c(t))dt h i Z h ′ i Z [ξ,η]M ′ Z M ′ 0 0 0 1 1 = ω(c(t))(Xf(c(t)),c′(t))dt= df(c(t))(c′(t))dt=f(c(1)) f(c(0))=f(m) f(m)=0. Z Z − − 0 0 This shows that [g,g] and hence that [g,g] = [g,g] . The inclusion (4.3) is a consequence of (4.2) and RemHark⊂3.2. ◦(cid:4) H ⊂ ◦ ◦