REFERE 10/88/37
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
CANONICAL GROUP ACTIONS
Jose F. Carinena
INTERNATIONAL ATOMIC ENERGY AGENCY
UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION
T
IC/88/3T
International Atomic Energy Agency and I. LOCALLY HAMILTONIAN DYNAMICAL SYSTEMS United nations Educational Scientific and Cultural Organization 1. Review of the theory of symplectic manifolds. In these notes (M,(*>) denotes a BYtnpleetic manifold: A/ is a finite-dimensional differen- tiate manifold and u is a nondegenerate 2-form which satiafiei du = 0 (that i&, u» e Zi(M)). INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS The dimension of M is necessarily even. For general resultB, reference textbooks are, for instance, [l] and [2]. The following well-known result characterises these manifolds completely, from the local [joint of view:
THEOREM I (DARBOUX). Around each point u e M Acre is a local chart (U:ip) such that if 4 - (»\--.>4w;pi,---,p CANONICAL GHOUP ACTIONS ' A dpi. (1.1) Recall that the nondegeneracy of w is equivalent, by definition, to the property that the maps utu : TVM —*T^M, given by: {wu(v),v') :=r wu(v,v'} (1,2) Jose F. Carinena * have maximal rank. International Centre for Theoretical Physics, Trieste, Italy. Now, the resulting map w : I'M •—* T' M is a base-presfrving Jibered mrtp, \.t\, the following diagram commutes: TM I- 1- M —^- M Tims it itnlutes a niap]>ing beLwten sections which, with a slight abuse of notation , we will alsrj write w, Now let H £ Cno(Af), the set of infinitoty dilTercntiabl*1 functions on A/, and lot. us ronfli'ler the 1-form dli. We define the llamiltonian vector field C// assorifit«cl wilh the llamiltuiiiaii // to l>e the unique vector fteid satisfying MIRAMARE - TRIESTE March l°8fi Nol.f that with Die usual definition of contraction of a p-form u by ft vector field ,V: [i(-VH(V, l'p-,):="(-V, >',,.., VP-,) we get. from (1.2): HV,,)v = dll. PROPOSITION 1. C\Hu = 0. PHOOF: For any r» (; A1")^/) we iiave C^" = t'(-V)rfo + rf[i(.Y)n]. Applying I his lirimnU 2 idrntjt.v of (tartan 1.o iht> prcs^nt caat-, we iinrncdiately get CyHuj ^ d ff ~ 0 | Permanent address: uepartamento de Fisica Teorica, Universidad de Zaragoaa, Zaragozn, Spain. This work vaa nroDented during Professor Carinena1 r, visit to the Universidad rte Costa Rica ar> a Visiting Scholar of the ICTP Office of External Act i vi t,i an . His nssociation with Costa Rica is made po:; r, 11 -1 r> ly <*erif"roun funding from tlif Italian Government, The proof of this proposition suggests a more general definition than that of "Hamiltoni.in COROLLARY. XH is an rde-Ai in XLH, in the sense of Lie algebras. field": we Bay that a vector field X € X(M,ui) is locally Hamilioman tfi(X)u G Z'(M). This ! The POMSOTI Anaciei of two functions fTg e C°°(A/) is the function {/,g] j^iveii hy: in equivalent, by Proposition 1, to the vanishing of Cxw, If i(,Y)j E B (A/), then X is TIamiltonian. We have: By the proof of I,h« previous theorem, we have: Here u'1 is an isomorphism of real vector spaces. PROPOSITION 2. In Darhoux coordinates; and from (t-3) we get at once the expression in Darboux roordinates: _ dH a en s (1.3) opt Sq1 oq' dpi oq* dpi tJpi vq' Here and in other parts of these notes, we use the Einstein convention for sums whenever it is convenient, to simplify the notation. iicm&rk, Tlie more general concept of a Poisson structure on a manifold Ai is derived as Examples, a) The classic example is the cotangent bundle T'{Ti.N). If ?r,..., qN arc cartesian follows. Let (/V, •) be An associative algebra of functions over M and let {-.-} ho a bilinear, coordinates on It", we write the induced coordinates in T'(RN) &»((',..,,«";pi , PN). aiitisymmfct.fic operation on N X N satisfying: Then (T'Tf." ,w) with u given by (LI) is a symplectic manifold. Note that w = ~~dO, with: i) {/. {ff- i§, {h,f)) + {h, {f,g}} = 0 (Jacobi iuentity) ii) {/,!7i t/iffi) -ffz+ 01 • {/-flj} (derivation property) (M) Then we say {•,-} defines a Paisson structure on A/. It is readily verified that the Poisson bracket defined earlier is a Poisson structure, if N = CfX>(M) and • denotes the ordinary More generally, if Q is the configuration apace of a mechanical system, we define the (commutative) product of funrlions. In this case, the Jacobi identity is assured by the property 1 N Liouville 1-form on Q by (1.4); (q t ,P\, • -,PN) now denotes the coordinates of a local rfw = U. chart oST'Q induced by a local chart of Q. We then call u := —d9 the lanonual igmplcctir form on T'Q. It is immediate that iLj ~ 0 and that u is nondegenerate. Whenever u> is a 2. Symmetries. differential, we say we have an etacl symplectic manifold. In T'Q we always use this structure Diffeomorphisms between symplectk manifolds induce transformations of locally Tlamil- of exact symplectic manifold. tonifin fields with respect tow into locally Hamillonian fields with refipect to /-.w := (F~1)*LJ, lltanmk, The necessity of considering more general symplectic manifolds than T'KN or even PROPOSITION 3. Let F: M -+ A/ be a difTcomorphism. Then T'Q arises from the process of "reduction" associated with the presence of constants of motion in mechanical systems, which define hypersurfaces in phase space thai in general are not exact symplectic manifolds. where t\w JS a notation for (F~ b) The tangent bundle TQ with local coordinates {g'.f') receives a symplectic structure via the Lagrangian L(q',v')- PROOF-: Let Y e .V(Af). Then by definition of contraction wt = TT*"rS—rrftf1 A (iv1 -f -T—rr—rdtj* A tiq1. tJVx yi'J I7U' i)/}1 by definition of F+u The nondegeneracy condition may in tliis case be written : Tims it is clear that i(X)u> - 0 => t(F.A')(F*w) = 0. I We say that a difleomorphism F: jl/ —* A/ is a sytnplcctomorphism if £'*<*> — w. We have finished the preliminaries. We now introduce some of tin- main definitions of THEOREM 2. If X,Y Remark. Denote by g the Lie algebra of G. It is clear that *„., : 0 e TrG — TUM. The map uf.V,,) = — dfa, or equivalently (l-fi) ) y I- Moreover, the minus sign has been introduced for A* to be a Lie algebra homomorphism, i.n. A - (Af Xj,]. Tli*: flow of the vector field X € -V(Jl/) is ^,ii) = *(exp("ia), u). We M] fl1 a [t is clear that the definition ia lioL unique and tlial two possible conioinriitum [ii.i|t|iingR ilif then have hy a "VoiLsLant": /;(«)= /a(u) + r(a) PROPOSITION 4, i) £x,^ = 0, Vn 6 S, if a"J on'r itG is a group of aymplectomorpliisms: whore r £ Q" by linearity. ii) Cx, H = 0, Va e 0: if and un(y ifG is a group of symmetries of H; We j>ropGse to identify the conditions under which fa is a Lie algebra liainoiiiorphism. iii) £jf4T =: 0, Vc € @T if anri only if G is a group of symnwirics of P. Hy (1.5) we have Condition (i) guarantees that the fundamental vector fields for a symplectic action n(G on "({/«./*}) = [.V.,A'»] = X[mM = ff(/[..t|) (Af,u) are always locally Harniltonian. If the stronger property that X\ 6 Xn{M,w), Vu 6 0 is satisfied, we say the action of (7 is HamiHoiiian or strongly cyinploctic. To characterize this last type of action, let us remark that (1.5) means that the mapping 1 X a := —w" o d is a Lie algebra homomorphism from C (M) into ,1'H(A/,U): where "(/') = T!i« liilirirar Hiitisymrnel.ric function -PH. On the other hand, we recall that A': 8 — ,Vr,w(Af, u), with X(a) = A'«, is a Lie algebra homomorphism. From Theorem 2, we si*e that we can define a new Lie algebra homoruorphism ((«.!>) ={U ft}- f[a.t] X so that the following diagram commutes: urrs llic ohstrurtiori to / hting a liomoniorphisni. If v/r /^ :— /a + ''(«), wo grl 0 s/e' We wi.sli Lu know wfirl IHT we ran ill gpJier.il re.drfine / in such a way tlint, £' ^ 0. where g' is the derived algebra a' = [p, a] and II'(M) denotes the first de Rham cohomology K.vaiMji/c. TIH: omiiomeiiluut mapping )fi a LJP algebra homomofpliism in the rtuic r>f i*xarl group, which has llu- Lie algebra structure transported from XLH(M, u>) via (*;..».,«) = <*«. *««(o» = -(*.*•>„• Before continuing, we note that ( must satisfy the equation where a; denotes, as usual, that the element a, is omitted. The linear operator i on C(g, a) := a 2 {([a, 4], e) + fl[t, e], a) + {([ n n 1 This follows from the equality J) (»,o) ;= (oEC"(|,a) :3y9eC - (B.o)such that a = W} = tni^-i, ^"(B.o) *- |o6C"(l.») :6a-0) = kerin. Tile elements of Z" are called n-cocycles, and tiiose of fi" are called Ji-coboundari<'8. and the Jacobi identities for {•,-} and [•,•]. Since fi1 = 0, we have B" C Z", The nth cohomology group //"(n,a) is definrd as Examples, a) In the specific case of the additive group R3 acting by translations on Q = R3, the corresponding action on 7"(R3) is evidently symplectic. To find the momentum mapping, note that the fundamental vector fields are X, = -a'—, with canonical liftings to T'(Tl3} given by "'' Kxiunpies. a) Let a := R, *(o) := 0 Va 6 0- In this case, if a g C'iG.Il). llit-ti (*«)(ai ,n;) = o([oi,a3)). Thus 7'(8,R) = (« 6 B' : (i([oi,oj]) = 0). If/3 € C'(g,R), ^ is a 2-cocyck if For o e R3, the function /. E C°°(T'R.3) is defined by ^([ai,aj],a3) + (3([iij,O3],o,) + /?([ii,,o1],a2) = 0- 5 /. = -i(Xt)B =a'Pi We conclude that ( may be identified with an element of Z (rJ,n.) and that {'1.2) may be rewritten 4^' ~ ^ + &*• So we have Using Sa( b) Let a := A"(Af), ^. Then 3 b) The natural action of the rotation group £0(3, R) on R also induces a symplectic action (Sa){a,b) = Cx.a(l>) - Cxt»{«)-<*([«,!> on T'lt.3, The fundamental vector fields on R3 are The elements of H'(B, AP(A/)) are those a for which 3^ £ A'(Af) with o(a) = £»./?. The elements of Z^($, \r(M)) are linear maps a: a —* AP(A/) satisfying £*.«(*)-£*,«( J hiq.f) = ni(j,-kij pt = ff • (f x p). maps n-cochains into n-cochains and n-coboundariefi into n-coboundaries. since S,t(Fa) = F(S^n) and induces therefore a morphism of cohomology groups: Identifying R3 witli its dual, we then have ^(?,P) = f 6 Moreowt, if 3. Theory of Central Extensions. B • 0 Let bF £ be Lie algebras. An extension of Q by b is another Lie algebra ( such that is exact, then 0 b in an exact sequence of Lie algebras. That is, t/b *£ g. We also require lhat t hp a central is exact. extension of fl, that. is+ b is Abelian and the image of b in t lies in the centre of r, and therefore An analogous and very remarkable property is that if IT: o' —• o is an equivariant map h is a trivial fl-moduler of g-modules, namely, *(n) o a = a o tf'(a) Va £ g, then for every positive integer n, there Given two extensions e, *', we say they are equivalent if there is a morphisiri x making the following diagram commute; exists a linear map tr.n ; //"(g, a') -. I!"(g, a), defined as follows: if a £ C"{fl, a'), let be given by 0 ——* b . t f g , 0 (*.no)(oi, o>,..., on) = ff[a(ai, a3,..., an)]. The mapping £,„ commutes with in, £.„ o <„ = in oir,n, and therefore o-.n(iT"(g, a')) C tl Z"(8, a) and ff.T,(fl (g, o')) Cfl"(l, a), which allows us to define a quotient map liy the "Five Lemma", x. must be an isomorphism. Given the central extension 0 • b " t > 0 0 Moreover, given an exart sequence of g-modules we have for each Linear sec Lion s of q (rjo s = ldfl) and fur all a, 6 £ JJ: 0 * a" —-—• a' —-—> a - 0 Identifying b with its image in f, we thus find an element £(fl.6) £ b sncfi that with equivariant mappings K and <7, it is easy to see that ({a,b) = [*[a),atb)\-8{[atb]). l 0 , //'(g, a") > // (j,o') //'(fl.a) It is easily seen frani Ihf Jacob* identity that is exact too, and furthermore, a linear map d: //'((, a) —• lP(g,a") can be found such thai A glance at Example (a) of §2 shows that £ defines a 2-cocyclc of a coliomology of g with values 0 — //'(9,o") — ll'iv.a') — rt'(fl.a) — /^(o,a") — //J(B.a') — U''(B,") in t), If we choose another linear section s', it differs from s by a linear mapping r: g —• b hi con set] IIOII re: is also exact. In fact, if ft e •Z'fs.a), then <*([y b is assort a tod to an (-lenient of //2(fl, b). Reriprornlly, consider ^ £ ^a(0i b)- We can define an extension of g by b in the din-ct sum of linear spares t$ ;= b © Q with the following Lie bracket: which, using the hypothesis of eqiiivariance of tr, may W rewritten for ^, ft £ ti; a, b ^ p. The J.arnbi ideniity is then a consequence of the coryrlr condition. = 0, We then have 0 ^^ b — . ce —^ 0 > 0 This gives a bilinear rnnp UJ : g x g —• a", l>y with r;^(^, «) := a. This defines a central1 extension, U£' is a cocyclr i*qnivalrnt to f, the central extensions tc' and Cr are equivalent. IixJred, 3r: 0 —•• b linear with £r{n, A) — fjn, fc) -f ?"([«, 6]). We define v: ff —• ^' by i.e. a 2-cochain u 6 ^*2fGt, fl")- Now it is just a matter of computation to check that u> is a 1 2-cocycle, 6w = 0, and that had we chosen a different linear section A1 for ff, h'[a) — h(a}+r[a), I hi restriction of \ to b is the identity and tj^ o\ ~ T^. \ is a Lit algehra itoiiiONior^bisin we would find a 2-cocyclc. w' differing from w by a 2-coboundary, u' — LJ + 6r. Consequently, the das§ [u] of u is well defined. Finally, in case the Ucocyrle a is a coboundary, a = 6fl with V([(A,O).(/I.*)1J)= v((«n, *),[«, !>])) = (£(„,*)+r(|n,l]),[«.4]) = [(A, n),(,i,%. . /? E C°(g,a), the equivariance of 9 E'ROPOSITION 5. Given an arbitrary fenlrai' ejtiension be a linear section for Jet (h« cocyde associated to a Jinear section s of 17; then the extension c( is erjuii-atent to the given extension t N itice, ncvertlieless, that we may define i. aympleclic action of rjf on (Al, w) by projection to fl and this action will be Hamiltonian with A" itselT, which ie ft Lie algebra homnmors»liismt PROOF: Define *:*(-.( by as comomentum mapping. We talk of a PoiBsoiiian or strongly HamiltomHn action [2] v((A,o)):=i(A)+ «(«)• whfn the comomentum mapping is surh a homomorphism. The moral is that one need only Then x| — I*!b and q O \ ~ r/^, Moreover, \ is a Lie algebra homomorphism: study Poissonian actions in order l,o inveutigaLe Hainiltonian actions, if one allowa £ to be b replaced by a central extension. . Let It2N = { (i1, • • ,xxNN ; ys ) } with the u^ual symplcctic form w = d-x* Ady,. 1 .«)).*((*<.*))] = (A+ *(*)] = [«(«), < Let O he. 11?^t acting by translations. To the fundamental vector field* A\ :s= — -—, K := and both expression cniiicidc, by the definition of (. 1 — -— correspond /j — r/j, f,+n — — x\ for 1 < i < JV. Tln-n l/«i/j+^} ~ —^ij for 1 < i,j < /V. In conclusion, we have proved: However, /[jf,,vj] — 0. This implies that in ord^r to obtain a Poissoniaii action, one must 2N THEOKEM 3. Tliere is ix liijective correspond™ re between the equivalence CJASSPS of central replace 0 = JX by the Lie algebra of the Hfusenbrrg group H?N+I- extensions of g k_v b and the eJerm-nis of //2(fl,b) when b is considered as a trivial 0-Jwdiife. 5, TIIP Connecting Hnmomurplkism. Now let F: g' —• g lie a Lie algebra inorpliism and let f be a central extension of a by b. We liavp scm that i\\e obstruction to Ihe comoinentmn mapping being a [Jc algebra ho- We may define an essentially unique "lilting" of extensions so that the Following diagram inotnorphisin is a measured by a cocycle, modulo a coboundary, we have interpretod this in commutes: tcriUH of the theory of extensions. U should be made clear, however, that not all extensione of g appear in the lifting, realized in the previous section, of the sequence 0 —+ IX —*CVf[M) —* .Vjt(JLf,u/) — 0, but only tlie image of //'(fi, B1{M)) in //2{BiR) under a certain hoinomor- jiliisrn known in general cohoniology theory as a "connecting homornorphism" We illustrate this phenomenon for the case of exact sytuplectic nianifolds. IS 0 , b ('onHJdc-r, then, the nianifotd (A/,— dO). For any a € 0, Cxt& = «« a closed l-ftsrin. I^t i 1 us assume that, the action ifi Ilaniiltouian. Then <>„ 6 ii {\1), The condition: Here tF is { (x,a) G rxj' : r/(r) = F(a) }n endowed with th* direct sum Lie algebra structure, F is defined by F(r,a) := a1, ami ^'(.E, «) :— a. Then (7/oF){i, n) = J)(X) and (f"oj/)(^,«) = F(o), so the diagram is commutative. The cocycle ^' wllich defines thr extension is obtained as follows IF* ifc a linear seciion beco of 0 in (, we define «'; g' —* i' by s'(ar) :— {* o F(a'),a'). Thrn wliich myans that rt: 0 — H](M) : a **-*<*& is an fletnent of Zx (0, By (Al)), wliriv 0 arts by Lie derivation. If we now roplacd & by 9' := fl + {3 with /i ft closed 1-form, we obtain or in other words, {' = F*(. so a* differs from a by a l-coboundary in this coliomolagy. Tims a UamilLoniAji action off? 4. Poisnoiiian Action*. on an i-xatt syniplectir manifold determines an element of //l (0, Ul[M)). Wo will write Now let ua nee how all this may be applied to our llamiltoninn formulation. \V« construct thr lifting : 0 R tx -^— fl > 0 Now consider tlip short pxact 0 . R. wliicb indufes a homomorphism: [f the top row is a split, pxact sequence, there is a linear section sx for c/x which is a hoino- morphism, and we may take as the ccmiornentum mapping / := X °sx • which is a Lie algebra ! homomorphism, OHierwise there is an obstruction to / being a homoirmrrihisni. In fact, lot s iiHe.n'(M)) -1 /; (B,R). 10 n THEOREM 4 (Kirillov, Kostant, Souriau). Every orbit of the coadjoint representation of a Lie Clearly dof = /i, So the previous equation may be rewritten: group G may be given a G-in variant symplectic structure in a c&nonkaJ way. This structure is given by PROPOSITION 6. The action of G on O is H&miHonian and strongly FJa Tli^refore 1 contlitton PROOF: From \i(Ya)u]Yb ^ u{Y«,Yb) - £M] = -¥*{. = -<&(n) we deduce that U =s la in the notation of \\l. Moreover, the comotncntuin mapping £a is a Lie algebra homomorphism: is salinfied if and only if A(o,6) = 0. But this is precisely the condition that the vector fields -» d/ (u) : B —• TJAf is tlie transpose of We first define equivarianec in a general context. Suppose that a Lie group G acts on two o the map />.„: T,,M -> B' if we klenlify 9" a r,. a". Indeed, if v e T M, we have manifolds M and N by the respective actions $ and *t. A differenliahle map F: -Af — /V is |ul U called equivariant if al any point u e Af: where 7 is an integral curve such that 7(0) — u, -y(O) — 1:. I'quivalently, the following diagram commutes: TllKOHEM 6. hcl G lie a I,ir group which acts in n strongly Uuniltimim manner on a sym- plrriic jnamfo/d (M,fi). Then: (1) The kernel fcer P,u i.s the symplcctir orthogonal lo (he tnngent sjiaceTu((7 u) to the orbit G 11 al ti £ M; (2) 'IVie imajfe f.uCr,, W) i« (he ajintfir'alor fl£ 0/ ifte suhaljebra gu of the isotrop.)' sudffroup lfG is connected, F is equivariant if and only if for all a € 8> the fundamental vector field I'ROOf: (1) We have on M and the fundamental vector field Z'a on N arc F-relatcd, namely, for any u € M. i' £ ker f'.u <=> (/ 0, Vrt € B * = 0, Vn ) = 0. Vu e B The argument is routine and we omit it. ('2} The range of P. is the annihilator of the kernel of the map a i-» d} (u) : S) —• TJAf. If G acts strongly symplectically on (Ai.Si), so that the comomentum mapping /- (t —* u a This kernel is exactly Bu 9in<"c <*/«(") = 0 is equivalent to A\,(u) = 0. I C°*(?rf) exists, the momentum mapping P ; M —• f' lias been defined so that This llworem has several consequences for the geometry of the foliation of g" liy C'«AiiG. (P(u),a) :=/„(«) VweA/,Vu€B- Nole, first of all, that the dimensions of the orhils depend on the rank of /'. This will he constant in the neighbourhood of a point if and only if the Drbits of nearby points have the We claim that the diagram same dimension. A point 11 € M is regular for P (that is, P is a submersion at Ihal poinl) if and only if M 9* the action of (7 on A/ is locally free iit a neighbourhood of u, thai is. (he isolropy suligroup (7ti is discrete. If M is a homogeneous space, it is the orbit of each of its points, so K is injective, consequently, the momentum mapping is a local diffeomorpliism from M to the corresponding commutes if and only if/ is a Lip algehra honiomorpliism. In general, thrte is an obstriiciion orbit tl P(u) in the coadjoint representation. Moreover. P. {M. ft) —» (C?,HJ) is symjilrrtic, to (7-equivariance measured by the degree lo which the action fails to he Poissonian. Kor. il Za, /?i are the fundamental vector fields corresponding to a,b e g, we liave More precisely, wp [lave the following: fi(^, Zi) = ~Ztf. = {/., h) = /(a t| = (Io,n o P = w(Y.A'b) o P. THEOREM 5. KG is conncrtFd, (he nmriiriKum fnafiimi.ir. cait be chosen l.o hf rrjiiivanajit if anj only if the action is Powsonian. Since the Za generate ,\'{M) and the V. generate X{P(u)), with F.(£.) = >'„, we conclude Ihal P*ui = il. Since Af and P(A/) have the some dimension, P is a local dilfeoinorphism. PROOF: Let a t—* 7>n l>e the fiinitamenlftt vector field associated with the action of i\. For any b 6 fl we have To summarize.", the symplec.tic homogeneous spaces of a Lie group G whose fraction is strongly llamiltonian. comprise the covering spaces of the coadjoint orbits of O in a*. 15 3, Group Cohomology and Equivalence of the Momentum Mapping. action of a Lie group G on a symplectic manifold determines an element [a] £ //^G7, fl"), such We now turn to the obstruction to the equivariance of P in terms of group cohomology. that the momentum mapping is equivariant if and only if [a] = 0. We observe that the following diagram commutes: Finally, we study the relation between the two cohomologies we have introduced. The mapping o:C-»j' induces a linear map «,,: T,G —• fl*, which may in turn be regarded as a bilinear map i:|x(-.R: (o.*) •-• ".;(«)*, where of course we identify T,G with J. It turns out that 6 is none other than A of §11.5. Indeed, let i E 9 and let g, = exptb in the corresponding one-param-'ter subgroup. Then r>") = /Ad,,-1"'") + °<(9i. a) To see this, notice that *, °*j-'u = *„ o if. Thus *,.,-iu*j-'u.e = *u.jif.c. and since Therefore (*,.Z.)(u) = *,;-,u{Z.lf-'u)) = ».Z.(u) and *„., o (Ad ga)t = ZM,,{K), we obtain the relation g.Z, = Z M ia wliioh is to say: Let us see what this means in our context, If we consider a Hamiltonian action of O on l/l, a symplectic manifold (A/,fl), then since Za is Hamtltonian with llamiltonian function — fa, 1 the vector field $f.Za is also Hamiitonian with Elamittonian function —/a oj^ . The relaLion as required, 1 we have just verified tells us that /„ o j" and /Ad,a differ by a constant function: 4. Elumonts of Rndurtion Tlieory. NoetherTs theorem establishes tliat every one-parameter suhgroup of canonical .symmel ry trans formal ions of a Hamiitonian //, whose generating vector field is llamiltoiiian. yields a where n(g,a) is linear in llu' second argument. constant of motion. This result is presented by Souriau [4] and Smale [5] as follows: Rephrasing this, we find that TliKORtM 7. If H is tnvnrianl under a Lie group of stmngly syrupleclic trHiisforinntiona of (.If.fl), the rTMirne/ltum maprJjn^ P: M —• 0" is ronslanf on ever.v integral rurvr of the vrctur field \'n assoriaferf (n // by i{T,r)Q - ~dll. where o: G -• g* is defined by a(g)a := {P(g -ij),a) = /o(j • u) = /Ad,-io(u) +<»(».«) = «CoAdj)P(u),a} + {«(?},a). => = ^{l'(M«)).a)h^ = F,,A = -Y.W = 0. Clearly a(g) measures the obstruction to the equivariance of P. [f we replace /„ by a /i = /a +*i(a). then a(j) is replaced by o'(j) = o(s) + M ) ~ (CoAd j)/j(a). l V\\F. range of P Lhus ronsists of constants of motion oftht* evoJulion governed by //. P~ {x) = Repeating the previous calculation, we immediately obtain: { u 6 M : P(u) =r x) is one of these "level swrfaces". [n the nw*i f«vourah|p CHPP, P":(-r) will be a submanifold of A/, for which r must be "regular", that ist the rank of P, must be ,,). (3.2) constant. The inclusion j: N —* M yields j*f} —: S, a 2-fortu which is clw^d but in general iiol, syniploctic (Jim N need not be evon)- For a Lie group (7 and a (7-iriodule A, we may define a cohomology by: LH us suppofie that dinifkprHu) is constant. If A\ V* £ knri], an Sinrf rfi: = 0 we nhiain [X.Y] 6 k*r 5^. Therefore kcr^ is iuvolulivr, ant! so (AM*) is fiiliiilcil inio leaves which art- integral fiiirfitc™ of D, It is natural loa.sk wlietbiT .V/ker!C is a nirinifohl; if so, wi1 rnny introdure a sytnplcrtii' form tf on it by [projection from L. The process of passing from {Af,l't) to (JV/krr D.IT) IS called ar^duction". T Then o *„ = 0 We set Z"(G,A) := (a € C"(G, .1) : tno = 0 ), tf"|G', A) •- {Vi En tUf- rase of P: M —» g", if r € 0* is regular, thr group Or = (lacing thr llamiltonian // and the one-parameter subgroup it gen^rati*s by a group (V: 16 17 Thus the fundamental fields are ith f/anujtonian actions on the same- symplrctk THEOREM 8, Let G, GG' be Lie groupsgp wi be the respective momentum mappings. m&nifold (M, £)) and If t P: M —> g'' and ff ; M —> B" be the respective m e orbits ofG. Thus 0, then Then P is constant on tjie orbits ofG' if and only if P' is constant on th the actions ofG *nd C commute. IV. EXAMPLES OF COADJOINT ORBITS 1. The Heitenberg Group H3. ; have Let G be the 3-diniensiona] Heisenberg group Ha, namely, C:= { I 0 1 y ] :i,y,r€lll. V 0 l) J A (global) chart {G, 4>) is given by Thus w = ^da I\dl3 on OT. 2. The Group of Plane Motions £V * I 0 1 y ! = {*.!/, 0- i } with the relations; iO 0 The Lie algebra of tlie group Ei is generated by { J, The composition law is U,P.] = P2. [J,Pj] = -n. I ( a>yi + l/a.-i The adjoint representation is given by Thus the left-invariant vector fields determined by ^-|0, ^\0- $T\Q are -'-f: •-•)•-(-::!)•-(! ID- 1 0 which satisfy the comniutation relations 0 «»i 0 sin* 1 0 0 / 1 0 0\ ">'l = ( oa 1 0 from which we obtain «i.,P, - 0 10 , Ad rxpait'i - Ad I = f 0 (1 1 0 0 0\ / 0 0 0\ \-a, 0 1/ ( I 0 0 0 0 0, adV = 0 0 0 , ad Z = 0. Thus = 0 cosj. -»in 0 1 0/ V-l 0 °/ 0 sin* cos To compute explicitly the fundamental fields of the coadjoint representation of thf Ileisrn- /l 0 o,\ /I - Letting («,/?,7> be coordinai™ in (C S R3. we have n f „ ~ c. f - a + h ;, PI. Pi) = U~ jj , CoAd exp rZ = In (CoAJ CoAd exP aX = I 0 ~ /» - »7, CoAd exp bY = < /» - 19 18 *?:;1-*]',! •' 3' we obtain the fundamental fields 4, Tlie Symplectic Group Sp{2n;Tl). As a final example, we consider the sympleclic group G = S(i('in, R) acting on the ma- v 2 dpi ' dpi' nifold M — R " \ {0}. (Tl)is is not the coadjoint action of G\ but we will show that it l Notice that is a PoisBonian action.) We write i = (i',,.. ,x";y ,y") € A-f, with symplectic form w = cfx' A r/j/1. [A^.AP,]^ — -Pl— .-»-] = p,- = Aft, Let ^ be a matrix ill the Lie. algebra .9p(2n,R). Its fundamental vector field is [AJ,.VPj] = [K_^Pl™,Pl-] = K_ = -^1. XA{z} = -a.,z'~ (,-,j= 1,2. ..-,2n). If we choose "cylindrira.]" coordinates (j,f*,d) given hy ;>, = /iconf?, p% — ps'inO, the fundamental vector fields become The 1-forni dj(A'j) is then + w(.V^) = -i(,iijI'^7)[d.-«Ad--° "] (<*= l,...,n) On the orbit Of := { (},p,B) : j £ II,-IT < 9 < w) till" symplrctk form u can he written in the form w = |(j, S) dj A Jfl, wliere 'lo determine, /j for which w(A"j) = -rf/^, we must solve tlie system ofe^ualions /(j) (^,^) (P, JVj (!) 00 p f> - ((j, /irasJ, nsin (?).[- sin 0 Pi cos0 /';, J]) p I' = ((j,p<-oa0,(ish\0), --siiiOP-, ros9/'i) = -s -ros-fl = -I. Kroni the first npiation (.'In is symmetric) we limj P P Tims LJ = -dj A 3. The Rotation Grmii> SO{Z). ami on substtl uliilK this in Ihe scrond equation, we get The Lie algebra of Ihe rotation group is (, Rip _ p g = so(.l) = { \ -z 0 : y ~z 0 Ui-ealling tliiit A (z sj>{2v,H) is of the form where we identify thcsr matrices witli the corresponding vectors JJ\I/,-) G H/\ With this identification, the adjoint action is the natural action of G on R.3 The fundamental vector fields are y^ — t$- and cyclic permutations. The rondjoint orbits are spheres, except for (0,0,0), which is a point orbit. Using spherical coordinate!! (r,S,ij>) in 9" 3! R'1. we obtain the Ihe ciiualinri [A'i) reduces to fundamental vector fieLJs on the orbit C}r: To conipule the xyinplei I ic. form LJ = f(0, A', siu - — V - - — This mav alsu br wrilieTl as so that .mil ninre .-1', + .1, = I), we Nil,I llial - H| cos^) = sinfl /,,(;) = -i;'.//U With (1.1) ami Ihus LJ = siu^rfO Arf^ mi any Or. •JO from which we easily derive VA,(r) = -JAz. Therefore, UA.IB) = (V/^J'JIV/a) = z'A'J'JJBt = t'A'JBz and, since A is Bympleclic, JA — -A'J and JA' — —A}; thue or indeed, {/Jl,/o) = - On th*1 other hand, From (4-3) we see that and both expressions coincide. We conclude thai the action of S(>{2n,R) on II2™ \ {0} is a Poissailiau action. Acknowledgements. These notes formed a series of lectures given by the author as a Visiting Scholar at the Univcrsily of Contu Rica in August. 1087, under the auspices of the International Centre of Theoretical Physics. The support of the ICT11 and the cooperation of the UCH is gratefully acknowledged. The final form of these notes profited from conversations hold with J. M. Gracia-Bondia and J. ('. Varilly on that occAsion. REFERENCES [1] R. Abraham ahd J B. Marsden, Foundation* of Mechanics, 2'"' edition, Benjamin. 1978. [2] P, LihprinaniL and C.-M. Marie, Symplcclic Gromdry and Avalxlicnl Mtfhamcs, lt<>iiM, 1987. [3] J. E. Uuinplireys, Introduction to Lie Algi-bras and Representation Theory, Springer, 1U7U. [4] J.-M. Souriau, Structure dti Systlmef Dynamiques, Dunod, 1969. [5] S. Srnale, Invent, Math. 10 (1970) 30-=i-331. [6) J. F. Carifie.na and L. A. Ibort, Nuovo Cim. 87B (1985) 41-40. [7] J, F. Carifiena and h A. lliort, "Non-canonical groups of transformations, anornulii-n and cohomology", J. Math. I'hys. 29 (198S), lo appear. [8] J. E. Marsden am! A. Weinst*in, llep. Math, Phys. 5 (1974) 121-130. 22