- Email: [email protected]
Hyper-lie poisson structures
HYPER-LIE
POISSON
BY PING
STRUCTURES
(*)
XU (I)
ABSTRACT. - The main purpose of the paper is to study hyperktiier structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkahler structures. Motivated by the work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras [IO], [ 1 I], we define hyper-Lie Poisson structures associated with a compact semi-simple Lie algebra and give criterion which implies their existence. We study an explicit example of a hyper-Lie Poisson structure, in which the moduli spaces of solutions to Nahm’s equations assocaited to Lie algebra su(2) are realized as hypersymplectic leaves and are related to the (co)adjoint orbits of 51(2. C). RESUME. - L’objet essentiel de cet article est 1’Ctude des structures hyper-klhleriennes du point de vue de la gComCtrie symplectique. Nous introduisons une notion de structure hypersymplectique qui englobe celle de structure hyper-klhlerienne. MotivCs par le travail de Kronheimer sur les orbites coadjointes des alg$bres de Lie semisimples [Kronheimer:LMS] [Kronheimer:.IDG1990], nous d&inissons des structures C> g une algebre de Lie semisimple compacte et donnons un critkre qui implique leur existence. Nous Ctudions un exemple explicite d’une telle structure >dans lequel les espaces de modules des solutions des Equations de Nahm associees B I’algtbre de Lie $42) sont rtalisCes comme feuilles hypersymplectiques et sont reliCes aux orbites coadjointes de a I( 2, C).
1. Introduction Due to its rich structure and close connection with gauge theory, hyperktiler manifolds have attracted increasing interest [l], [2], [7]. Roughly speaking, a hyperktihler manifold is a Riemannian manifold with three compatible complex structures I, J and K. The compatibility means I.hat I, J, K satisfy the quaternion identities I2 = J2 = K2 = IJK = -1, and the metric is ktihlerian with respect to I, J and K. While it is easy to find examples of Klhler manifolds, hyperkghler manifolds are in general more difficult to construct. The two main often used routes are twistor theory [13] and hyperktiler reduction [7], a generalization of Marsden-Weinstein reduction [ 121 in the hyper-context. To any Kghler manifold there associates a symplectic structure, namely its Ktihler form. For a hyperktihler manifold, there are three symplectic structures (or equivalently, Poisson structures) compaiible with one another in a certain sense (described in Section 2). However, there is an essential difference between Ktihler and hyperkghler manifolds. For (*) 1991 Mathematics (‘) Research partially
Subject Classijication. Primary 58 F 05. Secondary 53 B 35. supported by NSF grants DMS92-03398 and DMS95-04913.
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Kahler manifolds, there might be more than one metric compatible with the same symplectic structure. In contrast, hyperkahler structures are more rigid. Namely, the three symplectic structures completely determine the hyperkahler metric, whence the corresponding complex structures. This observation suggests another possible way of constructing a hyperkahler manifold, namely, by constructing three compatible symplectic structures on the manifold. By focusing on symplectic structures rather than the metrics, we arrive at a new way to define the hyperkahler condition. This leads to our definition of hypersymplectic manifolds. More precisely, a hypersymplectic manifold is a manifold which admits three symplectic structures satisfying the same compatibility condition as usual hyperkahler manifolds. Hypersymplectic manifolds and their basic properties will be studied in detail in Section 2. A powerful method of constructing symplectic manifolds in symplectic geometry is by means of Poisson manifolds. Every Poisson manifold admits a natural foliation, called the symplectic foliation, whose leaves are all symplectic [ 1.51. For example, symplectic structures on coadjoint orbits are induced from the Lie-Poisson structure on the Lie algebra dual JJ* [15]. It is reasonable to expect that there exist some examples of “hyper-Poisson manifolds”. By a hyper-Poisson manifold, we mean a manifold which admits three Poisson structures satisfying certain compatibility condition such that each leaf is hypersymplectic. Instead of developing g,eneral theory of hyper-Poisson structures, the present paper will be focused on finding some interesting examples. In particular, we will consider hyper-Lie Poisson structures, an analogue of Lie-Poisson structures in the hyper-context, which are presumably the simplest and most interesting examples. There are at least two reasons that hyper-Lie Poisson structures are important. The first one comes from our attempt to understand general hyperkahler reduction. According to the author’s knowledge, a successful reduction theory only exists at 0 so far. While LiePoisson structures (or their symplectic leaves) play an essential role in general symplectic reduction, carrying out general hyperkahler reduction will challenge our knowledge of hyper-Lie Poisson structures. In this aspect, an open question is
Open Problem: Suppose that X is a hyperkahler manifold, on which the Lie group G acts preserving the hyperkahler structure. Assume that the action is “good” enough so that the quotient X/G is a manifold. Then the three Poisson structures on X corresponding to the three Kahler forms will descend to three Poisson structures on the quotient X/G. What are the relation between these reduced Poisson structures, hyper-Poisson structures and general hyperkahl’er reduction? Understanding Kronheimer’s recent work on adjoint orbit hyperkahler structures provides another strong motivation. Using gauge theory and infinite dimensional hyperkahler reduction, Kronheimer proved that certain adjoint orbits of complex semi-simple Lie algebras admit hyperkahler metrics [lo], [ll]. Later on, his result was generalized by Biquard and Kovalev to arbitrary adjoint orbits [3], [9], and has been used to understand the Kostant-Sekiguchi correspondence [ 141. However, the hyperkahler metrics and symplectic structures obtained in this way are quite mysterious and elusive. Recall that an intrinsic reason for each coadjoint orbit to admit a symplectic structure is that the dual of any Lie algebra is a Poisson m,anifold, and each coadjoint orbit happens to be a symplectic leaf of this Lie-Poisson structure. Inspired by this fact as well as the results of Kronheimer and 4e
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others, it is quite reasonable to expect that there exists a hyper-Lie Poisson structure such that the orbits studied by Kronheimer et al. occur as its hyper-symplectic leaves. Then, this will provide US a natural source and symplectic explanation for those hyperk&Mer structures on adjoint orbits. To explore the connection between the work of Kronheimer and symplectic geometry was indeed the initial motivation for us to consider hyper-lie Poisson structures. In this paper, as an example, we will consider in detail a hyper-Lie Poisson structure associated with g = su(2). In the meantime, we will take some tentative steps toward hyper-Lie Poisson structures associated with general compact semi-simple Lie algebras. For this purpose, we will keep the discussion general from Section 2 through Section 3 while the last two sections will be devoted to the special case g = 42). To explain our approach. we need to rephrase the definition of hyperkahler manifolds in a way slightly different from the literature. Note that there is in fact no preferred choice of complex structures on a hyperkahler manifold. The bundle maps I’, J’ and K’ given by (I’, J’! K’) = (1, J, K)O, for any orthogonal matrix 0 E SO(3), will satisfy exactly the same quatemion relations. Therefore the map: 0 E SO(3) I’ assigns a complex structure to every orthogonal matrix in SO(3). In particular, under such an assignment, I, J and K are the complex structures corresponding to the identity matrix, the matrix of the cyclic permutation: {ei? e2, es} (e2: ea, el} and th e matrix of the cyclic permutation: {ei,ea:ea} {ea,ei,e:l}, respectively. Since a matrix in SO(3) can be naturally identified with a standard orthonormal basis in su(2) (such are also called frames in this paper), intrinsically we can think of a hyperkahler manifold as a manifold with a family of complex structures (or equivalently symplectic structures), parameterized by frames. This point of view is different from the conventional one, in which complex structures (or symplectic structures) on a hyperkahler manifold are considered to be parameterized by the unit sphere S2. This is the crux in our approach. Now the question arises in which space a hyper-Lie Poisson structure should live. To answer this question, we first recall that a Lie-Poisson space g* emerges as the target space of a momentum mapping of a hamiltonian G-space. A momentum mapping of a hyperklhler G-space X (i.e. a hyperkahler manifold X admitting a G-action which preservesthe hyperkahler structure) is usually considered, when the three complex structures 1, J, K are chosen, as a map from X to g” x g* x g* [7]. However, when I, J? K are replaced by any other three complex structures 1’: J’, K’ related by an orthogonal matrix in SO(3) as described earlier, the corresponding momentum mapping changes accordingly. Intrinsically, the momentum mapping of a hyperkahler G-manifold should therefore be considered as a map X -t L(g, 5u( 2)), where L( g: 5u( 2)) is the space of all linear maps from g to SU(2). It is therefore reasonable to expect that L(g, SU( 2)), as the target space of the momentum mapping of a hyperklhler G-manifold, should carry a hyper-Lie Poisson structure. Another possibly useful way to think of L(g, ~(2)) as a natural generalization of g” is to note that this space is obtained from g* = L(g, W) by replacing R’ by su(2). An elegant way of obtaining a family of Poisson structures on the spaceL( 8: 5u( 2)) goes as follows. The space L(g, su(2)) can be identified with g* x g* x g’, once an orthonormal basis of m(2), i .e., a frame, is fixed. If we can define a Poisson structure r on the space g’ x g* x 8’7 by pulling back 71;to L(g, su(2)) under such an identification, we then ANNALES
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obtain a Poisson structure on the space L(g, 5u( 2)). Th’ 1s construction in fact enables us to obtain a family of Poisson structures simply by varying the frames. Throughout the paper, we shall identify g* with g via the Killing form, hence g* x g* x g’ with g x g x g for simplicity. We note that any bivector field on g x g x g is determined by its corresponding brackets of all linear functions 1;‘, where lk(a, b, c) = (
2. Hypersymplectic
structures
The purpose of the present section is to introduce a notion called hypersymplectic structures, which includes hyperkahler manifolds as a special case and is much more natural from the viewpoint of symplectic geometry. Our definition of hyperkahler structures here is slightly different from the one in the literature, where complex structures and metrics have received much rnore attention. Our interests in this paper mainly lie in symplectic forms and their Poisson tensors. By n:(S), we denote the space of nondegenerate 2-forms on a manifold S, and l?+(~~T5’) the space of non-degenerate bivector fields on S. By K, we denote the map 0$(S) + lT+(A2TS), which is the inversion when elements in n”,(S) and I.+ (ARTS) are considered as bundle maps. For any w E 0:(S), we write w-l = K(W) E I’+.(A~TS). A su(2)-valued 2-form R on 5’ is said to be nondegenerate if the form WE= (<, 0) is 4e
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5 E mu,
where
the pairing
is with
respect to the
DEFINITION 2.1. - A hypersymplectic structure on a manifold 5’ is a closed non-degenerate tiU( 2)-valued two-form R such that < --i w;’ zs linear when 11<11= 1. That is, for any unit vectors
Suppose that R E O’(S) 6~ su(2) is orthonormal basis {el! e2, es} of su(2), as 0 = wiei + w2e2 + ~3.53, where wi) and 7r3 we denote their corresponding
a hypersymplectic structure on S. By choosing an also called a frczme in the sequel, n can be written w2 and wg are symplectic forms on S. By rl, ~~ bivector fields on S.
As usual, for each i, wt’ denotes the bundle map TS ---+ T*S given by {w,b(v): ?L} = w,(v, u), VU,V E TS, X) the vector field on S defined by wg(df) for any f E C-‘(S), and {f,g}i = Xjg for any f:g E C”(S). PROPOSITION
hypersymplectic
2.2. - Suppose that Cl = wlel structure on S. Then,
+ w~f-‘~ + wge3 E Q2(S) 8%su(2)
is a
(1).
(2)
[wf 0 (wjb)-‘1”
= -1,
where 1 denotes the identity map TS --+
ifi #j,
TS.
(59. [Ti. Tj] = 0, where the bracket
for
iI j,
any
is the :ichouten bracket on multivector~elds
[8],
Proof. - For any kr, k:t: k3 such that k: + ki + ki = 1, kter + kZe2 + k3e3 is a unit vector in su(2). Thus, it follows from definition that (klw;
= kl(wij-l
+ k2w; + k3w$l
+
k2(wi)-’
+ k3(wi)-1.
That is, 1 = (klw;
+ kpw; f k3w;)[kl(w;)-l
= (k; + k; -F k;) + k1k2[w;(w;)-1 + kh[w;(w;)-l Equations
(2) thus follow
+ w;(w;)-‘1
+ k2(4-l
+ k3(wi)-l]
+ w;(w;)-~] f k1k3[wf(w;)-1
+ w;(w;)-l].
immediately.
Also, from the argument above, we see that k.17~1+ k2n2 + k37r3 is still a Poisson tensor for any (k,, k2, k3) in the 2-sphere, hence for arbitrary k,! k2, kg as well. It thus follows q that [7rl, 7r2] = [7r2,7ra] = 1~1, “31 = 0. ANNALES
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An immediate consequence is the following: COROLLARY 2.3. - Let S be a hypersymplectic manifold with hypersymplectic form R E R2(S) 3 SU(~). YThenfor any <,Q E 8,
where rc = wE -l
and nV = w;,‘.
Sometimes the following equivalent version is more often used. PROPOSITION 2.4. - Equations (2) are equivalent to
w&q,
(3)
q)
%# 3,
= {f; .!?}i,
for any f, g E C”(S).
Proof. - It is quite obvious that [w,b0 (,;)-1]2
II= -1 *
(w;)-‘w~(wy
u
= +py
((w,“)-‘wp(w~)-ldf,
dg) = -((w”)-ldf,
dg)>
which is equivalent to WI(Xf; x;,
cl
= {f: 9);.
In fact, Equations (2,): or equivalently Equations (3) are also sufficient to construct a hypersymplectic structure on S. PROPOSITION 2.5. - Suppose that S is a manifold with three symplectic structures ~1, wp and w3 such that Equations (2) (or Equations (3)) hold. Then for any orthonormal basis {el, (22,ey} of SU(~), the su(2)-valued 2-f orm 0 = wlel + iu’2e2+ w2c2 dejines a hypersymplectic structure on S.
The proof is quite straightforward, and is left for the reader. To each hypersymplectic manifold, we associate a natural pseudo-metric as we will see below.
Let g : TS ---+ T*S be the bundle map given by g = w;(wyw;,
(4)
and I, J, K the bundle maps form TS to itself defined by J = ,9-lw;
I = flwf;
(5) THEOREM 2.6. -
K = .9-‘w;
and
(i) g is a pseudo-metric on S;
(ii) g can also be written as g = we-‘wi, (iii) I: .I>K satisfy the quaternion relation: I2
=
J2
=K2
or 9
= IJK=
=
w~(w~)-~w~;
-1.
Proof. - (1) It follows from Equations (2), by taking the dual, that [(W~ywZ”]’
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Hence, by using Equations (2) and (6) repeatedly, we have g* = -w~(~~)-‘w~ = [w~(w.$-l]w~ = -w~[(w~)-~w$ = [~~(w~)-‘]w~ = 9. That is, 9 is symmetric. Furthermore, it is evident that g is nondegenerate since ~1, w?. w3 are all nondegenerate. (2) It follows from Part (1) that g = .9” = -w~(w~)-‘ti~ equation can be obtained in a similar way.
= wF(wi)-‘wi.
The other
(3) Using Part (2), we have I = gplwF = [w~(w~)-~~$]~~w~ = (w~)-~LJ;. Similarly. .I = (wF)-‘wi and K = (LJ~)-~w!. Therefore, I2 = .J’ = K2 = -1. Furthermore, IJ = (w$~w;(w~)-~u; = [w~(w;)-lwyw~ = g-q = K. This concludes the proof.
0
It looks as if our definition of g depends on a particular choice of the frame. However, the following theorem indicates that 9 is in fact independent of frames up to a sign. THEOREM
pseudo-metrics
2.7. - If two frames coincide.
are of the same orientation,
then their corresponding
Proqf. - Let 5’= = {el $e2, es} and I = { el,, & eb} be any two frames. Suppose that (ei,ei.ej) = (e1,e2, es)0 for some orthogonal matrix 0 E SO(3). Let I, J! K be the induced almost complex structures on S corresponding to the frame F. We define I’, J’, K’ by the equation: (I’. J’, K’) = (I,
J, K)O.
It follows from the quaternion relation of I! .J, K that I’! J’, K’ same relation. Since wlel + wze2 + w2e2 = wiel, + wheh + wiei, (w~,~w~, w;) = (wl. w2. w,~)O. Hence, we have (7)
I’ = g-ywy,
.J’ = g-‘(w;)”
and
By using the quaternion relation of I’, J’, K’, (w~)“((w~)‘)-~(w~)~, which is g’ by definition.
we
also satisfy it follows
the that
K’ = g-+@ can easily
deduce
that g = Cl
Because of this result, we call g the pseudo-metric associated to the hypersymplectic structure despite of an ambiguity of signs. In particular, if g is positive (or negative) definite, the hypersymplectic structure becomes hyperkghler. We refer the reader to [l], [2], [6], [7] for the background on the subject of hyperkghler structures. The following result is well-known for hyperkshler structures, and is however still valid in our general context. Readers can find a proof in. for example, [2]. For completeness, we outline a proof here. THEOREM 2.8. - If S is a hypersymplectic manifold, I, ,JTK corresponding to any frame are integrable.
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- For any vector fields X: Y E X(S),
w&f,
Y) = g(JX, Y) = g(KIX,
y> =
w3(IX,
y>
Hence, we have the relation: XJw2
(8)
= IX Jwg
It follows that a complex vector field X is of type (1,O) with respect to I iff X Jw2 = iX Jug.
(9)
Suppose that X, Y are complex vector fields of type (1.0). In order to show that I is integrable it suffices to show that their bracket [X. Y] is of type (1,O) according to the Newlander-Nirenberg theorem. However, [X,Y]
Jwz
= Ls(Y
Jw2)
- Y J(Lxwz).
Now, Lxw2
=
(dL;y
+
=
d(iLX’W3)
LJ3&J2
= iLxw3,
and from Equation
(9), Y Jwq =
i(LyW3).
Thus, [X, Y] -1~2 = i[Lax(hyw3) Thus, I is integrable. Similarly,
- LY(L,~w~)]
= i([X, Y] Jug).
J and K are also integrable.
to check that ( slz,1), (~3~1) and all other similar Poisson-Nijenhuis structures in the sense of Kosmann-Scharzbach and Magri [8]. Nijenhuis structures are introduced by Kosmann-Scharzbach and Magri in the integrable systems. Therefore, it would be very interesting to explore the relation hypersymplectic manifolds and integrable systems. Remark. - It is not difficult
3. Hyper-Lie
0 pairs are Poissonstudy of between
Poisson structures
This section is devoted to the introduction of hyper-Lie Poisson structures. The main idea is to define on a suitable space A4 a family of Poisson structures parameterized by frames, which will coherently depend on the parameterization in a proper sense. We shall analyses the condition under which the induced symplectic foliations are independent of frames so that each leaf becomes hypersymplectic. To begin with, let g be a semisimple Lie algebra with Killing form (., .), and A a S2( g)-valued function on g x g x g, where S’(g) denotes the space of second order 4e S6RlE
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symmetric tensors on g. We note that any element in S*(g) can be naturally considered as a symmetric bilinear form on 9. So contracting with E: v E g, there corresponds to a function on g x g x g: A,:,, = CJA Jq. For any c E g, we denote by It, l:, 1: the linear functions on g x g x g defined by Ii(a, B, c) = ([: a), etc. Our first step is to define a Poisson structure on g x g x g. In order to do so (or even just to define a bivector field on g x g x g), it suffices to define its corresponding brackets among all linear functions 1;, i = 1,2,3, since they span the function space C”(g x g x g). DEFINITION 3.1. - Let A be a S’(g)-vatuedfunction defines a bivector field r on g x g x g.
on g x g x g. Thefollowing bracket
$7 $1 = q&1 {$q
= {&1:>
{@;j
= -q&,1
{l;:q
= {l&l:>
= $,q] = Zi,,]
G,“>$1 = +~.~] {I;, I;} = -{l,“, Z;} = A,,,.
Let G be a compact Lie group with Lie algebra g. Then G acts on g x g x g diagonally, with adjoint action on each factor. PROPOSITION 3.2. - The jZlowing
statements are equivalent:
(i) The bivector jield 7r is G-invariant; (ii) the map A : g x g x g -+ S’“(g) is G-equivariant, where G acts on S*(g) by the adjoint action; (iii>for
any I,rl,<
E 8, 5^fLc = Ab?l,C + A,,[,>+
Proof. - That (1) and (2) are equivalent is quite evident. (2) u (3). Suppose that A is G-equivariant. That is, A(gs) = A&A(s). It follows, by taking derivative, that for any < E g, {A = CCC&A.Hence for any qrl;C E 8, fA,,< = 71J&4 J< = qJaad
q The converse is also true by using the same argument backwards. The following theorem gives a necessary and sufficient condition for the bivector field 7r to be a Poisson tensor. As usual, for any f E C”(g x g x g) we write Xf for the vector field -ir#(df). THEOREM 3.3. - r (a: b, c) E g x g x 8,
is a Poisson tensor iff A is equivariant and at any point
(10)
< JXl;A
- Y/+A
= [c, [<,q]],
(11)
(J-G+
- ~1JXqA
= [b, [T <]I,
for any (, v E g. ANNALES
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Proof. - By z)ria, we denote the projection g x g x g --+ go given by pria(Ia, b, c) = a+ib. Similarly, ~7-13 denotes the projection from g x (I x g to g” given by pris(a, b, c) = a + ic. It is simple to see that Tp~i27r = Tprisr = 7ro. where 7rc is the Lie-Poisson tensor on 8’. which is identified with its dual as a real Lie algebra. Hence the Jacobi identity
{Ul, f*}9 f3) + C.P.= 0 holds if f?, i = 1,2,3, are linear functions of the form “i! Ii, or of the form I;! Ii, where c.p. stands for the cyclic permutation. It remains to check the following three cases: (1) fi = I1 fa = 1: and fa = $; (2) fl = I;, fa = 1; and fs = 1:; and (3) fi = I:, f2 = 1: and fs = I,.5’ It is simple to see that the Jacobi identity in Case (I) is equivalent to that YI is G-equivariant according to Proposition 3.2. As for Case (2),
Thus, the Jacobi identity follows iff Equation (10) holds. Similarly, equivalent to the Jacobi identity for Case (3). In the proof above., we have in fact shown the following:
Equation
(11) is 0
PROPOSITION 3.4. - Assume that both Equation (10) and Equation (11) hold for any EY17 E 8. Then, both ior : g x g x g --+ 8” and pr13 : g x g x g --+ 8” are Poisson maps, where gC, identi$ed with its dual, is equipped with the Lie Poisson structure as a real Lie algebra.
Consider the space ,V = L(g,su(2)), which consists of all linear maps from g to Here 542) is, only considered as a vector space without using any Lie algebra structure. M admits a natural G-action induced from the adjoint action on g. Whenever a frame 3, i.e., an orthonormal basis {el, e2, e3} of m(2), is chosen, M is identified with g* x g* x g”, which can also be identified with g x g x g using the Killing form on g. We shall denote such an identification M g x g x g by 9~. From now on, we will always identify g with its dual g*. Using the map Q~F, the Poisson structure r on g x g x g is pulled back to a Poisson structure 7r3 on hf. When the choice of frames 3 varies, we thus obtain a family of Poisson structures on M parameterized by frames. This is the very structure we are interested in. Corresponding to any frame 3 = {ei! e2, es}, there exist two frames 32 and 33 obtained by the cyclic permutations: {e2! es, ei}, and {e:s! ei, ea}, respectively. We also often use 31 to denote 3. The Poisson structures corresponding to 3i, 32 and .3Y are denoted by 1r3, ! ?r~, and ran. respectively. By choosing a frame, any vector-valued function F on g x g x g can be pulled back to a function FF on ,M via the map qF. If furthermore, F is invariant under the action of O(3), where O(3) acts on g x g x g by (a, b: c) (a, b. c)O for any (a, b,c) and 0 E O(3), F3 is independent of the frame 3, and therefore can be considered as a well-defined function on M. However, in most cases, the function F is only invariant under the S0(3)-action. In this case, the pull back FF depends on the orientation of 3. Whenever the orientation of frames is fixed, we shall still get a well-defined function on M.
m(2).
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In the sequel, we shall always assume that the S2(g)-valued function A on g x g x g is invariant under the SO(3:)-action, and therefore can be considered as a function on M when an orientation of fr.smes is fixed. THEOREM 3.5. - n = ry, el •MF~ e2 +I~F~ es does not depend on the choice offrames of the sameorientation, and therefore is a well-defined section of the vector bundle A~TM@~LL( 2). Proof. - Assume that 7 = {el,, e;! ei} is another frame and (el,: ek, ei) = (el, e2: e3)0 for some 0 E SO(3). It suffices to show that
or equivalently, (13)
({f:g}~,,(fr9}1,!{f,g}~)
for any f!g
E C-(M).
=
({f,g}3,,{f:g}3~,{f,S}rl)0
A4 can be identified with g x g x g under both 93 and 9~. If (a, b?c) and (a’: b’, c’) E g x g x g are, respectively, the coordinates of any point in M under these two identifications, they should be related by (a’: b’, c’) = (n, b! c)O. We assume that 0 = {(nij)}. A s an example, we shall show below that
(14)
{f!dZ
=
~llLf!d3-l
+
~2l{f.d.F~
+
a31{f,g}3z!
for f = (<>u) and g = (7, u). All the other casescan be proved similarly. In this case, it is simple to see that
([1?771>d(K?rll>b’) a. VI34 {.f,g}7,
=
(%lr”12r~13)
([&711:6’)
(K,VI:4 ([E:VI!4 =
(all,
a127
al3)
i
= ([E,VI*(4, = ([I>49(241 =
([I,
VI,
2QllU
([t?
71,
([ET
7715 4
42
b’)
l)U’
+
-
a’).
=
([<.
q].
=
([~,7/]!2UllU
-A,,71
-(Lt. ~1~a’)
+ alla
([C,rll,c’)
a11
o
al2
71,
U/j
-a,
2U1lU12b’
2fl>llU&’
+
~2l{fd7}3~
+ 2Ul2U13C’)
This completes the proof: ANNALES
SCIENTIFIQUES
DE L’kOLI!
NORMALE
SUPfiRlELiRE
+ a21b
-
U13
([I?717 b’) +
-
I(
-([I,
On the other hand, all{f,g}3~
U12
A,.,
0
Uf,)U’
-
a11
-t[bl:U’)
u’).
-
a3l{fd7)3~
a314
~UI~U&)
4%
4
)(
a13
) )
290
PING XU
An immediate consequence is the following: THEOREM 3.6. - Assume that Equations (I 0) and (11) hold. Then the Poisson tensors 7r3, r~ corresponding to (arty two frames commute.
Proof. - For any ki, ka, ks E 58such that k: + kz + k$ = 1, it follows from Theorem 3.5 that k1r3, + k27r~~+ ksn 3z is still a Poissontensor. Hence, 7r3,, ~~~ and 7r~~ all commute. Again according to Theorem 3.5, ~7 is a linear combination of 7r3-,, r3z and n-3s, and therefore commutes with 7r~(= 7r3, ). 0 Remark. - Under the assumption of Theorem 3.3, the Poisson structure 7r3 on M is G-invariant, and pri o $3 : 121-+ g, the composition of @3 with prl : 0 x g x g ----+ 0, the projection onto its first factor, is a G-equivariant momentum mapping. Similarly, 7r3F1 and 7r3> are G-invariant with equivariant momentum mappings pr2 o QF and rjr3 o QF, respectively. The rest of the section is devoted to the investigation of the symplectic foliation of 7r3. For simplicity, whenever a frame F is fixed, we shall omit the subscript 3- when denoting the Poisson structure under the circumstance without confusion. I.e., we will use 7ri to denote 7r~,. By Xj, for i = I! 2,3, we denote the vector field n#(df) for f E C-(M). THEOREM 3.7. - The symplectic foliation of n-g is independent of the choice offrames 3, and the induced family of symplectic structures on each leaf is hyper-symplectic ifffor any (a: 6, c) E 0 x 0 :< 0 and < E 0, the following system of equations for (u, 21,w) has a solution:
A#u + [v,c] - [w,b] = [<,a] -[u, c] + A#v + [w, a] = [<, b] (15)
[71,6]
- [v, u] + A#w = [I, c]
1 -[~,a]
- [v,b] - [w,c] = A#<,
where for any u E 0, .4#u is the g-valued function on M obtained by contracting with u, and A4 is identified with 0 x g x 0 under QT. Remark. - The first three equations can be written in terms of a single equation as
If such an A exists, we shall call the corresponding family of Poisson structures a hyper-Lie Poisson structure and their symplectic foliation hypersymplectic foliation. We need several lemmas before we can prove this theorem. The next lemma indicates that whether System (15) is solvable is independent of the choice of frames. Therefore, the statement in Theorem 3.7 is well justified. LEMMA 3.8. - For any fixed E E 0, (u, II, w) is a solution of System (15) for (a, 6, c) E 0 x 0 x 0 ifs ( u’, u’: ,u’) = (u, v, w)O is a solution of the same system for (u’,b’,c’), where 0 is any matrix in SO(3) and (u’,b’,c’) = (u,b:c)O. This can be proved by a straightforward verification, and is left to the reader. The proof of the following two lemmas is also quite straightforward from definition. 4e
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LEMMA 3.9. - For the Poisson structure 7r on g x g x g, the hamiltonian vector$elds of linear functions are gillen by
x1;= (-[ET4, -[Lb],-[<>c]), xl; = (-[I>b],[Ca],A#[), XI; = (-[I>cl:-A% [r,a])> for any < E g, where (a, b. c) is any point in g x g x g and the tangent space at this point is naturally identijied with g x g x g. LEMMA
3.10. - For an)’ [ E g and i:j
= 1: 2,3,
x;; = -x;,,
i fj;
x;; = -r^. E Proof of Theorem 3.7. - Assume that System (15) has a solution. We divide our proof into several steps. (1) 7g T*M
= &T’M
= T$~T*M.
From Lemma (3.10), it follows that X,T = -X:L
and Xs = -[
XF1 and X$ are in 7rzlT* M. Also, it is e&y to se: that Xi
= Xii.
Hence, both
= (A#<, -[i,c].
[<,b]). On
thk other hind, for any (u, 7;:UJ) E g x g x g g Ti,,6,Cj(g x \ x g), we have K$~(u, II! W) = (-[W
a] - [w, h] - [w, c]! -[u,
b] + [w, a] - A#w;
-[u,
cl + A%
+ [w, al)
by Lemma 3.9. It is equal to X$ if (u, w, w) is a solution of System (15). Hence, we have X$ E T$~T*M. This shows thit ng2T*M soCdo the other relations as well.
2 T$~T*M.
Similarly R$, T”M
& ns2T*M,
(2) The symplectic foliation of 7r~ does not depend on the choice of frames. Let 7 be another frarr.e, and XI its corresponding Poisson structure. According to Theorem 3.5, ~1 can be expressed as a linear combination of nap, 7r~~ and 7r~~. Therefore, it follows that nl#T*M C 7rfT’M from Step (1). According to Lemma 3.8 and Step (l), the symplectic foliations of 7r~,, i = 1, 2, 3, also coincide. Hence, exchanging 3 and 7, we obtain the other inclusion: ngT”M & *,#T*M. (3) Let wl, w2 and 1~3 be the symplectic structures on a hypersymplectic leaf corresponding to 1r3~, ~3~ and 7r~~. Then wl, w2, w3 satisfy Equations (3). Below we only prove the following equation: V f, 9 E Cm(M),
Wl(Xf2,Xy2)= {f>?l}l.
(16)
The other equations can also be proved similarly. .ANNALES
SCIENTIFIQLJES
DE L’6COL.E
NORMALE
SJPkRIEURE
292
PING
XU
In fact, it suffices to show this equation when f and g are linear functions. or g is 1: (for example f = Ii), we have LHS = wi (X$ : X,“)
If either f
(by Lemma 3.10)
= -wl(X:,;li,‘, = -(dz;> X,“) = -{g& = X;(g)
(by Lemma 3.10)
= -i(g) = {l&Ll>l. Similarly, Equation (16) holds if either f or g is 1;. It remains to check Equation (16) for f = I; and g = 1:. In this case, we know that Xg Jwi = (u, ‘u, w) according to the proof in Step (l), where (u, ZI, w) is a solution of Syttem (15) and is considered as a cotangent vector at (a, b, c) E g x 0 x 8. We also know that X; = (A#vl, -[?I! c], [v: b]). Therefore, ? Wl(X$.
x$
= (X$ _Iwl)(x;)
= (u,-4%)+ (‘u,-h cl)+ (w>[rl,bl) = (A#u
+ [u, c] - [w, b], 77) (by Equation (15))
= M:477) = (h a4 On the other hand, jl,“,lt}1 = Zij,l,El= ([~,<],a). completes the proof of Equation (16).
Hence, wl(Xs,X$) E
= {1,“,1;}1.
This
Finally, it is quite transparent from the proof above that the assumption in the statement El of Theorem 3.7 should also be necessary. To end this section, we give the following result which reveals the connection between the 5” (g)-valued function A and the induced pseudo-metric on the hypersyrnplectic leaves. THEOREM
ps:ucflo-metric dE,O
3.11. - Under the assumptions g on each hyper-symplectic
as in Theorem 3.3 and Theorem 3.7, the leaf is G-invariant, and for any < E 8,
= -4,~.
Proof. - Since the Poisson structures x3,, ~~~ and ~3~ are all G-invariant, so are their induced symplectic structures on each hyper-symplectic leaf. Hence, the pseudo-metric g is G-invariant. According to Equation (4), 9(i> 8 = -r1 kJ;i, 43 = -7r1(-dZ& 41;) = -{z;‘ll,“}l = -A<,<.
4e S6RIE - TOME 30 - 1997 -- No 3
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POISSON
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STRUCTURES
We have seen that the vector-valued function il plays a fundamental role in defining a hyper-Lie Poisson structure. The theorem above leads to some nondegenerate criterion that A should satisfy, i.e., A,,, = 0 iff [ = 0. The work of Kronheimer [lo], [II] very much supports the existence of A for compact semi-simple Lie algebras. A satisfactory solution to this problem should provide us a symplectic approach, and therefore an intrinsic explanation, on the existence of hyperkahler structures on adjoint orbits. The work on this project is still in progress. In the rest of the paper, instead we will consider the case that g = JZU(2). This case can be handled relatively more easily becauseof its special character as a three-dimensional Lie algebra. However, we shall see that certain nontrivial results, some of which are already quite striking, can be deduced even in such a simple case.
4. The case of g = su(2) From now on, we will work on the special case that g = 5u(2). In this case, a function A can be explicitly constructed on an open submanifold of M, and the corresponding hyper-Lie Poisson structures are studied under the general set-up in the previous section. By a, we denote the function on M defined by:
V(a:b,c)
@(a, b: c) = (a, [h cl),
E g x g x g.
Here again A4 is identified with g x g x g under some chosen frame. This equation defines the well known Lie algebra 3-cocycle corresponding to the Killing form. However, here we consider it as a function on g x g x g instead of A3g. Clearly, @ is independent of the choice of frames, provided that they have the same orientation. Hence, @ can still be considered as a well-defined function on M. Let M0 be the open submanifold of 111 consisting of all points where Q # 0. In other words, M0 consists of triples (a, 6, c) which are linearly independent. Let A : It40 c g x g x g + S’(g) be the map given by
(17)
A(u,
b: c) = $([u,
b] @ [u,b] + [b, c] @ [b, c] + [c, u] @ [c, a])>
V(u, b, c) E Mc,.
It is not difficult to check that the rhs of Equation (17) is invariant under the natural action of SO(3), so A can indeed be considered as a well-defined map from M, to S’(g). THEOREM
Proof.
4.1. - A is G-cquivariant
and satisfies
the condition
in Theorem
3.3
- That A is G-equivariant can be verified directly.
We note that A is uniquely characterized by the following relations:
(18)
uiA
= [b, cl>
for any (a, b, c) E A&,. Applying a J A = [b, c], we obtain
bJA
= [~,a]:
the vector field XI:
XpIA+uJX~;A
ANNALES
SCIENTIFIQUES
DE L’kOLII
NORMALE
and
= [Xpgb,c]
SUPtRIELJRE
cJA
= [u:b]
on both sides of the equation
+ [b,Xqc],
294
PING XU
where both sides are considered as a g-valued function on g x g x g. Using Lemma 3.9, we have
-[v, b] JA + a JXqA
= [[q, a], c] + [b>A%/].
Hence,
u JXz;A
= [[q, a], c] + [b! A#711 + A#[Q, b].
By contracting with < E g, it follows that
Thus
(a:(W+A - ‘I J&;-W = (r, [[rl,4: cl) - (rl,[[I,al,4) = b?K 771,4).
Using the other two identities in Equation (1S), similarly we deduce that
Since the Lie algebra g = su(2) is three dimensional and @(u,b, c) # 0, {a, b, c} constitutes a basis of su(2) at any point in M,. Equation (10) thus follows immediately, similarly for Equation (11). 0 In fact, A also satisfies the assumption as in Theorem 3.7. THEOREM 4.2. - For the function A dejned by Equation (17), System (15) always has
a solution for any < E g and (a, b, c) E MO. So MO has a hyper-Lie Poisson structure. In particular, its symplectic leaves are hyper-symplectic. Proof. - Fix any point (u, b, c) E Al,. Since su(2) is three-dimensional and System (15) is linear with respect to u, II, w, it suffices to prove this statement for any three linearly independent < E g = su(2). Therefore, it is sufficient to prove this for < = a, b, and c. For this purpose, one can check directly that u = u = 0, w = -b is a solution for < = a; u = -c, v = w = 0 is a solution for < = b; and u = 0, v = -a> w = 0 is a solution for < = c. 0 In fact, in this special case, the corresponding hypersymplectic foliation can be described quite explicitly. By X, we denote the gradient vector field of a, where M is equipped with the standard metric induced from the Killing form on g. As a frame is chosen and hf is identified with g x g x g, the vector field X at any point (a, b3c) can be written as X = ([kc], [wl,
(19)
hbl).
Since both the standard metric on M and the function @ are G-invariant, the gradient vector field X is also G-invariant. Therefore, it follows that [X,(1 = 0, de SGRIE - TOME
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THEOREM 4.3. - The symplectic foliation of r~ on M0 coincides with the orbits of the Lie algebra action of the direct product Lie algebra R x 5u( 2), with X and [, < E g being its generators.
Proof. - The symplectic distribution at any point (CL,b, c) is spanned by r$(T*M). By identifying g* with g, TTa,b,C,A $1 is identified with g x g x g as a vector space. To compute the symplectic distribution, it is sufficient to compute the image of a basis of g x g x g under the map :r$. Since g is three dimensional, {a, 6: C} can be considered as a basis of g. Hence, it suffices to do the computation for its corresponding basis in g x g x g. Using Lemma 3.9, we have 7r$(<:O,O) = -< = -([<>a], [<,b], [<,c]). It is also easy to see that r$(O, a, 0) = 6, K$(O, b!O) = -6 and rr,#(O,c:O) = X; n$(O,O,n) = c, n,“(O, 0, b) = -X and r$(O, 0, cj = -6. This concludes the proof of the theorem. 0 PROPOSITION 4.4. - The Lie algebra action defined as in Theorem 4.3 is locally free on MO, so its orbits are all 4-dimensional.
Proof. - If not, there is [ E g and lo E R not all zero, such that [ + kX = 0. If /C# 0, it follows that [b: c] = i[<, a]. H ence, @ = (a! [b, c]) = i (a, [I, a]) = 0. This contradicts to the definition of A&,. If k == 0, we have i = -([<, a], [I. b], [<: c]) = 0. It thus follows that < = 0 since {n,b,c} is a basis of g. 0 The coming result gives us a complete set of casimir functions for the hypersymplectic foliation. PROPOSITION 4.5. - The following functions (a,,b), (b,c): (c,u), (a,n) - (b;b) and (b, b) - (c, c) form a compr’eteset of casimirs for the Poisson structure TF on iV&,.
Proof. - It is simple to see that these functions are all G-invariant. To show that they are casimirs, it suffices to show that they are killed by the vector field X, which can be checked directly. It is also easy to see that these functions are all independent, so it follows 0 from dimension counting that this set of casimirs is complete. To end this section, we look at the induced metric on each hyper-symplectic leaf. The metric on the infinitesimal generators i of the G-action is already given by Theorem 3.11. In order to describe the metric, we only need to know its evaluation on the vector field X, which is the content of the following: PROPOSITION 4.6. g(X,X)
(20)
Proof. - We already know that X$
= -a.
= (A#<> -[<; c], [<: 61). Thus, w;X
= (0,O: a).
,4ccording to Lemma 3.9 and Lemma 3.10, we have w:X = (b, 0,O). Here, in both equations, the right hand sides are considered as elements in the cotangent space of 111, being identified with g x g x g. Therefore, 9(X,X) = -ni(&X.w~X) = 0 -xl((O,o,a):(b,O,O)) = --(&O,O,a),(b,O,O)) = -([c>u],b) = -a. The following consequence follows immediately from this result combining with Theorem 3.11. THEOREM 4.7. - When g := 5u( 2) and A is defined by Equation (17), each hypersymplectic leaf of h& is a 4-dimensional hyperkiihler manifold. ANNALES
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DE L’kC0I.E
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PING
5. Moduli
spaces of solutions
XU
to Nahm’s
equations
and (co)adjoint
orbits
This is a continuation of the last section. When M is identified with g x g x g by choosing a frame F’, the vector field X is written as X = ([h: c]. [c, a], [n, h]). Its flow is thus given by the following system of equations: a = [b. c] & = [c, a]
(21)
i c = [c&b]. Such a system is called Nahm’s equations. which was studied by Kronheimer [lo] modelled on the study of general Nahm’s equations made by Donaldson [5]. For any z E 41, we denote by ~~(3:) the flow generated by X through the point ;c. We denote by 5’ the set of all points .c E M such that the flow v+(x) converges as t ---+ --oc. 5’ has a natural foliation according to the limit points. Note that all the limit points are critical points of a. We denote by C the critic.31 set of a. The G-action on M leaves C invariant. For any orbit 0, we let Ss be the submanifold of S consisting of all points in S whose trajectory under the gradient vector field -Y converges to a point in 0, as t --co. so that in particular Sa be such a submanifold corresponding to the zero orbit. It is clear that S = U. 5’0, where the sum is over all the G-orbits in C. Kronheimer proved, using gauge theory, for a general semi-simple Lie algebra that certain 5’0 are hyperkahler manifolds and are diffeomorphic to adjoint orbits of gc [IO], [ 111. Below we will prove this result for the special case of su(2), as a consequence of the hyper-Lie Poisson structure on 5’. Our approach is quite elementary, and the family of symplectic structures on each leaf 5’0 is rather transparent. To start with, let us introduce a function F on JVf by F(n, b, c) = (0, a) + (b, b) + (c, c), where M is identified indeed a well-defined
with g x 0 x g by choosing function on M.
a frame. It is simple to see that F is
LEMMA 5.1. LSQ’ = (Ix11*. a7z.d LsF Proof. - The first identity the second one: we have LsF
follows
= Ls((n.a)
= f3D.
from a general property
of a gradient flow. As for
+ (b,b) + (c:c))
= 2(a, [b, c]) + 2(b: [c, CL]) + 2(c; [a‘: bl) = m. The following
result is crucial for characterizing
0
the elements in S.
PROPOSITION 5.2. - (i) If pt(x) converges as t -cm, x is either a critical point of @ or @(ix) > 0. In the latter case, we in fact have @(;pt(x)) > 0 for all t whenever pt (x) is dejined.
HYPER-LIE
POISSON
STRUCTURES
297
(ii) Zf pt (z) does not converge as t --CC or is not dejnedfor all t 5 0 (i.e., X is incomplete in the -CC direction), Q, cannot be always nonnegative along the$ow. Proof - By Lemma 5.1, $fD(yt(cz)) = LA-@ = llXl12 > 0. Hence. +(P~(z)) is an increasing function with respect to t. If cpt(z) converges as t --+ -x, the limit point must be a critical point. However,, the critical points of Q are defined by the system of equations:
[b:c]= [c,u]= [u:b]= 0. So @ vanishes at any critical point. This yields that @(.z) > 0. If Q(z) = 0, it follows that @(p,(n)) = 0, f or all t 5 0. By taking derivative, we have llXll = 0. Thus, :c is a critical point. In the case that a(:~) > 0, it is not difficult to see that (P(pt(:c)) has to stay positive for all t whenever pt(z) is defined, otherwise .c will be a critical point according to the same argument above. If yt(z) is not defined for all t 5 0, it must be unbounded as t approaches to a finite number. If pt(z) is defined for all t 5 0 but does not converge as t -x, it must be unbounded as t is sufficiently negative since @ is a real analytic function. In both cases, I-‘(p,-,(z)) d x, as t -. -X (X is either a positive number or SC>. Assume that Cp is always nonnegative along the flow. It follows from Lemma 5.1 that -$F(p,-,(.c)) = 6Q > 0. So F(pt(z)) 5 F(z) when t < 0, which is a contradiction. This concludes the proof. 0 By iZf+, we denote the submanifold of M consisting of all points where Q, is positive. The theorem above yields that S - C is contained in ,U+. Moreover, the vector field X is complete in S - C. It is clear that S - C is invariant under the G-action, hence invariant under the action of the product Lie algebra W x 5u(2) as defined in Theorem 4.3. In other words, S - C is a hq,per-Poisson submanifold of IV,,. To extend this hyper-Poisson structure to entire S, it suffices to extend 7rF to the critical set C. For this, one only needs to extend the vector valued function A to the critical set C. Since C is the limit set, a natural way to extend A is to take its limit along the flow X. This is in fact how we derive the formula below. When ?VI is identified with g x g x g under a chosen frame, a point 3’0 = (no, bo, c,,) E g x g x g is a critical point iff CLO:bo, CO are parallel. Hence, for any critical point, we can always choose a frame so that the critical point is of the form (a~, 0: 0) for some a0 E g under the identification: n/l Z g x g x g using this frame. Such a frame is called a standard frame. Clearly, the element a.0 is unique up to a sign. We then define the function A on C under a standard frame by:
where 1,: i = 1! 2,3, is an orthogonal basis for g g su(2). A is clearly well-defined eon C.
We also let A = 0 at 2 = 0.
To show that such an extension is smooth, we need to give an alternate description of S, which is much easier to deal with. For any given nontrivial critical point ~0, let Fr,, be a standard frame such that z. = (ao, 0: 0) E g x g x g when M is identified with g x g x g under Fzo. We denote, ANNALES
SCIENTIFIQUES
DE L’~COLE.
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PING XU
by L,, the subset of M consisting of all the points 2; = (a, b, c) E g x g x g (% ~$1under Fz,),
satisfying the condition: (CL6) = (b, c) = (a. C) = 0: (b,b) = (~4, ;;‘,,(b, b) = (~0, ao), { - .
(24)
It is easy to check that this definition is well-justified, i.e., does not depend on the choice of the standard frame Fr,. Also, we define Co as the subspaceof g x g x g consisting of all points (a, b, c) such that (u: b) = (b: c) = (a, c) = 0.
(25)
(u, u) = (b, b) = (c, c),
and
a 2 0.
It is clear that these relations are preserved under the transformation (a! b, c) (a’, b’; c’) = (a, b,c)O for any 0 E SO( 3). Therefore, Ca can also be considered as a subset of M. LEMMA
5.3. - For any nontrivial critical point :cg, C,, = SG.~“.
Proof. - C,, is obviously a closed submanifold of M. It is clear that if 2 E C,,, then cpt(z) will stay in C,, for all the t whenever the flow is defined, since X is tangent to c,,. Since the intersection of C,, with the hypersurface @ = 0 is contained in the critical set C, we conclude that @((p,(z)) > 0 if II: is not a critical point in C,,. Thus, according to Proposit:lon 5.2, cpt(z) exists for all f 5 0 and converges as t --+ --x. Let us assume that y is the limit point of cpt(x). Then y is a critical point and y E C,, . Assume that y = (v, V, W) under the standard frame &, . It is not difficult to seeby using Equation (24) that v = ul = 0 and (u, U) = (ua, ILO). The latter implies that y E G . x0. Hence C,, C SG,ro. Conversely, assumethat z is any point in SG,zo and cpt(z) y E G.zo as t --+ --3o. Fix a standard frame Jyz, so that under it za = (uo: 0,O). Then under this frame y =: (u, 0,O) with u E G . ao. Hence, (u~u) = (uo,uo). Suppose that il: = (a, b,c) under the frame &.,. From Proposition 4.5, it thus follows that (u, b) = (u, 0) = 0, (b, c) = (C),0) = 0, (a, c) = (u,O) = 0, ([I, 6) - (c! c) = 0 and (a, rz) - (b, b) = (u, U) = (ao, uo). That is, z is 0 in C,,. This completes the proof. From this lemma, ii. follows that for any ~0, y. E C, C,, and C,, are either disjoint, or equal. In the latter case, x0 and y. must lie in the same G-orbits of C. For this reason, we shall use Co to denote the space C,, for any 20 E 0. The lemma above shows that S0 = CO. In fact, this is also valid when 0 is the trivial orbit. LEMMA
5.4.
so = co. Proof. - That So 2, Co follows immediately from the fact that the functions: (a! b), (a, c), (b! c), (a! u) - (6 b), and (b, b) - (cI, c) are all preserved by the vector field X. As for the other direction, let us assumethat (a, b, c) is any nontrivial point in CO. By the definition of Co, we can write a = Xei, b = Xes and c = Xea, where X is a positive 4= SBR[E
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number and { er; ez, es} satisfies the standard su(2)-relation: [et, ea] = e3, etc. Therefore, it is not difficult to see that at = - &, /jt = - &, ct = - & is the flow through the point (a: b, c). Obviously, it goes to zero as t -cc. That is, (a, b, c) E S,. 0 Combining the two lemmas above, we have PROPOSITION 5.5. - For
Now we are ready to prove the smoothnessof the extension. THEOREM 5.6. - ~‘g is smooth when restricted to each So for any nontrivial G-orbit 0.
Proof. - It suffices to show that the extension A,,, as defined by Equation (23) is smooth for any < E g in a neighborhood of 0 in S0. Let T be the norm of the elements in 0. Then by Proposition 5.5, under a standard frame, points (n, 6, c) in Sc, are characterized by the equations: (a! b) = (b, c) = (a, c) = 0,
(b, b) = (C! 4,
(a, u) - (b; b) = T2.
Therefore, when a point (a: h, c) is in S0 but not in 0, we can always write a. = dmel, b = k2, I: = Xes, where X = v@$ and {el; e2, es} is an orthonormal basis of su(2) satisfying the standard relation. Write < =
Substituting [f + <,’ = (<. <) - [:,
we have
It is trivial to see that the extension of A as given by Equation (23) coincides with the equation above when X = 0. Since both X2 = (b, b) and E1 = H are smooth functions 0 on SO, A,,, is clearly smooth on So as well. For any nontrivial orbit (3 c C, it is obvious that So is invariant under the G-action, as well as that of the additive group W generated by X. Therefore, for any point x E So - 0, its hypersymplectic leaf .&, defined as in the previous section, is contained in SO. Since C, is a 4-dimensional manifold according to Proposition 4.4, it can be considered as an open neighborhood of 5 in SO. Clearly, So is a union of these leaves together with their boundary 0. As observed early, there is a standard frame 3 such that when M is identified with g x g x g under this frame, any point in 0 is written as z = (u:O,O) with a E g. In this way, 0 is naturally identified with a (co)adjoint orbit of g. Although there is an ambiguity for the choice of the frame F, such an adjoint orbit is uniquely determined, and the identification is unique up to a sign. In the following, we will fix any such a frame .Y=, and denote the Poisson structures 7r~~,7r~~ and 7r~~ simply by ~1, 7r2and 7r3,respectively, and the induced symplectic structures on any leaf by wl, w2 and wg for simplicity. THEOREM 5.7, - For any nontrivial G-orbit c> C C, the extended hyper-Poisson structure on S induces a hyperkiihler structure on So. Furthermore, if we choose a frame as in the ANNALES
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observation above, then 0 is a symplectic submantfold with respect to wl. In fact, it is a Kahler submanifold with respect to I and its Kahler metric coincides with the one on 0 when it is naturally identified vtith a (co)adjoint orbit. In the meantime, 0 is a lagrangian submantfold with respect to LU’Zand LU’~. Proof - It remains to consider points in 0. For any x E 0, one can directly verify that r#T~i’M = T,So, for i = 1, 2, 3, by using a local coordinate chart under a chosen standard frame 3. Hence, the bivector fields 7rlr,,%= 1,2.3, are all tangent to So and nondegenerate along 0. According to Theorem 4.2, the corresponding symplectic structures wi are all compatible along 6 by continuity, hence compatible in entire SO. Since the induced metric is negative definite on 5’0 - C, it is also negative definite along 0. Hence, the extended hyper-Poisson structure induces a hyperkghler structure on So. The rest of the conclusion can be verified directly: again by using local coordinates. 0 In fact, So is closely related to adjoint orbits of 8”. To see this, let 3 be any frame, and pr12 o qF : M --+ 8”’ the composition of the identification 9~ : M -+ g x g x JJ with the projection prla as defined in the proof of Theorem 3.3. According to Proposition 3.4, pr12 o qF is a Poisson map with respect to 7r~-. Hence, the image of So is a Poisson submanifold of 0”. The following theorem indicates that in generic case the image is in fact a single adjoint orbit. A general result was proved by Kronheimer [lo] using gauge theory. However, our proof in this special case here is quite elementary and only uses some well-known facts in symplectic geometry. By C?ia we denote the adjoint orbit in 8” containing the image (prr~ o PF) (0). THEOREM 5.8. - Zf 012 is a regular orbit in gC, then (~~12o Q,)(S,) = 012. In fact, prlz o QF : So C?12is a symplectic difleomorphism, where 5’0 is equipped with the symplectic structure induced from the Poisson structure ~3 and 012 is equipped with the (co)adjoint orbit symplectic structure.
We need some lemmas first. LEMMA 5.9. - Under the map ~7-12o @3 : SO ---+ gc, the inverse image of any bounded region is bounded.
Proof. - According to Proposition 5.5 and by the definition of Co, there is a standard frame 7, under which any point (a’, b’! c’) E g x g x g in So is characterized by
(a’: b’) = (b’: c’) = (a’, c’) = 0:
(b’, b’) = (2, c’), (a’, a’) - (b’, b’) = r2:
where r denotes the norm of elements in 0. Suppose that 0 = (aij) E SO(3) is the transformation matrix between the given frame 3 and this standard frame 7. That is, (a,b,c) = (a.‘: b’, c’)O. It is simple to see that (a,~) = ~~:rr” + (b’,b’), (b,b) = uf2r2 + (b’, b’), and (c,c) = &r2 + (6’,b’). Therefore, if (u,u) and (b,b) are bounded, (b’, b’) has to be bounded. This implies that (c, c) is bounded as well. 0 Two immediate consequencesare the following: COROLLARY 5.10. - The map pr12 o QJI : 5’0 -
gC is a proper map.
COROLLARY 5.11. - The image of ,570under pr12 o 9~ is closed. 4e
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Proof. - Using the map !l?~, we may identify M with g x g x g. Suppose this identification ((L,, b,, cn) is a sequence in So such that a, -+ a0 and We need to show that (ao, bO) E (p ~12 0 Ql,)(S,). It follows from Lemma is bounded. Therefore there exists a convergent subsequence Cam, whose limit by CO. Then, (aa, bo, CO) is in SO since So is closed. This concludes the proof.
301 that under b,, ---+ bo. 5.9 that c,, is denoted 0
Proof of Theorem 5.8. - Again, let us identify A4 with g x g x g by qF. It is easy to see is the tangent vector, at a + ib, generated by the adjoint action ad,,,. Hence, the projection of entire flow: (priz o QF)(cpt(x)) lies in a single orbit C for all t. Since cpt(z) converges to 0 as t goes to -00, then (priz o Q,)(0) C C. Then-fore, 012 C c. Since 0 12 is a regular orbit, it thus follows that 012 = C. This means that the image (prip o \JrF)(S,) is contained in (3,, (this part of the argument is due to Kronheimer [lo]).
that at any point (a: b, c) E g x g x g, T(pr12)X
On the other hand, according to Corollary 5.11, (pria o q~)(So) is a closed Poisson submanifold in 012. Hence, it must be the entire orbit 0i2. That is, (prla o qF) : SC? 012 is onto. This map is automatically a submersion since it is a Poisson map. By dimension counting, it must be a local diffeomorphism. However? by Corollary 5.10, it is a proper map. Therefore, it must be a covering. Since 0i2 is simply connected, c3i2 is thus a diffeomorphism. 0 (m2 0 93) : so Finally, we will show that So - (0) is diffeomorphic to the nilpotent orbit of s l( 2, C). THEOREM 5.12. - So - {O:I is a hypersymplectic leaf of MO, and therefore is a hyperkahler manifold. For any frame 3’, pr12 o 9~ is a symplectic diffeomorphism between So - (0) and the nilpotent orbit of sltI2, C), where So - (0) is equipped with the symplectic structure corresponding to 7r3 and the nilpotent orbit is equipped with the standard coadjoint symplectic structure.
Proof. - It is clear that St, - (0) is a union of hypersymplectic leaves since it is invariant under both the G-action and the flow of X. Each hypersymplectic leaf is 4-dimensional, and therefore must be open in So - (0). Since So - (0) itself is connected, it must be a single hypersymplectic leaf. Thus, it is hyperkahler according to Theorem 4.7. By Proposition 5.5, under the identification $3 : M g x g x g, a point (a! b, c) E g x g x g is in So - (0) iff a = Xei, b = Xe2 and c = Xes for some standard orthonormal basis {ei, es, ea} of su(2), and X > 0. Hence, its image under pr12: n + ib, is clearly in the nilpotent orbit of sI(2, C). A similar argument as in Corollary 5.1 I shows that (priz o q~)(Sa - (0)) is in fact closed. Hence it has to be the whole nilpotent orbit since prl2 o 9~ is a Poisson map. Finally, it is quite obvious that prl2 o q3 is injective on So - (0). In fact, we always have 0 c = [a, b]/m. This concludes our proof. Remark. - As we have seen, (co)adjoint orbits of 51(2, C) are related to the points in MO which have bounded trajectories (in the ---co direction) under the gradient vector field X and are contained in M+. However, according to Theorem 4.7, there are other hypersymplectic leaves of MO which are contained in &i- = Ma - M+. It would be interesting to explore further the geometric structures for those leaves, and in particular the connection with the hyperkahler metrics on the cotangent bundles of hermitian symmetric spaces of noncompact type studied recently by Biquard and Gauduchon [4]. ANNALES
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ACKNOWLEDGEMENTS
The author would like to thank Jorgen Andersen, Sean Bates, Olivier Biquard, Ranee Brylinski, Hansjijrg Geiges, Victor Cuillemin, Nigel Hitchin, Peter Kronheimer, Oscar ’ >arcia-Prada, Yvette Kosmann-Schwarzbach and Alan Weinstein for useful discussions and email correspondences. Special thanks go to Ludmil Katzarkov and Tony Pantev for calling his attention to the papers [lo], [I 11. Most of this work was carried out when the author was a member of MSRI, where the research was supported by NSF Grant DMS9022140. In addition, the author wishes to thank the Center of Mathematics at Zhejiang University, China, for its hospitality while part of this work being done. REFERENCES [I] [2] [3] [4] [5] [6] [7] [8]
[9] [IO]
[I l] [12]
[13] [14] [15]
M. F. ATIYAH, Hyer-Kiihler manifolds (Collection: Comple.~ Geometry and Analysis, Springer-Verlag Lecture Notes in Mathematics, Vol. 1422, 1990, pp. l-13). M. F. ATIYAH and N. J. HITCHIN, The geometry and dynamics of magnetic monopoles, Princeton University Press, 1988. 0. BIQUARD, SW les Pquutions de Nahm et la structure de Poisson des algebres de Lie semi-simples complexes (Math. Ann. Vol. 304, 1996, pp. 253-276). 0. BIQLJARD and P. G~\L~~L~~HoN, HFperkiihler metrics on cotangent bundles of hermitian symmetric spaces, preprint. S. K. DONALDSON. Nohm’s equations and the classification of monopoles (Commun. Math. Phy.s. Vol. 96, 1984, pp. 387-407) N. J. HITCHIPZ, Hyerkdhler mantj%lds (Siminaire Bourbaki 44’ annCe, No. 748, AstPrisyue, Vol. 206, 1992, pp. 137-166). N. J. HITCHIN, A. K.~RL.HEDE, U. LINSTROM and M. ROCEK, H>jperkiihler metrics and super.ymmetr?j, Commun. Math. Phy. Vol. 108, 1987. pp. 535-589. Y. KOSMANN-SCHL+RKZBACH and F. ~UAGRI, Poisson-Nijenhuis structures (Ann. Inst. H. Poincare’ Phys. Theor. Vol. 53, 1990. pp. 35-81). A. G. KO~ALEV, Nuhm’s equations und complex adjoint orbits (Quurt. J. Muth. Osford Ser. (2) Vol. 47, 1996, pp. 41-58). P. B. KRONHEIMER. A hyper-Kiihlerian structure on coadjoint orbits of a semisimple complex group (J. of LMS. Vol. 42, 1990, pp. 193-208). P. B. KRONHEIMER, Iristantons and the geometry of the nilpotent variety (J. Difl Geom. Vol. 32, 1990, pp. 473-490). J. MARSDEN and A. WEINSTEIN, Reduction of vymplectic manifolds with symmetry (Rep. Math. Phys. Vol. 5, 1974, pp. 121-129). R. PENROSE, Nonlinear gravitons and curved twistor theory, Gen. Ralotiv. Grav. Vol. 7, 1976, pp. 31-52. M. VERGNE, Instuntonr et correspondance de Kostant-Sekiguchi, (C. R. Acad. Sci. Paris., t. 320, SCrie I, 1995. pp. 901-906). A. WEINSTEIN, The local structure of Poisson manifolds. (J. Di# Geom. Vol. 18, 1983, pp. 523-557). (Manuscript
PING XlJ
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA. E-mail: [email protected]
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13, 1995; 7, 1996.)
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XU (I)
ABSTRACT. - The main purpose of the paper is to study hyperktiier structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkahler structures. Motivated by the work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras [IO], [ 1 I], we define hyper-Lie Poisson structures associated with a compact semi-simple Lie algebra and give criterion which implies their existence. We study an explicit example of a hyper-Lie Poisson structure, in which the moduli spaces of solutions to Nahm’s equations assocaited to Lie algebra su(2) are realized as hypersymplectic leaves and are related to the (co)adjoint orbits of 51(2. C). RESUME. - L’objet essentiel de cet article est 1’Ctude des structures hyper-klhleriennes du point de vue de la gComCtrie symplectique. Nous introduisons une notion de structure hypersymplectique qui englobe celle de structure hyper-klhlerienne. MotivCs par le travail de Kronheimer sur les orbites coadjointes des alg$bres de Lie semisimples [Kronheimer:LMS] [Kronheimer:.IDG1990], nous d&inissons des structures C
1. Introduction Due to its rich structure and close connection with gauge theory, hyperktiler manifolds have attracted increasing interest [l], [2], [7]. Roughly speaking, a hyperktihler manifold is a Riemannian manifold with three compatible complex structures I, J and K. The compatibility means I.hat I, J, K satisfy the quaternion identities I2 = J2 = K2 = IJK = -1, and the metric is ktihlerian with respect to I, J and K. While it is easy to find examples of Klhler manifolds, hyperkghler manifolds are in general more difficult to construct. The two main often used routes are twistor theory [13] and hyperktiler reduction [7], a generalization of Marsden-Weinstein reduction [ 121 in the hyper-context. To any Kghler manifold there associates a symplectic structure, namely its Ktihler form. For a hyperktihler manifold, there are three symplectic structures (or equivalently, Poisson structures) compaiible with one another in a certain sense (described in Section 2). However, there is an essential difference between Ktihler and hyperkghler manifolds. For (*) 1991 Mathematics (‘) Research partially
Subject Classijication. Primary 58 F 05. Secondary 53 B 35. supported by NSF grants DMS92-03398 and DMS95-04913.
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Kahler manifolds, there might be more than one metric compatible with the same symplectic structure. In contrast, hyperkahler structures are more rigid. Namely, the three symplectic structures completely determine the hyperkahler metric, whence the corresponding complex structures. This observation suggests another possible way of constructing a hyperkahler manifold, namely, by constructing three compatible symplectic structures on the manifold. By focusing on symplectic structures rather than the metrics, we arrive at a new way to define the hyperkahler condition. This leads to our definition of hypersymplectic manifolds. More precisely, a hypersymplectic manifold is a manifold which admits three symplectic structures satisfying the same compatibility condition as usual hyperkahler manifolds. Hypersymplectic manifolds and their basic properties will be studied in detail in Section 2. A powerful method of constructing symplectic manifolds in symplectic geometry is by means of Poisson manifolds. Every Poisson manifold admits a natural foliation, called the symplectic foliation, whose leaves are all symplectic [ 1.51. For example, symplectic structures on coadjoint orbits are induced from the Lie-Poisson structure on the Lie algebra dual JJ* [15]. It is reasonable to expect that there exist some examples of “hyper-Poisson manifolds”. By a hyper-Poisson manifold, we mean a manifold which admits three Poisson structures satisfying certain compatibility condition such that each leaf is hypersymplectic. Instead of developing g,eneral theory of hyper-Poisson structures, the present paper will be focused on finding some interesting examples. In particular, we will consider hyper-Lie Poisson structures, an analogue of Lie-Poisson structures in the hyper-context, which are presumably the simplest and most interesting examples. There are at least two reasons that hyper-Lie Poisson structures are important. The first one comes from our attempt to understand general hyperkahler reduction. According to the author’s knowledge, a successful reduction theory only exists at 0 so far. While LiePoisson structures (or their symplectic leaves) play an essential role in general symplectic reduction, carrying out general hyperkahler reduction will challenge our knowledge of hyper-Lie Poisson structures. In this aspect, an open question is
Open Problem: Suppose that X is a hyperkahler manifold, on which the Lie group G acts preserving the hyperkahler structure. Assume that the action is “good” enough so that the quotient X/G is a manifold. Then the three Poisson structures on X corresponding to the three Kahler forms will descend to three Poisson structures on the quotient X/G. What are the relation between these reduced Poisson structures, hyper-Poisson structures and general hyperkahl’er reduction? Understanding Kronheimer’s recent work on adjoint orbit hyperkahler structures provides another strong motivation. Using gauge theory and infinite dimensional hyperkahler reduction, Kronheimer proved that certain adjoint orbits of complex semi-simple Lie algebras admit hyperkahler metrics [lo], [ll]. Later on, his result was generalized by Biquard and Kovalev to arbitrary adjoint orbits [3], [9], and has been used to understand the Kostant-Sekiguchi correspondence [ 141. However, the hyperkahler metrics and symplectic structures obtained in this way are quite mysterious and elusive. Recall that an intrinsic reason for each coadjoint orbit to admit a symplectic structure is that the dual of any Lie algebra is a Poisson m,anifold, and each coadjoint orbit happens to be a symplectic leaf of this Lie-Poisson structure. Inspired by this fact as well as the results of Kronheimer and 4e
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others, it is quite reasonable to expect that there exists a hyper-Lie Poisson structure such that the orbits studied by Kronheimer et al. occur as its hyper-symplectic leaves. Then, this will provide US a natural source and symplectic explanation for those hyperk&Mer structures on adjoint orbits. To explore the connection between the work of Kronheimer and symplectic geometry was indeed the initial motivation for us to consider hyper-lie Poisson structures. In this paper, as an example, we will consider in detail a hyper-Lie Poisson structure associated with g = su(2). In the meantime, we will take some tentative steps toward hyper-Lie Poisson structures associated with general compact semi-simple Lie algebras. For this purpose, we will keep the discussion general from Section 2 through Section 3 while the last two sections will be devoted to the special case g = 42). To explain our approach. we need to rephrase the definition of hyperkahler manifolds in a way slightly different from the literature. Note that there is in fact no preferred choice of complex structures on a hyperkahler manifold. The bundle maps I’, J’ and K’ given by (I’, J’! K’) = (1, J, K)O, for any orthogonal matrix 0 E SO(3), will satisfy exactly the same quatemion relations. Therefore the map: 0 E SO(3) I’ assigns a complex structure to every orthogonal matrix in SO(3). In particular, under such an assignment, I, J and K are the complex structures corresponding to the identity matrix, the matrix of the cyclic permutation: {ei? e2, es} (e2: ea, el} and th e matrix of the cyclic permutation: {ei,ea:ea} {ea,ei,e:l}, respectively. Since a matrix in SO(3) can be naturally identified with a standard orthonormal basis in su(2) (such are also called frames in this paper), intrinsically we can think of a hyperkahler manifold as a manifold with a family of complex structures (or equivalently symplectic structures), parameterized by frames. This point of view is different from the conventional one, in which complex structures (or symplectic structures) on a hyperkahler manifold are considered to be parameterized by the unit sphere S2. This is the crux in our approach. Now the question arises in which space a hyper-Lie Poisson structure should live. To answer this question, we first recall that a Lie-Poisson space g* emerges as the target space of a momentum mapping of a hamiltonian G-space. A momentum mapping of a hyperklhler G-space X (i.e. a hyperkahler manifold X admitting a G-action which preservesthe hyperkahler structure) is usually considered, when the three complex structures 1, J, K are chosen, as a map from X to g” x g* x g* [7]. However, when I, J? K are replaced by any other three complex structures 1’: J’, K’ related by an orthogonal matrix in SO(3) as described earlier, the corresponding momentum mapping changes accordingly. Intrinsically, the momentum mapping of a hyperkahler G-manifold should therefore be considered as a map X -t L(g, 5u( 2)), where L( g: 5u( 2)) is the space of all linear maps from g to SU(2). It is therefore reasonable to expect that L(g, SU( 2)), as the target space of the momentum mapping of a hyperklhler G-manifold, should carry a hyper-Lie Poisson structure. Another possibly useful way to think of L(g, ~(2)) as a natural generalization of g” is to note that this space is obtained from g* = L(g, W) by replacing R’ by su(2). An elegant way of obtaining a family of Poisson structures on the spaceL( 8: 5u( 2)) goes as follows. The space L(g, su(2)) can be identified with g* x g* x g’, once an orthonormal basis of m(2), i .e., a frame, is fixed. If we can define a Poisson structure r on the space g’ x g* x 8’7 by pulling back 71;to L(g, su(2)) under such an identification, we then ANNALES
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obtain a Poisson structure on the space L(g, 5u( 2)). Th’ 1s construction in fact enables us to obtain a family of Poisson structures simply by varying the frames. Throughout the paper, we shall identify g* with g via the Killing form, hence g* x g* x g’ with g x g x g for simplicity. We note that any bivector field on g x g x g is determined by its corresponding brackets of all linear functions 1;‘, where lk(a, b, c) = (
2. Hypersymplectic
structures
The purpose of the present section is to introduce a notion called hypersymplectic structures, which includes hyperkahler manifolds as a special case and is much more natural from the viewpoint of symplectic geometry. Our definition of hyperkahler structures here is slightly different from the one in the literature, where complex structures and metrics have received much rnore attention. Our interests in this paper mainly lie in symplectic forms and their Poisson tensors. By n:(S), we denote the space of nondegenerate 2-forms on a manifold S, and l?+(~~T5’) the space of non-degenerate bivector fields on S. By K, we denote the map 0$(S) + lT+(A2TS), which is the inversion when elements in n”,(S) and I.+ (ARTS) are considered as bundle maps. For any w E 0:(S), we write w-l = K(W) E I’+.(A~TS). A su(2)-valued 2-form R on 5’ is said to be nondegenerate if the form WE= (<, 0) is 4e
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5 E mu,
where
the pairing
is with
respect to the
DEFINITION 2.1. - A hypersymplectic structure on a manifold 5’ is a closed non-degenerate tiU( 2)-valued two-form R such that < --i w;’ zs linear when 11<11= 1. That is, for any unit vectors
Suppose that R E O’(S) 6~ su(2) is orthonormal basis {el! e2, es} of su(2), as 0 = wiei + w2e2 + ~3.53, where wi) and 7r3 we denote their corresponding
a hypersymplectic structure on S. By choosing an also called a frczme in the sequel, n can be written w2 and wg are symplectic forms on S. By rl, ~~ bivector fields on S.
As usual, for each i, wt’ denotes the bundle map TS ---+ T*S given by {w,b(v): ?L} = w,(v, u), VU,V E TS, X) the vector field on S defined by wg(df) for any f E C-‘(S), and {f,g}i = Xjg for any f:g E C”(S). PROPOSITION
hypersymplectic
2.2. - Suppose that Cl = wlel structure on S. Then,
+ w~f-‘~ + wge3 E Q2(S) 8%su(2)
is a
(1).
(2)
[wf 0 (wjb)-‘1”
= -1,
where 1 denotes the identity map TS --+
ifi #j,
TS.
(59. [Ti. Tj] = 0, where the bracket
for
iI j,
any
is the :ichouten bracket on multivector~elds
[8],
Proof. - For any kr, k:t: k3 such that k: + ki + ki = 1, kter + kZe2 + k3e3 is a unit vector in su(2). Thus, it follows from definition that (klw;
= kl(wij-l
+ k2w; + k3w$l
+
k2(wi)-’
+ k3(wi)-1.
That is, 1 = (klw;
+ kpw; f k3w;)[kl(w;)-l
= (k; + k; -F k;) + k1k2[w;(w;)-1 + kh[w;(w;)-l Equations
(2) thus follow
+ w;(w;)-‘1
+ k2(4-l
+ k3(wi)-l]
+ w;(w;)-~] f k1k3[wf(w;)-1
+ w;(w;)-l].
immediately.
Also, from the argument above, we see that k.17~1+ k2n2 + k37r3 is still a Poisson tensor for any (k,, k2, k3) in the 2-sphere, hence for arbitrary k,! k2, kg as well. It thus follows q that [7rl, 7r2] = [7r2,7ra] = 1~1, “31 = 0. ANNALES
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An immediate consequence is the following: COROLLARY 2.3. - Let S be a hypersymplectic manifold with hypersymplectic form R E R2(S) 3 SU(~). YThenfor any <,Q E 8,
where rc = wE -l
and nV = w;,‘.
Sometimes the following equivalent version is more often used. PROPOSITION 2.4. - Equations (2) are equivalent to
w&q,
(3)
q)
%# 3,
= {f; .!?}i,
for any f, g E C”(S).
Proof. - It is quite obvious that [w,b0 (,;)-1]2
II= -1 *
(w;)-‘w~(wy
u
= +py
((w,“)-‘wp(w~)-ldf,
dg) = -((w”)-ldf,
dg)>
which is equivalent to WI(Xf; x;,
cl
= {f: 9);.
In fact, Equations (2,): or equivalently Equations (3) are also sufficient to construct a hypersymplectic structure on S. PROPOSITION 2.5. - Suppose that S is a manifold with three symplectic structures ~1, wp and w3 such that Equations (2) (or Equations (3)) hold. Then for any orthonormal basis {el, (22,ey} of SU(~), the su(2)-valued 2-f orm 0 = wlel + iu’2e2+ w2c2 dejines a hypersymplectic structure on S.
The proof is quite straightforward, and is left for the reader. To each hypersymplectic manifold, we associate a natural pseudo-metric as we will see below.
Let g : TS ---+ T*S be the bundle map given by g = w;(wyw;,
(4)
and I, J, K the bundle maps form TS to itself defined by J = ,9-lw;
I = flwf;
(5) THEOREM 2.6. -
K = .9-‘w;
and
(i) g is a pseudo-metric on S;
(ii) g can also be written as g = we-‘wi, (iii) I: .I>K satisfy the quaternion relation: I2
=
J2
=K2
or 9
= IJK=
=
w~(w~)-~w~;
-1.
Proof. - (1) It follows from Equations (2), by taking the dual, that [(W~ywZ”]’
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Hence, by using Equations (2) and (6) repeatedly, we have g* = -w~(~~)-‘w~ = [w~(w.$-l]w~ = -w~[(w~)-~w$ = [~~(w~)-‘]w~ = 9. That is, 9 is symmetric. Furthermore, it is evident that g is nondegenerate since ~1, w?. w3 are all nondegenerate. (2) It follows from Part (1) that g = .9” = -w~(w~)-‘ti~ equation can be obtained in a similar way.
= wF(wi)-‘wi.
The other
(3) Using Part (2), we have I = gplwF = [w~(w~)-~~$]~~w~ = (w~)-~LJ;. Similarly. .I = (wF)-‘wi and K = (LJ~)-~w!. Therefore, I2 = .J’ = K2 = -1. Furthermore, IJ = (w$~w;(w~)-~u; = [w~(w;)-lwyw~ = g-q = K. This concludes the proof.
0
It looks as if our definition of g depends on a particular choice of the frame. However, the following theorem indicates that 9 is in fact independent of frames up to a sign. THEOREM
pseudo-metrics
2.7. - If two frames coincide.
are of the same orientation,
then their corresponding
Proqf. - Let 5’= = {el $e2, es} and I = { el,, & eb} be any two frames. Suppose that (ei,ei.ej) = (e1,e2, es)0 for some orthogonal matrix 0 E SO(3). Let I, J! K be the induced almost complex structures on S corresponding to the frame F. We define I’, J’, K’ by the equation: (I’. J’, K’) = (I,
J, K)O.
It follows from the quaternion relation of I! .J, K that I’! J’, K’ same relation. Since wlel + wze2 + w2e2 = wiel, + wheh + wiei, (w~,~w~, w;) = (wl. w2. w,~)O. Hence, we have (7)
I’ = g-ywy,
.J’ = g-‘(w;)”
and
By using the quaternion relation of I’, J’, K’, (w~)“((w~)‘)-~(w~)~, which is g’ by definition.
we
also satisfy it follows
the that
K’ = g-+@ can easily
deduce
that g = Cl
Because of this result, we call g the pseudo-metric associated to the hypersymplectic structure despite of an ambiguity of signs. In particular, if g is positive (or negative) definite, the hypersymplectic structure becomes hyperkghler. We refer the reader to [l], [2], [6], [7] for the background on the subject of hyperkghler structures. The following result is well-known for hyperkshler structures, and is however still valid in our general context. Readers can find a proof in. for example, [2]. For completeness, we outline a proof here. THEOREM 2.8. - If S is a hypersymplectic manifold, I, ,JTK corresponding to any frame are integrable.
ANNALES
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then the almost complex structures
286 Proox
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- For any vector fields X: Y E X(S),
w&f,
Y) = g(JX, Y) = g(KIX,
y> =
w3(IX,
y>
Hence, we have the relation: XJw2
(8)
= IX Jwg
It follows that a complex vector field X is of type (1,O) with respect to I iff X Jw2 = iX Jug.
(9)
Suppose that X, Y are complex vector fields of type (1.0). In order to show that I is integrable it suffices to show that their bracket [X. Y] is of type (1,O) according to the Newlander-Nirenberg theorem. However, [X,Y]
Jwz
= Ls(Y
Jw2)
- Y J(Lxwz).
Now, Lxw2
=
(dL;y
+
=
d(iLX’W3)
LJ3&J2
= iLxw3,
and from Equation
(9), Y Jwq =
i(LyW3).
Thus, [X, Y] -1~2 = i[Lax(hyw3) Thus, I is integrable. Similarly,
- LY(L,~w~)]
= i([X, Y] Jug).
J and K are also integrable.
to check that ( slz,1), (~3~1) and all other similar Poisson-Nijenhuis structures in the sense of Kosmann-Scharzbach and Magri [8]. Nijenhuis structures are introduced by Kosmann-Scharzbach and Magri in the integrable systems. Therefore, it would be very interesting to explore the relation hypersymplectic manifolds and integrable systems. Remark. - It is not difficult
3. Hyper-Lie
0 pairs are Poissonstudy of between
Poisson structures
This section is devoted to the introduction of hyper-Lie Poisson structures. The main idea is to define on a suitable space A4 a family of Poisson structures parameterized by frames, which will coherently depend on the parameterization in a proper sense. We shall analyses the condition under which the induced symplectic foliations are independent of frames so that each leaf becomes hypersymplectic. To begin with, let g be a semisimple Lie algebra with Killing form (., .), and A a S2( g)-valued function on g x g x g, where S’(g) denotes the space of second order 4e S6RlE
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symmetric tensors on g. We note that any element in S*(g) can be naturally considered as a symmetric bilinear form on 9. So contracting with E: v E g, there corresponds to a function on g x g x g: A,:,, = CJA Jq. For any c E g, we denote by It, l:, 1: the linear functions on g x g x g defined by Ii(a, B, c) = ([: a), etc. Our first step is to define a Poisson structure on g x g x g. In order to do so (or even just to define a bivector field on g x g x g), it suffices to define its corresponding brackets among all linear functions 1;, i = 1,2,3, since they span the function space C”(g x g x g). DEFINITION 3.1. - Let A be a S’(g)-vatuedfunction defines a bivector field r on g x g x g.
on g x g x g. Thefollowing bracket
$7 $1 = q&1 {$q
= {&1:>
{@;j
= -q&,1
{l;:q
= {l&l:>
= $,q] = Zi,,]
G,“>$1 = +~.~] {I;, I;} = -{l,“, Z;} = A,,,.
Let G be a compact Lie group with Lie algebra g. Then G acts on g x g x g diagonally, with adjoint action on each factor. PROPOSITION 3.2. - The jZlowing
statements are equivalent:
(i) The bivector jield 7r is G-invariant; (ii) the map A : g x g x g -+ S’“(g) is G-equivariant, where G acts on S*(g) by the adjoint action; (iii>for
any I,rl,<
E 8, 5^fLc = Ab?l,C + A,,[,>+
Proof. - That (1) and (2) are equivalent is quite evident. (2) u (3). Suppose that A is G-equivariant. That is, A(gs) = A&A(s). It follows, by taking derivative, that for any < E g, {A = CCC&A.Hence for any qrl;C E 8, fA,,< = 71J&4 J< = qJaad
q The converse is also true by using the same argument backwards. The following theorem gives a necessary and sufficient condition for the bivector field 7r to be a Poisson tensor. As usual, for any f E C”(g x g x g) we write Xf for the vector field -ir#(df). THEOREM 3.3. - r (a: b, c) E g x g x 8,
is a Poisson tensor iff A is equivariant and at any point
(10)
< JXl;A
- Y/+A
= [c, [<,q]],
(11)
(J-G+
- ~1JXqA
= [b, [T <]I,
for any (, v E g. ANNALES
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Proof. - By z)ria, we denote the projection g x g x g --+ go given by pria(Ia, b, c) = a+ib. Similarly, ~7-13 denotes the projection from g x (I x g to g” given by pris(a, b, c) = a + ic. It is simple to see that Tp~i27r = Tprisr = 7ro. where 7rc is the Lie-Poisson tensor on 8’. which is identified with its dual as a real Lie algebra. Hence the Jacobi identity
{Ul, f*}9 f3) + C.P.= 0 holds if f?, i = 1,2,3, are linear functions of the form “i! Ii, or of the form I;! Ii, where c.p. stands for the cyclic permutation. It remains to check the following three cases: (1) fi = I1 fa = 1: and fa = $; (2) fl = I;, fa = 1; and fs = 1:; and (3) fi = I:, f2 = 1: and fs = I,.5’ It is simple to see that the Jacobi identity in Case (I) is equivalent to that YI is G-equivariant according to Proposition 3.2. As for Case (2),
Thus, the Jacobi identity follows iff Equation (10) holds. Similarly, equivalent to the Jacobi identity for Case (3). In the proof above., we have in fact shown the following:
Equation
(11) is 0
PROPOSITION 3.4. - Assume that both Equation (10) and Equation (11) hold for any EY17 E 8. Then, both ior : g x g x g --+ 8” and pr13 : g x g x g --+ 8” are Poisson maps, where gC, identi$ed with its dual, is equipped with the Lie Poisson structure as a real Lie algebra.
Consider the space ,V = L(g,su(2)), which consists of all linear maps from g to Here 542) is, only considered as a vector space without using any Lie algebra structure. M admits a natural G-action induced from the adjoint action on g. Whenever a frame 3, i.e., an orthonormal basis {el, e2, e3} of m(2), is chosen, M is identified with g* x g* x g”, which can also be identified with g x g x g using the Killing form on g. We shall denote such an identification M g x g x g by 9~. From now on, we will always identify g with its dual g*. Using the map Q~F, the Poisson structure r on g x g x g is pulled back to a Poisson structure 7r3 on hf. When the choice of frames 3 varies, we thus obtain a family of Poisson structures on M parameterized by frames. This is the very structure we are interested in. Corresponding to any frame 3 = {ei! e2, es}, there exist two frames 32 and 33 obtained by the cyclic permutations: {e2! es, ei}, and {e:s! ei, ea}, respectively. We also often use 31 to denote 3. The Poisson structures corresponding to 3i, 32 and .3Y are denoted by 1r3, ! ?r~, and ran. respectively. By choosing a frame, any vector-valued function F on g x g x g can be pulled back to a function FF on ,M via the map qF. If furthermore, F is invariant under the action of O(3), where O(3) acts on g x g x g by (a, b: c) (a, b. c)O for any (a, b,c) and 0 E O(3), F3 is independent of the frame 3, and therefore can be considered as a well-defined function on M. However, in most cases, the function F is only invariant under the S0(3)-action. In this case, the pull back FF depends on the orientation of 3. Whenever the orientation of frames is fixed, we shall still get a well-defined function on M.
m(2).
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In the sequel, we shall always assume that the S2(g)-valued function A on g x g x g is invariant under the SO(3:)-action, and therefore can be considered as a function on M when an orientation of fr.smes is fixed. THEOREM 3.5. - n = ry, el •MF~ e2 +I~F~ es does not depend on the choice offrames of the sameorientation, and therefore is a well-defined section of the vector bundle A~TM@~LL( 2). Proof. - Assume that 7 = {el,, e;! ei} is another frame and (el,: ek, ei) = (el, e2: e3)0 for some 0 E SO(3). It suffices to show that
or equivalently, (13)
({f:g}~,,(fr9}1,!{f,g}~)
for any f!g
E C-(M).
=
({f,g}3,,{f:g}3~,{f,S}rl)0
A4 can be identified with g x g x g under both 93 and 9~. If (a, b?c) and (a’: b’, c’) E g x g x g are, respectively, the coordinates of any point in M under these two identifications, they should be related by (a’: b’, c’) = (n, b! c)O. We assume that 0 = {(nij)}. A s an example, we shall show below that
(14)
{f!dZ
=
~llLf!d3-l
+
~2l{f.d.F~
+
a31{f,g}3z!
for f = (<>u) and g = (7, u). All the other casescan be proved similarly. In this case, it is simple to see that
([1?771>d(K?rll>b’) a. VI34 {.f,g}7,
=
(%lr”12r~13)
([&711:6’)
(K,VI:4 ([E:VI!4 =
(all,
a127
al3)
i
= ([E,VI*(4, = ([I>49(241 =
([I,
VI,
2QllU
([t?
71,
([ET
7715 4
42
b’)
l)U’
+
-
a’).
=
([<.
q].
=
([~,7/]!2UllU
-A,,71
-(Lt. ~1~a’)
+ alla
([C,rll,c’)
a11
o
al2
71,
U/j
-a,
2U1lU12b’
2fl>llU&’
+
~2l{fd7}3~
+ 2Ul2U13C’)
This completes the proof: ANNALES
SCIENTIFIQUES
DE L’kOLI!
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SUPfiRlELiRE
+ a21b
-
U13
([I?717 b’) +
-
I(
-([I,
On the other hand, all{f,g}3~
U12
A,.,
0
Uf,)U’
-
a11
-t[bl:U’)
u’).
-
a3l{fd7)3~
a314
~UI~U&)
4%
4
)(
a13
) )
290
PING XU
An immediate consequence is the following: THEOREM 3.6. - Assume that Equations (I 0) and (11) hold. Then the Poisson tensors 7r3, r~ corresponding to (arty two frames commute.
Proof. - For any ki, ka, ks E 58such that k: + kz + k$ = 1, it follows from Theorem 3.5 that k1r3, + k27r~~+ ksn 3z is still a Poissontensor. Hence, 7r3,, ~~~ and 7r~~ all commute. Again according to Theorem 3.5, ~7 is a linear combination of 7r3-,, r3z and n-3s, and therefore commutes with 7r~(= 7r3, ). 0 Remark. - Under the assumption of Theorem 3.3, the Poisson structure 7r3 on M is G-invariant, and pri o $3 : 121-+ g, the composition of @3 with prl : 0 x g x g ----+ 0, the projection onto its first factor, is a G-equivariant momentum mapping. Similarly, 7r3F1 and 7r3> are G-invariant with equivariant momentum mappings pr2 o QF and rjr3 o QF, respectively. The rest of the section is devoted to the investigation of the symplectic foliation of 7r3. For simplicity, whenever a frame F is fixed, we shall omit the subscript 3- when denoting the Poisson structure under the circumstance without confusion. I.e., we will use 7ri to denote 7r~,. By Xj, for i = I! 2,3, we denote the vector field n#(df) for f E C-(M). THEOREM 3.7. - The symplectic foliation of n-g is independent of the choice offrames 3, and the induced family of symplectic structures on each leaf is hyper-symplectic ifffor any (a: 6, c) E 0 x 0 :< 0 and < E 0, the following system of equations for (u, 21,w) has a solution:
A#u + [v,c] - [w,b] = [<,a] -[u, c] + A#v + [w, a] = [<, b] (15)
[71,6]
- [v, u] + A#w = [I, c]
1 -[~,a]
- [v,b] - [w,c] = A#<,
where for any u E 0, .4#u is the g-valued function on M obtained by contracting with u, and A4 is identified with 0 x g x 0 under QT. Remark. - The first three equations can be written in terms of a single equation as
If such an A exists, we shall call the corresponding family of Poisson structures a hyper-Lie Poisson structure and their symplectic foliation hypersymplectic foliation. We need several lemmas before we can prove this theorem. The next lemma indicates that whether System (15) is solvable is independent of the choice of frames. Therefore, the statement in Theorem 3.7 is well justified. LEMMA 3.8. - For any fixed E E 0, (u, II, w) is a solution of System (15) for (a, 6, c) E 0 x 0 x 0 ifs ( u’, u’: ,u’) = (u, v, w)O is a solution of the same system for (u’,b’,c’), where 0 is any matrix in SO(3) and (u’,b’,c’) = (u,b:c)O. This can be proved by a straightforward verification, and is left to the reader. The proof of the following two lemmas is also quite straightforward from definition. 4e
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LEMMA 3.9. - For the Poisson structure 7r on g x g x g, the hamiltonian vector$elds of linear functions are gillen by
x1;= (-[ET4, -[Lb],-[<>c]), xl; = (-[I>b],[Ca],A#[), XI; = (-[I>cl:-A% [r,a])> for any < E g, where (a, b. c) is any point in g x g x g and the tangent space at this point is naturally identijied with g x g x g. LEMMA
3.10. - For an)’ [ E g and i:j
= 1: 2,3,
x;; = -x;,,
i fj;
x;; = -r^. E Proof of Theorem 3.7. - Assume that System (15) has a solution. We divide our proof into several steps. (1) 7g T*M
= &T’M
= T$~T*M.
From Lemma (3.10), it follows that X,T = -X:L
and Xs = -[
XF1 and X$ are in 7rzlT* M. Also, it is e&y to se: that Xi
= Xii.
Hence, both
= (A#<, -[i,c].
[<,b]). On
thk other hind, for any (u, 7;:UJ) E g x g x g g Ti,,6,Cj(g x \ x g), we have K$~(u, II! W) = (-[W
a] - [w, h] - [w, c]! -[u,
b] + [w, a] - A#w;
-[u,
cl + A%
+ [w, al)
by Lemma 3.9. It is equal to X$ if (u, w, w) is a solution of System (15). Hence, we have X$ E T$~T*M. This shows thit ng2T*M soCdo the other relations as well.
2 T$~T*M.
Similarly R$, T”M
& ns2T*M,
(2) The symplectic foliation of 7r~ does not depend on the choice of frames. Let 7 be another frarr.e, and XI its corresponding Poisson structure. According to Theorem 3.5, ~1 can be expressed as a linear combination of nap, 7r~~ and 7r~~. Therefore, it follows that nl#T*M C 7rfT’M from Step (1). According to Lemma 3.8 and Step (l), the symplectic foliations of 7r~,, i = 1, 2, 3, also coincide. Hence, exchanging 3 and 7, we obtain the other inclusion: ngT”M & *,#T*M. (3) Let wl, w2 and 1~3 be the symplectic structures on a hypersymplectic leaf corresponding to 1r3~, ~3~ and 7r~~. Then wl, w2, w3 satisfy Equations (3). Below we only prove the following equation: V f, 9 E Cm(M),
Wl(Xf2,Xy2)= {f>?l}l.
(16)
The other equations can also be proved similarly. .ANNALES
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In fact, it suffices to show this equation when f and g are linear functions. or g is 1: (for example f = Ii), we have LHS = wi (X$ : X,“)
If either f
(by Lemma 3.10)
= -wl(X:,;li,‘, = -(dz;> X,“) = -{g& = X;(g)
(by Lemma 3.10)
= -i(g) = {l&Ll>l. Similarly, Equation (16) holds if either f or g is 1;. It remains to check Equation (16) for f = I; and g = 1:. In this case, we know that Xg Jwi = (u, ‘u, w) according to the proof in Step (l), where (u, ZI, w) is a solution of Syttem (15) and is considered as a cotangent vector at (a, b, c) E g x 0 x 8. We also know that X; = (A#vl, -[?I! c], [v: b]). Therefore, ? Wl(X$.
x$
= (X$ _Iwl)(x;)
= (u,-4%)+ (‘u,-h cl)+ (w>[rl,bl) = (A#u
+ [u, c] - [w, b], 77) (by Equation (15))
= M:477) = (h a4 On the other hand, jl,“,lt}1 = Zij,l,El= ([~,<],a). completes the proof of Equation (16).
Hence, wl(Xs,X$) E
= {1,“,1;}1.
This
Finally, it is quite transparent from the proof above that the assumption in the statement El of Theorem 3.7 should also be necessary. To end this section, we give the following result which reveals the connection between the 5” (g)-valued function A and the induced pseudo-metric on the hypersyrnplectic leaves. THEOREM
ps:ucflo-metric dE,O
3.11. - Under the assumptions g on each hyper-symplectic
as in Theorem 3.3 and Theorem 3.7, the leaf is G-invariant, and for any < E 8,
= -4,~.
Proof. - Since the Poisson structures x3,, ~~~ and ~3~ are all G-invariant, so are their induced symplectic structures on each hyper-symplectic leaf. Hence, the pseudo-metric g is G-invariant. According to Equation (4), 9(i> 8 = -r1 kJ;i, 43 = -7r1(-dZ& 41;) = -{z;‘ll,“}l = -A<,<.
4e S6RIE - TOME 30 - 1997 -- No 3
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We have seen that the vector-valued function il plays a fundamental role in defining a hyper-Lie Poisson structure. The theorem above leads to some nondegenerate criterion that A should satisfy, i.e., A,,, = 0 iff [ = 0. The work of Kronheimer [lo], [II] very much supports the existence of A for compact semi-simple Lie algebras. A satisfactory solution to this problem should provide us a symplectic approach, and therefore an intrinsic explanation, on the existence of hyperkahler structures on adjoint orbits. The work on this project is still in progress. In the rest of the paper, instead we will consider the case that g = JZU(2). This case can be handled relatively more easily becauseof its special character as a three-dimensional Lie algebra. However, we shall see that certain nontrivial results, some of which are already quite striking, can be deduced even in such a simple case.
4. The case of g = su(2) From now on, we will work on the special case that g = 5u(2). In this case, a function A can be explicitly constructed on an open submanifold of M, and the corresponding hyper-Lie Poisson structures are studied under the general set-up in the previous section. By a, we denote the function on M defined by:
V(a:b,c)
@(a, b: c) = (a, [h cl),
E g x g x g.
Here again A4 is identified with g x g x g under some chosen frame. This equation defines the well known Lie algebra 3-cocycle corresponding to the Killing form. However, here we consider it as a function on g x g x g instead of A3g. Clearly, @ is independent of the choice of frames, provided that they have the same orientation. Hence, @ can still be considered as a well-defined function on M. Let M0 be the open submanifold of 111 consisting of all points where Q # 0. In other words, M0 consists of triples (a, 6, c) which are linearly independent. Let A : It40 c g x g x g + S’(g) be the map given by
(17)
A(u,
b: c) = $([u,
b] @ [u,b] + [b, c] @ [b, c] + [c, u] @ [c, a])>
V(u, b, c) E Mc,.
It is not difficult to check that the rhs of Equation (17) is invariant under the natural action of SO(3), so A can indeed be considered as a well-defined map from M, to S’(g). THEOREM
Proof.
4.1. - A is G-cquivariant
and satisfies
the condition
in Theorem
3.3
- That A is G-equivariant can be verified directly.
We note that A is uniquely characterized by the following relations:
(18)
uiA
= [b, cl>
for any (a, b, c) E A&,. Applying a J A = [b, c], we obtain
bJA
= [~,a]:
the vector field XI:
XpIA+uJX~;A
ANNALES
SCIENTIFIQUES
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and
= [Xpgb,c]
SUPtRIELJRE
cJA
= [u:b]
on both sides of the equation
+ [b,Xqc],
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PING XU
where both sides are considered as a g-valued function on g x g x g. Using Lemma 3.9, we have
-[v, b] JA + a JXqA
= [[q, a], c] + [b>A%/].
Hence,
u JXz;A
= [[q, a], c] + [b! A#711 + A#[Q, b].
By contracting with < E g, it follows that
Thus
(a:(W+A - ‘I J&;-W = (r, [[rl,4: cl) - (rl,[[I,al,4) = b?K 771,4).
Using the other two identities in Equation (1S), similarly we deduce that
Since the Lie algebra g = su(2) is three dimensional and @(u,b, c) # 0, {a, b, c} constitutes a basis of su(2) at any point in M,. Equation (10) thus follows immediately, similarly for Equation (11). 0 In fact, A also satisfies the assumption as in Theorem 3.7. THEOREM 4.2. - For the function A dejned by Equation (17), System (15) always has
a solution for any < E g and (a, b, c) E MO. So MO has a hyper-Lie Poisson structure. In particular, its symplectic leaves are hyper-symplectic. Proof. - Fix any point (u, b, c) E Al,. Since su(2) is three-dimensional and System (15) is linear with respect to u, II, w, it suffices to prove this statement for any three linearly independent < E g = su(2). Therefore, it is sufficient to prove this for < = a, b, and c. For this purpose, one can check directly that u = u = 0, w = -b is a solution for < = a; u = -c, v = w = 0 is a solution for < = b; and u = 0, v = -a> w = 0 is a solution for < = c. 0 In fact, in this special case, the corresponding hypersymplectic foliation can be described quite explicitly. By X, we denote the gradient vector field of a, where M is equipped with the standard metric induced from the Killing form on g. As a frame is chosen and hf is identified with g x g x g, the vector field X at any point (a, b3c) can be written as X = ([kc], [wl,
(19)
hbl).
Since both the standard metric on M and the function @ are G-invariant, the gradient vector field X is also G-invariant. Therefore, it follows that [X,(1 = 0, de SGRIE - TOME
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THEOREM 4.3. - The symplectic foliation of r~ on M0 coincides with the orbits of the Lie algebra action of the direct product Lie algebra R x 5u( 2), with X and [, < E g being its generators.
Proof. - The symplectic distribution at any point (CL,b, c) is spanned by r$(T*M). By identifying g* with g, TTa,b,C,A $1 is identified with g x g x g as a vector space. To compute the symplectic distribution, it is sufficient to compute the image of a basis of g x g x g under the map :r$. Since g is three dimensional, {a, 6: C} can be considered as a basis of g. Hence, it suffices to do the computation for its corresponding basis in g x g x g. Using Lemma 3.9, we have 7r$(<:O,O) = -< = -([<>a], [<,b], [<,c]). It is also easy to see that r$(O, a, 0) = 6, K$(O, b!O) = -6 and rr,#(O,c:O) = X; n$(O,O,n) = c, n,“(O, 0, b) = -X and r$(O, 0, cj = -6. This concludes the proof of the theorem. 0 PROPOSITION 4.4. - The Lie algebra action defined as in Theorem 4.3 is locally free on MO, so its orbits are all 4-dimensional.
Proof. - If not, there is [ E g and lo E R not all zero, such that [ + kX = 0. If /C# 0, it follows that [b: c] = i[<, a]. H ence, @ = (a! [b, c]) = i (a, [I, a]) = 0. This contradicts to the definition of A&,. If k == 0, we have i = -([<, a], [I. b], [<: c]) = 0. It thus follows that < = 0 since {n,b,c} is a basis of g. 0 The coming result gives us a complete set of casimir functions for the hypersymplectic foliation. PROPOSITION 4.5. - The following functions (a,,b), (b,c): (c,u), (a,n) - (b;b) and (b, b) - (c, c) form a compr’eteset of casimirs for the Poisson structure TF on iV&,.
Proof. - It is simple to see that these functions are all G-invariant. To show that they are casimirs, it suffices to show that they are killed by the vector field X, which can be checked directly. It is also easy to see that these functions are all independent, so it follows 0 from dimension counting that this set of casimirs is complete. To end this section, we look at the induced metric on each hyper-symplectic leaf. The metric on the infinitesimal generators i of the G-action is already given by Theorem 3.11. In order to describe the metric, we only need to know its evaluation on the vector field X, which is the content of the following: PROPOSITION 4.6. g(X,X)
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Proof. - We already know that X$
= -a.
= (A#<> -[<; c], [<: 61). Thus, w;X
= (0,O: a).
,4ccording to Lemma 3.9 and Lemma 3.10, we have w:X = (b, 0,O). Here, in both equations, the right hand sides are considered as elements in the cotangent space of 111, being identified with g x g x g. Therefore, 9(X,X) = -ni(&X.w~X) = 0 -xl((O,o,a):(b,O,O)) = --(&O,O,a),(b,O,O)) = -([c>u],b) = -a. The following consequence follows immediately from this result combining with Theorem 3.11. THEOREM 4.7. - When g := 5u( 2) and A is defined by Equation (17), each hypersymplectic leaf of h& is a 4-dimensional hyperkiihler manifold. ANNALES
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5. Moduli
spaces of solutions
XU
to Nahm’s
equations
and (co)adjoint
orbits
This is a continuation of the last section. When M is identified with g x g x g by choosing a frame F’, the vector field X is written as X = ([h: c]. [c, a], [n, h]). Its flow is thus given by the following system of equations: a = [b. c] & = [c, a]
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i c = [c&b]. Such a system is called Nahm’s equations. which was studied by Kronheimer [lo] modelled on the study of general Nahm’s equations made by Donaldson [5]. For any z E 41, we denote by ~~(3:) the flow generated by X through the point ;c. We denote by 5’ the set of all points .c E M such that the flow v+(x) converges as t ---+ --oc. 5’ has a natural foliation according to the limit points. Note that all the limit points are critical points of a. We denote by C the critic.31 set of a. The G-action on M leaves C invariant. For any orbit 0, we let Ss be the submanifold of S consisting of all points in S whose trajectory under the gradient vector field -Y converges to a point in 0, as t --co. so that in particular Sa be such a submanifold corresponding to the zero orbit. It is clear that S = U. 5’0, where the sum is over all the G-orbits in C. Kronheimer proved, using gauge theory, for a general semi-simple Lie algebra that certain 5’0 are hyperkahler manifolds and are diffeomorphic to adjoint orbits of gc [IO], [ 111. Below we will prove this result for the special case of su(2), as a consequence of the hyper-Lie Poisson structure on 5’. Our approach is quite elementary, and the family of symplectic structures on each leaf 5’0 is rather transparent. To start with, let us introduce a function F on JVf by F(n, b, c) = (0, a) + (b, b) + (c, c), where M is identified indeed a well-defined
with g x 0 x g by choosing function on M.
a frame. It is simple to see that F is
LEMMA 5.1. LSQ’ = (Ix11*. a7z.d LsF Proof. - The first identity the second one: we have LsF
follows
= Ls((n.a)
= f3D.
from a general property
of a gradient flow. As for
+ (b,b) + (c:c))
= 2(a, [b, c]) + 2(b: [c, CL]) + 2(c; [a‘: bl) = m. The following
result is crucial for characterizing
0
the elements in S.
PROPOSITION 5.2. - (i) If pt(x) converges as t -cm, x is either a critical point of @ or @(ix) > 0. In the latter case, we in fact have @(;pt(x)) > 0 for all t whenever pt (x) is dejined.
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(ii) Zf pt (z) does not converge as t --CC or is not dejnedfor all t 5 0 (i.e., X is incomplete in the -CC direction), Q, cannot be always nonnegative along the$ow. Proof - By Lemma 5.1, $fD(yt(cz)) = LA-@ = llXl12 > 0. Hence. +(P~(z)) is an increasing function with respect to t. If cpt(z) converges as t --+ -x, the limit point must be a critical point. However,, the critical points of Q are defined by the system of equations:
[b:c]= [c,u]= [u:b]= 0. So @ vanishes at any critical point. This yields that @(.z) > 0. If Q(z) = 0, it follows that @(p,(n)) = 0, f or all t 5 0. By taking derivative, we have llXll = 0. Thus, :c is a critical point. In the case that a(:~) > 0, it is not difficult to see that (P(pt(:c)) has to stay positive for all t whenever pt(z) is defined, otherwise .c will be a critical point according to the same argument above. If yt(z) is not defined for all t 5 0, it must be unbounded as t approaches to a finite number. If pt(z) is defined for all t 5 0 but does not converge as t -x, it must be unbounded as t is sufficiently negative since @ is a real analytic function. In both cases, I-‘(p,-,(z)) d x, as t -. -X (X is either a positive number or SC>. Assume that Cp is always nonnegative along the flow. It follows from Lemma 5.1 that -$F(p,-,(.c)) = 6Q > 0. So F(pt(z)) 5 F(z) when t < 0, which is a contradiction. This concludes the proof. 0 By iZf+, we denote the submanifold of M consisting of all points where Q, is positive. The theorem above yields that S - C is contained in ,U+. Moreover, the vector field X is complete in S - C. It is clear that S - C is invariant under the G-action, hence invariant under the action of the product Lie algebra W x 5u(2) as defined in Theorem 4.3. In other words, S - C is a hq,per-Poisson submanifold of IV,,. To extend this hyper-Poisson structure to entire S, it suffices to extend 7rF to the critical set C. For this, one only needs to extend the vector valued function A to the critical set C. Since C is the limit set, a natural way to extend A is to take its limit along the flow X. This is in fact how we derive the formula below. When ?VI is identified with g x g x g under a chosen frame, a point 3’0 = (no, bo, c,,) E g x g x g is a critical point iff CLO:bo, CO are parallel. Hence, for any critical point, we can always choose a frame so that the critical point is of the form (a~, 0: 0) for some a0 E g under the identification: n/l Z g x g x g using this frame. Such a frame is called a standard frame. Clearly, the element a.0 is unique up to a sign. We then define the function A on C under a standard frame by:
where 1,: i = 1! 2,3, is an orthogonal basis for g g su(2). A is clearly well-defined eon C.
We also let A = 0 at 2 = 0.
To show that such an extension is smooth, we need to give an alternate description of S, which is much easier to deal with. For any given nontrivial critical point ~0, let Fr,, be a standard frame such that z. = (ao, 0: 0) E g x g x g when M is identified with g x g x g under Fzo. We denote, ANNALES
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by L,, the subset of M consisting of all the points 2; = (a, b, c) E g x g x g (% ~$1under Fz,),
satisfying the condition: (CL6) = (b, c) = (a. C) = 0: (b,b) = (~4, ;;‘,,(b, b) = (~0, ao), { - .
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It is easy to check that this definition is well-justified, i.e., does not depend on the choice of the standard frame Fr,. Also, we define Co as the subspaceof g x g x g consisting of all points (a, b, c) such that (u: b) = (b: c) = (a, c) = 0.
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(u, u) = (b, b) = (c, c),
and
a 2 0.
It is clear that these relations are preserved under the transformation (a! b, c) (a’, b’; c’) = (a, b,c)O for any 0 E SO( 3). Therefore, Ca can also be considered as a subset of M. LEMMA
5.3. - For any nontrivial critical point :cg, C,, = SG.~“.
Proof. - C,, is obviously a closed submanifold of M. It is clear that if 2 E C,,, then cpt(z) will stay in C,, for all the t whenever the flow is defined, since X is tangent to c,,. Since the intersection of C,, with the hypersurface @ = 0 is contained in the critical set C, we conclude that @((p,(z)) > 0 if II: is not a critical point in C,,. Thus, according to Proposit:lon 5.2, cpt(z) exists for all f 5 0 and converges as t --+ --x. Let us assume that y is the limit point of cpt(x). Then y is a critical point and y E C,, . Assume that y = (v, V, W) under the standard frame &, . It is not difficult to seeby using Equation (24) that v = ul = 0 and (u, U) = (ua, ILO). The latter implies that y E G . x0. Hence C,, C SG,ro. Conversely, assumethat z is any point in SG,zo and cpt(z) y E G.zo as t --+ --3o. Fix a standard frame Jyz, so that under it za = (uo: 0,O). Then under this frame y =: (u, 0,O) with u E G . ao. Hence, (u~u) = (uo,uo). Suppose that il: = (a, b,c) under the frame &.,. From Proposition 4.5, it thus follows that (u, b) = (u, 0) = 0, (b, c) = (C),0) = 0, (a, c) = (u,O) = 0, ([I, 6) - (c! c) = 0 and (a, rz) - (b, b) = (u, U) = (ao, uo). That is, z is 0 in C,,. This completes the proof. From this lemma, ii. follows that for any ~0, y. E C, C,, and C,, are either disjoint, or equal. In the latter case, x0 and y. must lie in the same G-orbits of C. For this reason, we shall use Co to denote the space C,, for any 20 E 0. The lemma above shows that S0 = CO. In fact, this is also valid when 0 is the trivial orbit. LEMMA
5.4.
so = co. Proof. - That So 2, Co follows immediately from the fact that the functions: (a! b), (a, c), (b! c), (a! u) - (6 b), and (b, b) - (cI, c) are all preserved by the vector field X. As for the other direction, let us assumethat (a, b, c) is any nontrivial point in CO. By the definition of Co, we can write a = Xei, b = Xes and c = Xea, where X is a positive 4= SBR[E
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number and { er; ez, es} satisfies the standard su(2)-relation: [et, ea] = e3, etc. Therefore, it is not difficult to see that at = - &, /jt = - &, ct = - & is the flow through the point (a: b, c). Obviously, it goes to zero as t -cc. That is, (a, b, c) E S,. 0 Combining the two lemmas above, we have PROPOSITION 5.5. - For
Now we are ready to prove the smoothnessof the extension. THEOREM 5.6. - ~‘g is smooth when restricted to each So for any nontrivial G-orbit 0.
Proof. - It suffices to show that the extension A,,, as defined by Equation (23) is smooth for any < E g in a neighborhood of 0 in S0. Let T be the norm of the elements in 0. Then by Proposition 5.5, under a standard frame, points (n, 6, c) in Sc, are characterized by the equations: (a! b) = (b, c) = (a, c) = 0,
(b, b) = (C! 4,
(a, u) - (b; b) = T2.
Therefore, when a point (a: h, c) is in S0 but not in 0, we can always write a. = dmel, b = k2, I: = Xes, where X = v@$ and {el; e2, es} is an orthonormal basis of su(2) satisfying the standard relation. Write < =
Substituting [f + <,’ = (<. <) - [:,
we have
It is trivial to see that the extension of A as given by Equation (23) coincides with the equation above when X = 0. Since both X2 = (b, b) and E1 = H are smooth functions 0 on SO, A,,, is clearly smooth on So as well. For any nontrivial orbit (3 c C, it is obvious that So is invariant under the G-action, as well as that of the additive group W generated by X. Therefore, for any point x E So - 0, its hypersymplectic leaf .&, defined as in the previous section, is contained in SO. Since C, is a 4-dimensional manifold according to Proposition 4.4, it can be considered as an open neighborhood of 5 in SO. Clearly, So is a union of these leaves together with their boundary 0. As observed early, there is a standard frame 3 such that when M is identified with g x g x g under this frame, any point in 0 is written as z = (u:O,O) with a E g. In this way, 0 is naturally identified with a (co)adjoint orbit of g. Although there is an ambiguity for the choice of the frame F, such an adjoint orbit is uniquely determined, and the identification is unique up to a sign. In the following, we will fix any such a frame .Y=, and denote the Poisson structures 7r~~,7r~~ and 7r~~ simply by ~1, 7r2and 7r3,respectively, and the induced symplectic structures on any leaf by wl, w2 and wg for simplicity. THEOREM 5.7, - For any nontrivial G-orbit c> C C, the extended hyper-Poisson structure on S induces a hyperkiihler structure on So. Furthermore, if we choose a frame as in the ANNALES
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observation above, then 0 is a symplectic submantfold with respect to wl. In fact, it is a Kahler submanifold with respect to I and its Kahler metric coincides with the one on 0 when it is naturally identified vtith a (co)adjoint orbit. In the meantime, 0 is a lagrangian submantfold with respect to LU’Zand LU’~. Proof - It remains to consider points in 0. For any x E 0, one can directly verify that r#T~i’M = T,So, for i = 1, 2, 3, by using a local coordinate chart under a chosen standard frame 3. Hence, the bivector fields 7rlr,,%= 1,2.3, are all tangent to So and nondegenerate along 0. According to Theorem 4.2, the corresponding symplectic structures wi are all compatible along 6 by continuity, hence compatible in entire SO. Since the induced metric is negative definite on 5’0 - C, it is also negative definite along 0. Hence, the extended hyper-Poisson structure induces a hyperkghler structure on So. The rest of the conclusion can be verified directly: again by using local coordinates. 0 In fact, So is closely related to adjoint orbits of 8”. To see this, let 3 be any frame, and pr12 o qF : M --+ 8”’ the composition of the identification 9~ : M -+ g x g x JJ with the projection prla as defined in the proof of Theorem 3.3. According to Proposition 3.4, pr12 o qF is a Poisson map with respect to 7r~-. Hence, the image of So is a Poisson submanifold of 0”. The following theorem indicates that in generic case the image is in fact a single adjoint orbit. A general result was proved by Kronheimer [lo] using gauge theory. However, our proof in this special case here is quite elementary and only uses some well-known facts in symplectic geometry. By C?ia we denote the adjoint orbit in 8” containing the image (prr~ o PF) (0). THEOREM 5.8. - Zf 012 is a regular orbit in gC, then (~~12o Q,)(S,) = 012. In fact, prlz o QF : So C?12is a symplectic difleomorphism, where 5’0 is equipped with the symplectic structure induced from the Poisson structure ~3 and 012 is equipped with the (co)adjoint orbit symplectic structure.
We need some lemmas first. LEMMA 5.9. - Under the map ~7-12o @3 : SO ---+ gc, the inverse image of any bounded region is bounded.
Proof. - According to Proposition 5.5 and by the definition of Co, there is a standard frame 7, under which any point (a’, b’! c’) E g x g x g in So is characterized by
(a’: b’) = (b’: c’) = (a’, c’) = 0:
(b’, b’) = (2, c’), (a’, a’) - (b’, b’) = r2:
where r denotes the norm of elements in 0. Suppose that 0 = (aij) E SO(3) is the transformation matrix between the given frame 3 and this standard frame 7. That is, (a,b,c) = (a.‘: b’, c’)O. It is simple to see that (a,~) = ~~:rr” + (b’,b’), (b,b) = uf2r2 + (b’, b’), and (c,c) = &r2 + (6’,b’). Therefore, if (u,u) and (b,b) are bounded, (b’, b’) has to be bounded. This implies that (c, c) is bounded as well. 0 Two immediate consequencesare the following: COROLLARY 5.10. - The map pr12 o QJI : 5’0 -
gC is a proper map.
COROLLARY 5.11. - The image of ,570under pr12 o 9~ is closed. 4e
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Proof. - Using the map !l?~, we may identify M with g x g x g. Suppose this identification ((L,, b,, cn) is a sequence in So such that a, -+ a0 and We need to show that (ao, bO) E (p ~12 0 Ql,)(S,). It follows from Lemma is bounded. Therefore there exists a convergent subsequence Cam, whose limit by CO. Then, (aa, bo, CO) is in SO since So is closed. This concludes the proof.
301 that under b,, ---+ bo. 5.9 that c,, is denoted 0
Proof of Theorem 5.8. - Again, let us identify A4 with g x g x g by qF. It is easy to see is the tangent vector, at a + ib, generated by the adjoint action ad,,,. Hence, the projection of entire flow: (priz o QF)(cpt(x)) lies in a single orbit C for all t. Since cpt(z) converges to 0 as t goes to -00, then (priz o Q,)(0) C C. Then-fore, 012 C c. Since 0 12 is a regular orbit, it thus follows that 012 = C. This means that the image (prip o \JrF)(S,) is contained in (3,, (this part of the argument is due to Kronheimer [lo]).
that at any point (a: b, c) E g x g x g, T(pr12)X
On the other hand, according to Corollary 5.11, (pria o q~)(So) is a closed Poisson submanifold in 012. Hence, it must be the entire orbit 0i2. That is, (prla o qF) : SC? 012 is onto. This map is automatically a submersion since it is a Poisson map. By dimension counting, it must be a local diffeomorphism. However? by Corollary 5.10, it is a proper map. Therefore, it must be a covering. Since 0i2 is simply connected, c3i2 is thus a diffeomorphism. 0 (m2 0 93) : so Finally, we will show that So - (0) is diffeomorphic to the nilpotent orbit of s l( 2, C). THEOREM 5.12. - So - {O:I is a hypersymplectic leaf of MO, and therefore is a hyperkahler manifold. For any frame 3’, pr12 o 9~ is a symplectic diffeomorphism between So - (0) and the nilpotent orbit of sltI2, C), where So - (0) is equipped with the symplectic structure corresponding to 7r3 and the nilpotent orbit is equipped with the standard coadjoint symplectic structure.
Proof. - It is clear that St, - (0) is a union of hypersymplectic leaves since it is invariant under both the G-action and the flow of X. Each hypersymplectic leaf is 4-dimensional, and therefore must be open in So - (0). Since So - (0) itself is connected, it must be a single hypersymplectic leaf. Thus, it is hyperkahler according to Theorem 4.7. By Proposition 5.5, under the identification $3 : M g x g x g, a point (a! b, c) E g x g x g is in So - (0) iff a = Xei, b = Xe2 and c = Xes for some standard orthonormal basis {ei, es, ea} of su(2), and X > 0. Hence, its image under pr12: n + ib, is clearly in the nilpotent orbit of sI(2, C). A similar argument as in Corollary 5.1 I shows that (priz o q~)(Sa - (0)) is in fact closed. Hence it has to be the whole nilpotent orbit since prl2 o 9~ is a Poisson map. Finally, it is quite obvious that prl2 o q3 is injective on So - (0). In fact, we always have 0 c = [a, b]/m. This concludes our proof. Remark. - As we have seen, (co)adjoint orbits of 51(2, C) are related to the points in MO which have bounded trajectories (in the ---co direction) under the gradient vector field X and are contained in M+. However, according to Theorem 4.7, there are other hypersymplectic leaves of MO which are contained in &i- = Ma - M+. It would be interesting to explore further the geometric structures for those leaves, and in particular the connection with the hyperkahler metrics on the cotangent bundles of hermitian symmetric spaces of noncompact type studied recently by Biquard and Gauduchon [4]. ANNALES
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ACKNOWLEDGEMENTS
The author would like to thank Jorgen Andersen, Sean Bates, Olivier Biquard, Ranee Brylinski, Hansjijrg Geiges, Victor Cuillemin, Nigel Hitchin, Peter Kronheimer, Oscar ’ >arcia-Prada, Yvette Kosmann-Schwarzbach and Alan Weinstein for useful discussions and email correspondences. Special thanks go to Ludmil Katzarkov and Tony Pantev for calling his attention to the papers [lo], [I 11. Most of this work was carried out when the author was a member of MSRI, where the research was supported by NSF Grant DMS9022140. In addition, the author wishes to thank the Center of Mathematics at Zhejiang University, China, for its hospitality while part of this work being done. REFERENCES [I] [2] [3] [4] [5] [6] [7] [8]
[9] [IO]
[I l] [12]
[13] [14] [15]
M. F. ATIYAH, Hyer-Kiihler manifolds (Collection: Comple.~ Geometry and Analysis, Springer-Verlag Lecture Notes in Mathematics, Vol. 1422, 1990, pp. l-13). M. F. ATIYAH and N. J. HITCHIN, The geometry and dynamics of magnetic monopoles, Princeton University Press, 1988. 0. BIQUARD, SW les Pquutions de Nahm et la structure de Poisson des algebres de Lie semi-simples complexes (Math. Ann. Vol. 304, 1996, pp. 253-276). 0. BIQLJARD and P. G~\L~~L~~HoN, HFperkiihler metrics on cotangent bundles of hermitian symmetric spaces, preprint. S. K. DONALDSON. Nohm’s equations and the classification of monopoles (Commun. Math. Phy.s. Vol. 96, 1984, pp. 387-407) N. J. HITCHIPZ, Hyerkdhler mantj%lds (Siminaire Bourbaki 44’ annCe, No. 748, AstPrisyue, Vol. 206, 1992, pp. 137-166). N. J. HITCHIN, A. K.~RL.HEDE, U. LINSTROM and M. ROCEK, H>jperkiihler metrics and super.ymmetr?j, Commun. Math. Phy. Vol. 108, 1987. pp. 535-589. Y. KOSMANN-SCHL+RKZBACH and F. ~UAGRI, Poisson-Nijenhuis structures (Ann. Inst. H. Poincare’ Phys. Theor. Vol. 53, 1990. pp. 35-81). A. G. KO~ALEV, Nuhm’s equations und complex adjoint orbits (Quurt. J. Muth. Osford Ser. (2) Vol. 47, 1996, pp. 41-58). P. B. KRONHEIMER. A hyper-Kiihlerian structure on coadjoint orbits of a semisimple complex group (J. of LMS. Vol. 42, 1990, pp. 193-208). P. B. KRONHEIMER, Iristantons and the geometry of the nilpotent variety (J. Difl Geom. Vol. 32, 1990, pp. 473-490). J. MARSDEN and A. WEINSTEIN, Reduction of vymplectic manifolds with symmetry (Rep. Math. Phys. Vol. 5, 1974, pp. 121-129). R. PENROSE, Nonlinear gravitons and curved twistor theory, Gen. Ralotiv. Grav. Vol. 7, 1976, pp. 31-52. M. VERGNE, Instuntonr et correspondance de Kostant-Sekiguchi, (C. R. Acad. Sci. Paris., t. 320, SCrie I, 1995. pp. 901-906). A. WEINSTEIN, The local structure of Poisson manifolds. (J. Di# Geom. Vol. 18, 1983, pp. 523-557). (Manuscript
PING XlJ
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA. E-mail: [email protected]
4e SBRIE -TOME
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received November revised November
13, 1995; 7, 1996.)