CAMPINAS
Instituto de Matemática, Estatística e
Computação Científica
EDER DE MORAES CORREA
Integrable systems in coadjoint orbits and
applications
Sistemas integráveis em órbitas coadjuntas e
aplicações
Campinas
2017
Integrable systems in coadjoint orbits and applications
Sistemas integráveis em órbitas coadjuntas e aplicações
Tese apresentada ao Instituto de Matemática, Estatística e Computação Científica da Uni-versidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Doutor em Matemática.
Thesis presented to the Institute of Mathe-matics, Statistics and Scientific Computing of the University of Campinas in partial ful-fillment of the requirements for the degree of Doctor in Mathematics.
Orientador: Luiz Antonio Barrera San Martin
Coorientador: Lino Anderson da Silva Grama
Este exemplar corresponde à versão
fi-nal da Tese defendida pelo aluno Eder
de Moraes Correa e orientada pelo
Prof. Dr. Luiz Antonio Barrera San
Martin.
Campinas
2017
Ficha catalográfica
Universidade Estadual de Campinas
Biblioteca do Instituto de Matemática, Estatística e Computação Científica Ana Regina Machado - CRB 8/5467
Correa, Eder de Moraes,
C817i CorIntegrable systems in coadjoint orbits and applications / Eder de Moraes Correa. – Campinas, SP : [s.n.], 2017.
CorOrientador: Luiz Antonio Barrera San Martin. CorCoorientador: Lino Anderson da Silva Grama.
CorTese (doutorado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica.
Cor1. Lie, Teoria de. 2. Geometria simplética. 3. Sistemas hamiltonianos. 4. Calabi-Yau, Variedades de. 5. Geometria diferencial. I. San Martin, Luiz Antonio Barrera,1955-. II. Grama, Lino Anderson da Silva,1981-. III.
Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. IV. Título.
Informações para Biblioteca Digital
Título em outro idioma: Sistemas integráveis em órbitas coadjuntas e aplicações Palavras-chave em inglês: Lie theory Symplectic geometry Hamiltonian systems Calabi-Yau manifolds Differential geometry
Área de concentração: Matemática Titulação: Doutor em Matemática Banca examinadora:
Lino Anderson da Silva Grama [Coorientador] Paulo Regis Caron Ruffino
Diego Sebastian Ledesma Ivan Struchiner
Alesia Mandini
Data de defesa: 26-06-2017
Programa de Pós-Graduação: Matemática
pela banca examinadora composta pelos Profs. Drs.
Prof(a). Dr(a). LINO ANDERSON DA SILVA GRAMA
Prof(a). Dr(a). PAULO REGIS CARON RUFFINO
Prof(a). Dr(a). DIEGO SEBASTIAN LEDESMA
Prof(a). Dr(a). IVAN STRUCHINER
Prof(a). Dr(a). ALESSIA MANDINI
Acknowledgements
It is a pleasure to thank Professor Luiz Antonio Barrera San Martin, Professor Lino Anderson da Silva Grama and Professor Leo T. Butler for the support, guidance and encouragement. I want to thank my family and friends for supporting me all the time, I am deeply indebted to Tami Holanda, for all her support and care.
I would also like to thank CNPq and CAPES for providing the financial support for this work.
Resumo
Esta tese tem como objetivo estudar sistemas Hamiltonianos integráveis em órbitas coadjuntas e tópicos relacionados às suas aplicações. Este trabalho se divide essencialmente em duas partes que podem ser brevemente descritas da seguinte forma. Na primeira parte estudamos a construção de sistemas Hamiltonianos integráveis de Gelfand-Tsetlin em órbitas coadjuntas de grupos de Lie compactos clássicos. Para sistemas do tipo Gelfand-Tsetlin construímos uma formulação via matriz de Lax que nos permite recuperar as quantidades conservadas do sistema como invariantes espectrais de uma matriz de Lax apropriada. Ainda no contexto de sistemas Hamiltonianos, fornecemos uma descrição completa das funções que definem os sistemas integráveis de Gelfand-Tsetlin-Molev para dois exemplos concretos de dimensão baixa. Na segunda parte deste trabalho estudamos a construção de métricas de Calabi-Yau no fibrado canônico de variedades flag complexas. Utilizando ferramentas da teoria de representações para álgebras de Lie e a técnica de Calabi ansatz, descrevemos vários exemplos não triviais de variedades de Calabi-Yau completas não compactas. A principal motivação para o desenvolvimento deste trabalho são as relações entre variedades flag complexas, variedades tóricas, teoremas de convexidade para aplicações momento e simetria do espelho (mirror symmetry). A conexão entre as duas partes brevemente descritas aqui se dá no contexto das fibrações Lagrangianas especiais. A construção de tais exemplos de fibrações são extremamente importantes para o entendimento da dualidade entre geometria simplética e geometria complexa proposta pela simetria do espelho.
Palavras-chave: teoria de Lie, geometria Simplética, geometria Complexa, sistemas
Abstract
The purpose of this thesis is to study Hamiltonian integrable systems in coadjoint orbits and topics related to their applications. This work is essentially divided in two parts which can be briefly described as follows. In the first part we study the construction of Gelfand-Tsetlin integrable systems in coadjoint orbits of classical compact Lie groups. For Gelfand-Tsetlin integrable systems we provide a Lax matrix formulation which allows us recovering the conserved quantities of the system as spectral invariants associated to a suitable Lax matrix. Still within the context of Hamiltonian systems, we also provide a complete description of the functions which compose Gelfand-Tsetlin-Molev integrable systems for two low dimensional concrete examples. In the second part of this work we study the construction of Calabi-Yau metrics on canonical bundles of complex flag manifolds. By means of tools provided by the Lie algebra representation theory and the Calabi ansatz technique, we describe a huge family of non trivial examples of complete non compact Calabi-Yau manifolds. The main motivations for developing this work are the relationship between complex flag manifolds, toric manifolds, convexity theorems for moment maps and mirror symmetry. The connection between these two parts briefly described here is the background of special Lagrangian fibrations. The construction of examples of such kind of fibrations are extremely important to understand the duality between symplectic geometry and complex geometry proposed by mirror symmetry.
Keywords: Lie theory, Symplectic geometry, Complex geometry, Hamiltonian systems,
1 Introduction 10
2 Generalities on symplectic geometry 17
2.1 Hamiltonian G-spaces . . . 17
2.2 Hamiltonian systems . . . 24
2.3 Lax pair and Hamiltonian systems . . . 26
2.4 Collective Hamiltonians . . . 31
2.5 Thimm’s trick . . . 35
2.6 Gelfand-Tsetlin integrable systems . . . 37
2.6.1 Gelfand-Tsetlin Systems for adjoint orbits of U(N) . . . 38
2.6.2 Gelfand-Tsetlin Systems for adjoint orbits of SO(N) . . . 42
3 Lax formalism and Gelfand-Tsetlin integrable systems 48 3.1 Lax equation and collective Hamiltonians . . . 49
3.2 Thimm’s trick and spectral invariants . . . 51
3.3 Liouville’s theorem and Lax formalism for Gelfand-Tsetlin integrable systems 62 4 K¨ahler structure on coadjoint orbits 67 4.1 Generalities about coadjoint orbits . . . 67
4.2 Compact real form of simple Lie algebras . . . 68
4.3 Generalities about Lie group decompositions . . . 69
4.4 Symplectic structure on coadjoint orbits . . . 71
4.5 Complex and K¨ahler structures on coadjoint orbits . . . 73
5 Generalities on K¨ahler-Einstein manifolds 83 5.1 Holomorphic line bundles . . . 83
5.2 K¨ahler manifolds and Einstein metrics . . . 86
6 Calabi ansatz technique and complex flag manifolds 90 6.1 Calabi-Yau metrics on canonical bundles of flag manifolds . . . 91
Appendices 119 A Necessary condition for integrability of collective Hamiltonian systems 120
A.1 Basic notations and conventions . . . 120
A.2 Moment map and some of its properties . . . 121
A.3 Moment map, symplectic slice and transversality . . . 126
A.4 Collective Hamiltonians and necessary condition for integrability . . . 128
A.5 Liouville’s theorem . . . 135
A.6 Arnold’s theorem . . . 141
B Integrable systems in regular adjoint orbits of compact symplectic Lie group 143 B.1 Generalities about H and sp(N ) . . . 145
B.2 An overview about the Gelfand-Tsetlin-Molev integrable system . . . 149
B.2.1 Multiplicity free action and representation theory . . . 149
B.2.2 Generalities about Yangians and twisted Yangians . . . 158
B.2.3 Quantities in involution via classical limit procedure . . . 163
B.3 The Gelfand-Tsetlin-Molev integrable systems . . . 165
B.4 Applications in low dimensional symplectic Lie groups . . . 169
B.4.1 Integrable system in regular orbits of Sp(2) . . . 169
B.4.2 Integrable system in regular orbits of Sp(3) . . . 173
C Projective algebraic realization of coadjoint orbits 177 C.1 Representation theory of simple Lie algebras . . . 177
C.2 Analytic projective subvarieties and algebraic realization . . . 180
C.3 Projective embedding of flag manifolds . . . 184
C.4 Borel-Weil theorem and Kodaira embedding for flag manifolds . . . 191
D Holomorphic line bundles and Calabi ansatz technique 201 D.1 Local K¨ahler potential and Chern class of line bundles . . . 202
D.1.1 Preliminary generalities . . . 202
D.1.2 Local potential and representation theory . . . 210
D.1.3 Chern class for holomorphic line bundles over flag manifolds . . . 219
D.2 Calabi ansatz technique on K¨ahler-Einstein Fano manifolds . . . 243
D.2.1 Ricci-flat K¨ahler metrics on canonical bundles of K¨ahler-Einstein Fano manifolds . . . 243
Chapter
1
Introduction
In this work we study Hamiltonian systems in coadjoint orbits and Ricci-flat metrics on canon-ical bundles of complex flag manifolds. Our motivations are essentially the study of convexity properties of moment maps and the construction of completely integrable Hamiltonian sys-tems. One of the first problems related to the convexity properties of moment maps was solved by Horn [84] in 1954. The problem studied in [85] can be briefly described as fol-lows. If we consider Hn
λ as being the set of the n × n square Hermitian matrix with the same
spectrum λ = (λ1 ≥ . . . ≥ λn), how does the image of the map Φ: Hλn → Rn, such that
Φ: A → (A11, . . . , Ann), look like? The answer provided by Horn was that Φ(Hλn) = ∆(λ) is
the convex hull of the set defined by the points
λσ (1), . . . , λσ (n) ∈ Rn, with σ ∈ Sn,
where Sn denotes the permutation group of the set {1, . . . ,n}. Although Horn’s solution for
the above problem does not require the usage of symplectic geometry tools, it has the flavor of Hamiltonian Lie group action, moment maps and adjoint orbits as we will see forward. In 1974 Kostant [104] established a convex theorem for coadjoint orbits of compact Lie groups which generalizes Horn’s result. Roughly speaking, Kostant’s convexity theorem states that for a compact Lie group G with Lie algebra g, after fixed a maximal torus T ⊂ G with Lie algebra t, then the image of the projection π : g∗ → t∗ of any coadjoint orbit O ⊂ g∗ is a
convex set, namely, if we denote by µ = π |O the restriction of the projection over O, we have
µ : O → t∗,
and µ(O) = ∆T, where ∆T is the convex hull of µ(OT), here OT ⊂ O denotes the set of fixed points of O under the action of T . If we take a look at the problems which we have briefly described, we see that, from a more general stand point, the above convexity theorems gather together elements of the symplectic geometry and Lie theory.
In the general setting of Hamiltonian Lie group actions of compact Lie groups on sympletc manifolds, Atiyah [6] and independently Guillemin and Sternberg [71], almost simultaneously in 1982, provided a generalization of Kostant’s convexity theorem in the setting of compact symplectic manifolds having a Hamiltonian toric action (abelian convexity theorems). They showed that the image of a symplectic manifold under the moment map is also a convex poly-tope (more precisely, it is the convex hull of the image of the set of fixed points of the manifold
under the torus action).
In 1982 a non-abelian convexity theorem was established by Guillemin and Sternberg [71] in the context of compact K¨ahler manifolds, this theorem was generalized by Kirwan [97] two years latter. In the non-abelian setting an alternative proof for the non-abelian convexity the-orem also can be found in [149].
Beyond the context of Hamiltonian Lie group actions on symplectic manifolds, the process to assign convex bodies (polytopes) to compact symplectic manifolds leads us to the following question: What kind of geometric information are encoded in these convex bodies? In order to understand the relevance of this question we need to look more closely the especial setting of toric manifolds.
A toric manifold is defined by a 2n-dimensional compact symplectic manifold endowed with an effective Hamiltonian action of a n-dimensional compact torus. In this context the abelian convexty theorem allows us to classify toric manifolds by means of the convex bodies called Delzant polytopes [39]. Therefore, for toric manifolds we have the following correspondence
Toric manifolds ←→ Delzant polytopes
As we can see in [39] the above correspondence shows us that the symplectic and complex structures of a toric manifold are completely determined, up to isomorphism, by its associated Delzant polytope. In 1994, V. Guillemin [66] showed that not just the symplectic geometry and complex geometry of a toric manifold are determined by its polytope, but also, to a cer-tain extent, its K¨ahler geometry.
The interplay between symplectic and complex geometry observed in the setting of toric K¨ahler manifolds becomes more interesting under the Ricci-flatness assumption, namely when we consider toric Calabi-Yau manifolds. In fact, once the moment map associated to a toric Calabi-Yau manifold provides a Lagrangian torus fibration over its Delzant polytope, the envi-ronment of toric Calabi-Yau manifolds plays an important role in the study of mirror symmetry [105], more precisely on its geometric formulation called Strominger-Yau-Zaslow conjecture [155].
Besides the above facts concerned with toric Calabi-Yau manifolds, another important fea-ture of toric manifods is its relation with Hamiltonian integrable systems. Since the moment map associated to a toric manifold defines a Hamiltonian integrable system, the common back-ground of convexity theorems makes the seek for Hamiltonian integrable systems in coadjoint orbits a very interesting problem.
In 1983 Guillemin and Sternberg [76] showed how to employ Thimm’s trick [158] in order to obtain quantities in involution defined by collective Hamiltonians [77] in coadjoint orbits of compact semisimple Lie groups. Furthermore, they also showed that for coadjoint orbits of the compact unitary Lie group U(n) the set of Poisson commuting functions provided by Thimm’s trick in fact defines a completely integrable system, this kind of integrable systems were named by them as Gelfand-Tsetlin integrable systems.
A remarkable feature of the Gelfand-Tsetlin integrable systems is their connection with repre-sentation theory of Lie groups, Lie algebras and geometric quantization, see for example [72] and [74]. Since Gelfand-Tsetlin integrable systems were introduced, many other results con-cerned with Hamiltonian systems defined by collective Hamiltonians have been established, for instance, in [74] Guillemin and Sternberg studied in a general setting the constraints to get integrability for Hamiltonian systems composed by collective Hamiltonians. They showed that the necessary condition for integrability of collective Hamiltonian systems is that the space of invariant smooth functions in the manifold needs to be an abelian algebra with re-spect to the Poisson bracket induced by the symplectic form. Moreover, they concluded that this last context turns out to be the case for coadjoint orbits of U(n) and SO(n), see [135] for an exposition about Gelfand-Tsetlin systems in coadjoint orbits of the compact Lie group SO(n). For coadjoint orbits of the classical compact Lie group Sp(n) the quantities in involution ob-tained by Thimm’s trick are not enough to get integrability, thus we do not have Gelfand-Tsetlin integrable systems defined in coadjoint orbits of Sp(n). Actually, these coadjoint orbits do not satisfy the necessary condition for integrability established in [74]. In spite of this, in a recent work [79] M. Harada showed how to construct integrable systems in regular coadjoint orbits of the compact symplectic Lie group by means of a different approach which involves Thimm’s trick and classical limit procedure. These integrable systems were named by her as Gelfand-Tsetlin-Molev integrable systems. The ideas involved in Harada’s construction come from geometric quantization procedure and representation theory [121], [122] and [124]. Another interesting application of Gelfand-Tsetlin integrable systems is in mirror symmetry. In [131] T. Nishinou, Y. Nohara and K. Ueda showed how to degenerate Gelfand-Tsetlin in-tegrable systems defined in coadjoint orbits of the compact unitary Lie group into moment maps defined in toric manifolds. This procedure allowed them to associate to Gelfand-Tsetlin integrable systems potential functions, which encode information of the Lagrangian Floer ho-mology.
Coadjoint orbits ! Toric manifolds
The relation between coadjoint orbits and toric manifolds by means of toric degeneration makes clear how the understanding of the geometry of toric manifolds can be useful to under-stand the geometry of coadjoint orbits, other important results concerned with this relation are [93] and [80], where many tools of algebraic geometry also are employed to the degener-ating process.
Inspired by the background which we have described, the main goal of this work is to provide an extensive discussion about integrable systems in coadjoint orbits and related topics. This thesis can be divided in two main parts which can be described as follows. In the first part, Chapter 2 and Chapter 3, we deal with issues related to the construction of the Gelfand-Tsetlin integrable systems [76] in coadjoint orbits of classical compact Lie groups and their formu-lation in terms of Lax matrix. In the second part, Chapter 4, Chapter 5 and Chapter 6, we deal with issues related to the K¨ahler geometry of coadjoint orbits, holomorphic line bundles, K¨ahler-Einstein metrics and Ricci-flat metrics.
In Chapter 2 we review the basic background of Hamiltonian G-spaces, Hamiltonian systems and Lax formalism for Hamiltonian systems. After that we review some properties of collective Hamiltonians [77] and discuss how to apply Thimm’s trick [158] to get quantities in involu-tion in coadjoint orbits. We finish Chapter 2 by describing the construcinvolu-tion of Gelfand-Tsetlin systems in coadjoint orbits of the classical compact Lie groups U(n) and SO(n). Questions related to the necessary condition for integrability of Hamiltonian systems composed by col-lective Hamiltonians are discussed in Appendix A, following the approach of [74]. We also provide a complete description for the construction of the Gelfand-Tsetlin-Molev systems [79] in Appendix B. In this appendix we also describe the Gelfand-Tsetlin-Molev systems for the concrete cases of regular orbits of Sp(3) and Sp(2). The computations which we provide for this two examples also are new in the literature.
In Chapter 3 we gather together the ideas involved in the construction of Gelfand-Tsetlin inte-grable systems and the concept of a Lax pair. A Lax pair L, P consist of two matrices, functions on the phase space (M, ω) of the system, such that the Hamiltonian evolution equation of mo-tion associated to a Hamiltonian H ∈ C∞(M), may be written as zero curvature equation
dL dt +
L, P = 0
The notion of Lax pair is a new emergent language used in the studies of integrable systems, and one of the most important feature of this concept is its relation with the classical r-matrix [31]. The concept of classical r-matrix was introduced in the late 1970’s by Sklyanin [151] as a part of a vast research program launched by L. D. Faddeev which culminated in the discovery of the Quantum Inverse Scattering Method and of Quantum Groups [50]. Motivated by the work [79] and its relation with quantum groups, we propose a new approach for Gelfand-Tsetlin integrable systems by means of the following result
Theorem A. Let (O(Λ), ωO (Λ), G, Φ) be a Hamiltonian G-space defined by an adjoint orbit
O (Λ) = Ad(G)Λ, whereG = U(N ) or SO(N ). Then there exists a Lax pair L, P : O(Λ) → gl(r, R) satisfying
dL dt +
L, P = 0,
such that the spectral invariants of L define an integrable system in O(Λ). Furthermore, this integrable system coincides with the Gelfand-Tsetlin integrable system.
Although the integrability condition ensures the existence of a Lax pair for integrable systems, it is not clear what would be a suitable choice for such a pair, since we do not have uniqueness. Therefore, the above result provides a canonical way to assign a Lax pair to Gelfand-Tsetlin integrable systems for adjoint orbits of U(N ) and SO(N ). The ideas involved in our construc-tion are quite natural owing to the underlying matrix-nature which we have in the context of coadjoint orbits of classical compact Lie groups. Furthermore, all information about the Gelfand-Tsetlin pattern are encoded in the set of spectral invariants of the matrix L. Besides the different approach which we provide in this work for Gelfand-Tsetlin integrable systems, we hope that the content which we have developed may help to establish new connections
between Gelfand-Tsetlin integrable systems and associated topics like quantum groups, Yang-Baxter equation [12, p. 13-16] and geometric quantization.
In Chapter 4 we provide a Lie-theoretical description of the geometric structures which we have on coadjoint orbits, namely symplectic structure, complex structure and K¨ahler struc-ture. Our approach is intended to explain how these geometric structures are connected with elements of Lie theory. In Appendix C we also provide a complete description of the projective algebraic realization of coadjoint orbits from the Lie theory stand point.
In Chapter 5 we review some basic facts concerned with holomorphic line bundles and K¨ahler-Einstein metrics. The idea is to establish the basic language to be used in Chapter 6 and in Appendix D.
Chapter 6 is devoted to study Ricci-flat metrics defined on the canonical bundle of complex flag manifolds. Thus our main task is to study the special case of the Einstein equation in K¨ahler manifolds
Ric(ω) = 0
In 1979, in an important paper [26], Calabi introduced a technique to construct K¨ahler-Einstein metrics on the total space of Holomorphic vector bundles over K¨ahler-Einstein manifolds. This technique is known as the Calabi ansatz technique. By means of this method Calabi pro-vided Ricci-flat metrics for two important classes of manifolds, cotangent bundles of projective spaces and canonical bundle of K¨ahler-Einstein manifolds. The basic idea of Calabi’s technique is to use the Hermitian vector bundle structure over a K¨ahler-Einstein manifold and the Ein-stein condition on the base manifold to reduce the Ricci-flat condition, which is generally a Monge-Amp`ere equation, to an ordinary differential equation [100].
Since complex flag manifolds are K¨ahler-Einstein Fano manifolds we can apply the Calabi ansatz to obtain Ricci-flat metrics defined on the total space of their canonical bundle. By means of Calabi’s technique and the description of K¨ahler potential for invariant K¨ahler met-rics for flag manifolds developed by H. Azad [8], we prove the following result
Theorem B.Let (XP, ωXP) be a complex flag manifold associated to P = PΘ ⊂ GC, such that
dimC(XP) = n, then the canonical bundle KXP admits a complete Ricci-flat K¨ahler metric with
K¨ahler form given by
ωCY = (2πu + C)n+11 π∗ωXP − n+11 (2πu + C)−n+1n i∇ξ ∧ ∇ξ ,
where C > 0 is some positive constant and u : KXP →R≥0is given by u([д, ξ ]) = |ξ |2, ∀[д, ξ ] ∈
KXP. Furthermore, the above K¨ahler form is completely determined by the quasi-potential
φ : GC→ Rgiven explicitly by
φ(д) = 2π1 log Y
α ∈Σ\Θ
for every д ∈ GC. Therefore, (K
XP, ωCY) is a (complete) noncompact Calabi-Yau manifold with
Calabi-Yau metric ωCY completely determined by Θ ⊂ Σ.
One of the most important feature of Theorem B is that it allows us to assign to each subset Θ ⊂ Σ a complete noncompact Calabi-Yau manifold for which we have the Calabi-Yau metric completely determined by elements of the Lie theory. From Theorem B we obtain the follow-ing results which provide new explicit examples of Ricci-flat K¨ahler metrics
Proposition B1. The total space of the canonical bundle KW6 over the Wallach space W6 =
SL(3, C)/B admits a Calabi-Yau metric ωCY (locally) defined by
ωCY = (2π |ξ |2+ C)14ωW6− 14(2π |ξ |2+ C)−34i∇ξ ∧ ∇ξ ,
for some C > 0, such that
ωW6 = 1 πhi∂∂ log 1 + 2 X k=1
|zk|2 + i∂∂ log 1 + |z3|2+ |z1z3−z2|2i, and
∇ξ = dξ + 2ξh∂ log 1 + P2k=1|zk|2 + ∂ log 1 + |z3|2+ |z1z3−z2|2i.
Proposition B2. The total space of the canonical bundle KGr(2,C4) over the complex
Grass-mannian Gr(2, C4) admits a complete Calabi-Yau metric ωCY (locally) described by
ωCY = (2π |ξ |2+ C)15ωGr(2,C4)− 15(2π |ξ |2+ C)−45i∇ξ ∧ ∇ξ ,
for some C > 0, such that
ωGr(2,C4) = 2 πi∂∂ log1 + 4 X k=1 |zk|2+ det z1 z3 z2 z4 2, and ∇ξ = dξ + 4ξ ∂ log1 + Pk=14 |zk|2+ det z1 z3 z2 z4 2.
Theorem C.Consider GC = SL(n +1, C) and B ⊂ GC = SL(n +1, C) (Borel subgroup), then the total space of the canonical bundle KXBover the complex full flag manifold XB = SL(n+1, C)/B
admits a complete Calabi-Yau metric ωCY (locally) described by
ωCY = (2π |ξ |2+ C)n(n+1)+22 ωX
B − n(n+1)+22 (2π |ξ |2+ C)
−n(n+1)+2n(n+1) i∇ξ ∧ ∇ξ ,
for some C > 0, such that
• ωXB =
n
X
k=1
hδB, h∨αki
2π i∂∂ log1 + XI,I
0,k
∆I(k)(n−(z))
• ∇ξ = dξ + n X k=1 hδB, hα∨ kiξ ∂ log1 + X I,I0,k ∆I(k)(n−(z)) 2. (Vertical)
It is worth to point out that most of the well known examples of Ricci-flat metrics defined in the total space of the canonical bundle of Fano manifolds obtained by means of Calabi’s technique are toric manifolds, e.g. CPn, therefore, the above results provide a new class of
non-toric examples.
Our motivations to set up the above results are [26], [143, p. 108], [100] and [60], see also [138], [55], [112] and [23]. We hope to apply the above description on the study of special Lagrangian submanifolds in the total space of the canonical bundle over complex flag manifolds, and also analyse its relations with integrable systems defined on flag manifolds, e.g. Gelfand-Tsetlin integrable systems, these ideas are based on [60] and [95].
Introduction Part I Chapters 2 and 3 Part II Chapters 4,5 and 6 Appendices A and B Appendices A-D Appendices C and D
Figure 1.1: In Chapters 2 and 3 we study the Lax formulation of Gelfand-Tsetlin integrable systems. Further results concerned with integrable systems in coadjoint orbits also can be found in Appendices A and B; In Chapters 4, 5 and 6 we study the construction of Ricci-flat metrics on canonical bundles of complex flag manifolds by means of Calabi’s technique. Further results concerned with projective embedding of coadjoint orbits, holomorphic line bundles over complex flag manifolds and Calabi’s technique can be found in Appendices C and D.
Chapter
2
Generalities on symplectic geometry
The purpose of this chapter is to provide the basic background about Hamiltonian Lie group actions, Hamiltonian systems and Lax matrices. The idea is to review some basic results in order to apply it on the study of Gelfand-Tsetlin integrable systems in coadjoint orbits. The main geometric objects which we will consider throughout this chapter are symplectic mani-folds and Lie groups.
Our main references for the next sections are [141], [28], [75] and [4]. For the approach of Gelfand-Tsetlin integrable systems we will follow [76] and [136], further results concerned with the elements involved in the construction of Gelfand-Tsetlin integrable systems also can be found in [158], [74], [75] and [77]. Besides, we also provide an extensive discussion about the necessary condition for integrability of Hamiltonian systems defined by collective Hamil-tonians in Appendix A.
2.1
Hamiltonian G-spaces
Let M be a smooth manifold, a symplectic structure on M is a closed 2-form ω ∈ Ω2(M) which
satisfies ωn
, 0, i.e. ω is a nondegenerate form. Notice that this last condition implies that dim(M) = 2n.
Definition 2.1.1. A symplectic manifold is a pair (M, ω) composed by a smooth manifold M endowed with a symplectic structure ω ∈ Ω2(M).
There are three basic examples of symplectic manifolds which are specially important for the study of symplectic geometry and applications in physics, these examples can be briefly de-scribed as follows
Example 2.1.1. Consider the smooth manifold M = R2n. By fixing coordinates (q,p) = (q1, . . . , qn, p1, . . . , pn) on R2n we can define ω0= n X i=1 dpi∧dqi.
From this we obtain a symplectic manifold defined by (R2n, ω0). As we will see afterwards this
Example 2.1.2. Let N be a n-dimensional manifold and consider the manifold defined by M = T N∗. Since we have a projection map
π : T∗N → N ,
we can define on M = T∗N the following 1-form
θα(v) = α (π∗v) (tautological one-form),
for every α ∈ T∗N and v ∈ T
α(T∗N ), here π∗: T (T∗N ) → T N denotes the pushforward of π.
The 1-form θ described above allows us to define a 2-form ω ∈ Ω2(T∗N ) by
ω(u,v) = dθ (u,v).
A straightforward calculation shows that the pair (M = T∗N , ω = dθ ) defines a symplectic
manifold, see [141, p. 370] for a complete exposition. The symplectic manifold defined by the cotangent bundle of a smooth manifold is the mathematical model for the phase space of dynamical systems, see for example [141, p. 428].
Example 2.1.3. Let G be a Lie group with Lie algebra g. Consider the coadjoint action of G on g∗ defined by the coadjoint representation Ad∗: G → GL(g∗), where
Ad∗(д)ϕ = ϕ ◦ Ad(д−1),
for every д ∈ G and ϕ ∈ g∗. For a fixed ϕ ∈ g∗, consider the manifold defined by the coadjoint
orbit O(ϕ), namely
O (ϕ) =nAd∗(д)ϕ ∈ g∗
д∈Go, and take the 2-form ωO (ϕ) ∈Ω2(O (ϕ)) defined by1
ωO (ϕ)(ad∗(X )µ, ad∗(Y )µ) = −µ([X,Y]),
for every µ ∈ O(ϕ) and X,Y ∈ g. We can check that the pair (O(ϕ), ωO (ϕ)) defines a symplectic
manifold, see [141, p. 377] for more details. Coadjoint orbits are important for the study of dynamical systems with symmetry. In this work several results will be covered for the case when G is a compact Lie group.
An important result concerned with the local geometry of symplectic manifolds is the follow-ing theorem [141, p. 356, Theorem 8.1.5]
Theorem 2.1.2. (Darboux) Let (M, ω) be a 2n-dimensional symplectic manifold. For every x ∈ M there exists an open neighbourhood x ∈ U ⊂ M with coordinates (q,p) such that
ω|U = n X i=1 dpi ∧dqi, 1Here we have ad∗(X )µ = d dt
the coordinates (q,p,U ) are called Darboux’s coordinates.
Now let us collect some basic facts about symplectic manifolds and vector fields. Let (M, ω) be a symplectic manifold. For X ∈ Γ(TM) we have that
• X is a symplectic vector field if ιXω is a closed 1-form;
• X is a Hamiltonian vector field if ιXω is an exact 1-form.
From the above facts we have the following definition
Definition 2.1.3. Let (M, ω) be a symplectic manifold. Given a Lie group action τ : G → Diff(M) with associated infinitesimal action δτ : g → Γ(TM), we say that
• τ is a symplectic action if ιδτ (X )ω is a closed 1-form for every X ∈ g;
• τ is a Hamiltonian action if ιδτ (X )ω is an exact 1-form for every X ∈ g.
Now suppose we have a Hamiltonian action τ : G → Diff(M), keeping the above notation, in this case we have a map Φ∗: g → C∞(M), such that2
Φ∗(X ) = hΦ, Xi and dhΦ, Xi + ι
δτ (X )ω = 0,
thus if we fix a basis {Xi}for g, by definition, we can solve the equations
dhΦ, Xii+ ιδτ (Xi)ω = 0,
for every i = 1, . . . , dim(g), therefore we obtain a map Φ: (M, ω) → g∗, such that
Φ =X
i
hΦ, XiiXi∗,
here {X∗
i} denotes the dual basis of {Xi}, the map Φ is called moment map. Moment maps play an important role in the study of Hamiltonian Lie group actions, we have the following important result [141, p. 493]
Proposition 2.1.4. Let (M, ω) be a symplectic manifold and τ : G → Diff(M) be a smooth action. Then the action τ is Hamiltonian if and only if it admits a moment map Φ: (M, ω) → g∗.
In the context of Hamiltonian Lie group actions we have the following important definition Definition 2.1.5. Let (M, ω) be a symplectic manifold and τ : G → Diff(M) be a Hamiltonian action. We say that the moment map Φ associated to τ is equivariant if it satisfies
Φ(τ (д)p) = Ad∗(д)Φ(p),
for every p ∈ M, X ∈ g and д ∈ G. 2We also denote Φ∗(X ) = Φ
In what follows we fix a basic data which is the content of the following definition
Definition 2.1.6. A Hamiltonian G-space (M, ω,G, Φ) is composed by:
• A symplectic manifold (M, ω) and a connected Lie group G, with Lie algebra g.
• A left Hamiltonian Lie group action τ : G → Diff(M), with associated infinitesimal action δτ : g → Γ(T M).
• A moment map Φ: (M, ω) → g∗.
Remark 2.1.1. Let (M, ω,G, Φ) be a Hamiltonian G-space as above. In this setting we can define cΦ: G → g∗by
cΦ(д) = Φ(τ (д)p) − Ad∗(д)(Φ(p)),
for p ∈ M. Since M is connected cΦ: G → g∗ does not depend on p ∈ M, thus c is a measure for
the moment map to fail the equivariance property. Moreover, we have the following property cΦ(дh) = cΦ(д) + Ad∗(д)cΦ(h),
for every h,д ∈ G. It follows that cΦ: G → g∗defines a 1-cocycle for the representation Ad∗: G →
GL(g∗), see for example [141, p. 495-496] or [145].
From the above comments we have the following result
Theorem 2.1.7. Let (M, ω,G, Φ) be a Hamiltonian G-space. If G is a compact Lie group, then there exists an equivariant moment mapΦ: (M, ω) → ge ∗.
Proof. The proof can be found in [133, p. 150], see also [145].
Remark 2.1.2. Unless otherwise stated, for all HamiltonianG-spaces (M, ω,G, Φ) we will assume Φ: (M, ω) → g∗ as being an equivariant moment map.
We are interested in studying Hamiltonian G-spaces given by (O(λ), ωO (λ), G, Φ), where:
• (O(λ), ωO (λ)) is the Symplectic manifold defined by the coadjoint orbit of a compact and
connected Lie group G, i.e.
O (λ) =nAd∗(д)λ ∈ g∗
д∈Go,
and the symplectic structure is given by the Kirillov-Kostant-Souriau 2-form ωO (λ)(ad∗(X )ξ , ad∗(Y )ξ ) = −ξ ([X,Y]),
for every ξ ∈ O(λ), X,Y ∈ g. Notice that the manifold O(λ) is exactly the integral manifold through of λ ∈ g∗, which integrates the distribution defined by the infinitesimal
• The Hamiltonian Lie group action τ : G → Diff(O(λ)), is given by the natural restriction of the coadjoint action of G on g∗over O(λ). If we consider the equation
dhΦ, X i + ιδτ (X )ωO (λ)= 0,
a straightforward calculation shows us that the Ad∗-equivariant moment map
Φ: (O(λ), ωO (λ)) → g∗,
is given by the natural inclusion map O(λ) ,→ g∗.
In the above context we will fix the Lie group G as being one of the classical compact Lie groups SO(N ), U(N ), or Sp(N ) = U(N , H). By fixing an Ad-invariant inner product on the Lie algebra of these classical compact Lie groups, we obtain an isomorphism g g∗which allows
us to identify adjoint and coadjoint orbits. It follows that we can consider O (λ) O(Λ) =nAd(д)Λ ∈ g
д∈Go,
where Λ = diag(iλ1, . . . , iλN) ∈ g, and Ad(д)X = дXд−1, for every д ∈ G and every X ∈ g.
In general the invariant symplectic structure for adjoint orbits of SO(N ), U(N ), or Sp(N ) = U(N, H), has the following expression
ωO (Λ)(ad(X )Z, ad(Y )Z ) = −2πic Tr(Z[X,Y]),
for every Z ∈ O(Λ), and X,Y ∈ g. The constant factor c > 0, in the above expression is usually taken as c = 2N , for U(N ), c = N − 1, for SO(N ) and c = 2(N + 1), for Sp(N ), see for example [141, p. 248].
Remark 2.1.3. The above manifolds defined by coadjoint orbits are also called generalized flag manifolds, we will approach this topic in Chapter 4. For the purpose of this first part of the work the basic background which we have described so far is enough to approach Hamiltonian systems in coadjoint orbits of classical compact Lie groups.
In what follows we will provide a brief description of all adjoint orbits associated to classical compact Lie groups, moreover, we will describe some basic low dimensional examples which motivate the study of adjoint orbits in differential geometry.
Example 2.1.4. Given Λ ∈ u(N ), such that
Λ = diag(iλ1, . . . , iλN), with λ1 = · · · = λn1 k1 > λn1+1 = · · · = λn2 k2 > · · · > λnr −1 = · · · = λN kr ,
O (Λ) = U(N )/U(k1) × · · · × U(kr).
An important example of such a manifold is the complex Grassmannian of r-planes in CN,
which is given by
Gr(r, CN) = U(N )/U(N − r ) × U(r ),
for the particular case when r = 1, we have the complex projective space Gr(1, CN) = CPN −1.
Example 2.1.5. Given Λ ∈ so(2N + 1) = so(2N + 1, C) ∩ u(2N + 1), such that Λ = diag(iλ1, . . . , iλN, −iλ1, . . . , −iλN, 0),
with λ1 = · · · = λn1 k1 > λn1+1 = · · · = λn2 k2 > · · · > λnr −1+1 = · · · = λN kr = 0,
for Pki = N . The adjoint orbit O(Λ) ⊂ so(2N + 1) is identified with the homogeneous space
O (Λ) = SO(2N + 1)/U(k1) × · · · × U(kr−1) × SO(2kr+ 1),
here we are following the convention of [146, p. 118-120] for classical compact Lie groups. For the case N = 1, if we take Λ ∈ so(3), such that
Λ = diag(iλ1, −iλ1, 0),
with λ1 > 0, the manifold O(Λ) ⊂ so(3) is given by
O (Λ) = SO(3)/SO(2),
i.e. in this case we have O(Λ) = S2, in fact it is the unique sphere which admits a symplectic
structure since H2(Sn) = {0}, for n , 2.
Example 2.1.6. Given Λ ∈ sp(N ) = sp(2N , C) ∩ u(2N ), [146, p. 118-120], such that Λ = diag(iλ1, . . . , iλN, −iλ1, . . . , −iλN),
with λ1 = · · · = λn1 k1 > λn1+1 = · · · = λn2 k2 > · · · > λnr −1+1 = · · · = λN kr = 0,
for Pki = N . The adjoint orbit O(Λ) ⊂ sp(N ) is identified with the homogeneous space
O (Λ) = Sp(N )/U(k1) × · · · × U(kr−1) × Sp(kr).
Λ = diag(iλ1, . . . , iλN, −iλ1, . . . , −iλN),
with λ1 > λ2 = · · · = λN = 0, the adjoint orbit is given by
O (Λ) = Sp(N )/U(1) × Sp(N − 1) CP2N −1,
this manifold is well known by its relations with the quaternionic projective space HPN −1,
[167], [142]. This relation can be described as follows
HPN −1 Sp(N )/Sp(1) × Sp(N − 1) and S2 Sp(1) × Sp(N − 1)/U(1) × Sp(N − 1),
from these we have a S2-bundle
S2 ,→ CP2N −1 →HPN −1.
For the case N = 2, the above ideas lead us to the following diagram of sphere bundles over the fiber bundle described above
S1 S3 S7
S2 CP3 HP1= S4
the S3-bundle over S4is a very interesting sphere bundle, see for example [36], [65], [120].
Example 2.1.7. Given Λ ∈ so(2N ) = so(2N, C) ∩ u(2N ), such that Λ = diag(iλ1, . . . , iλN, −iλ1, . . . , −iλN),
with λ1 = · · · = λn1 k1 > λn1+1 = · · · = λn2 k2 > · · · > λnr −1+1 = · · · = λN kr = 0,
for Pki = N . The adjoint orbit O(Λ) ⊂ so(2N ) is identified with the homogeneous space
O (Λ) = SO(2N )/U(k1) × · · · × U(kr−1) × SO(2kr),
here again we are following the conventions of [146, p. 118-120]. A particular interesting case of the above orbit is given by O(Λ) ⊂ so(4), where
Λ = diag(iλ1, iλ2, −iλ1, iλ2),
with λ1 > λ2. In this case the manifold is given by the homogeneous space
O (Λ) = SO(4)/U(1) × U(1), here we can use the fact that Spin(4) SU(2) × SU(2), and
SO(4) Spin(4)/{±1}, in order to write
O (Λ) CP1×CP1 S2×S2,
this four manifold is the Hirzebruch surface Σ0 = CP1 × CP1, see for instance [15, p. 191],
further details about the geometry of this kind of four manifold can be found in [64], [111]. These examples which we have briefly described above compose the basic setting which we will approach in the first part of this work, further results concerned with manifolds defined by coadjoint orbits will be given in Chapter 4.
2.2
Hamiltonian systems
Once we have described the manifolds which we are interested in, now we will collect some basic results and definitions in the context of Hamiltonian systems. It is worth to point out that our exposition about Hamiltonian systems will not be extensive, thus in what follows we will just cover some basic generalities about this topic.
Let (M, ω) be a symplectic manifold, given H ∈ C∞(M), since the symplectic structure provides
an isomorphism between the tangent and cotangent bundle of M, we can take XH ∈ Γ(T M),
such that
dH + ιXHω = 0,
from this we can associate to each smooth function a dynamical system through the ordinary differential equation defined by XH, namely
dφt
dt =XH(φt).
The equation above is usually called the equation of motion associated to H ∈ C∞(M). If we
take Darboux coordinates (q,p), we have the following local expression for the Hamiltonian vector field XH ∈Γ(T M)
XH = (∂piH )∂qi − (∂qiH )∂pi,
therefore the dynamical system is (locally) defined by equations dqi dt = ∂H ∂pi and dpi dt =− ∂H ∂qi. From the above comments we have the following definition
Definition 2.2.1. A Hamiltonian system is defined by (M, ω, H ), where (M, ω) is a symplectic manifold and H ∈ C∞(M).
In the context of the above definition the smooth function H ∈ C∞(M) is called the
Hamilto-nian of the system (M, ω, H ).
A Hamiltonian system (M, ω, H ) can be alternatively described in the following way, associ-ated to any symplectic manifold (M, ω) we have a natural Poisson structure ·, · M, induced by the symplectic structure [28, p. 108]. Given F ∈ C∞(M) the evolution equation of motion
is given by
d
dt(F ◦ φt) = H, F M(φt),
where φt is the Hamiltonian flow of XH ∈Γ(T M), and H, F M = ω(XH, XF).
By means of the underlying Poisson structure of (M, ω), the equations which locally define the Hamiltonian system can be rewritten as follows
dqi
dt = H, q
i M, and dpdt =i H,pi M.
Definition 2.2.2. (Liouville integrability) Let (M, ω, H ) be a Hamiltonian system, we say that such a system is integrable if there exists H1, . . . , Hn: (M, ω) → R, such that Hi ∈C∞(M), for each i = 1, . . . ,n = 12dim(M), satisfying
• Hi, Hj M = 0, for all i, j = 1, . . . ,n,
• dH1∧. . . ∧ dHn, 0, in an open dense subset of M,
In the above definition of integrability we have H = Hi, for some i = 1, . . . ,n, we will fix
H = H1.
When the integrability condition holds the equation of motion associated to the Hamiltonian H ∈ C∞(M) can be solved by “quadrature”. Actually, Liouville’s theorem, see [141, p. 585] or
Appendix A.5, states that we can take a canonical transformation (qi, pi) 7→ (ψi, Hi), such that
(locally) we have
ω =X
i
dHi∧dψi,
and the equations of motion on this new coordinate system are given by dψi dt = H,ψ i M = ∂H∂H i = Ci, and dHi dt = H, H i M = 0,
here Ci depends on just of H = (H1, . . . , Hn), thus it is constant on time. The solution by
quadrature methods is locally given by
ψi(t ) = ψi(0) + tCi and Hi(t ) = Hi(0),
Remark 2.2.1. Under the assumption of compacity and connectedness on the level manifolds of the function
H = (H1, . . . , Hn) : (M, ω) → Rn,
we can obtain through of Arnold’s theorem, see for example [141, p. 586-595], local coordinates (I, θ ), called action and angle coordinates. By means of these coordinates we can show that the set of regular, compact and connected fibers of the above map has the structure of a locally trivial torus bundle over an open set of Rn, see [46].
2.3
Lax pair and Hamiltonian systems
In this section we will introduce some basic ideas about the concept of Lax pair and describe its relation to the study of integrability in the context of Hamiltonian systems. More details about this topic can be found in [12], [141, p. 578].
Let (M, ω) be a symplectic manifold. A Lax pair is given by a pair of matrix-valued smooth functions
L, P : (M, ω) → Mr×r(R) End(Rr).
It will be convenient to denote End(Rr) = gl(r, R), namely, we will consider the underlying
natural Lie algebra structure induced by the commutator on End(Rr).
The matrix-valued function L is called Lax matrix and the matrix-valued function P is called auxiliary matrix. We say that a Hamiltonian system (M, ω, H ) admits a Lax pair if the equation of motion associated to H ∈ C∞(M) is equivalent to the equation
dL dt +
L, P = 0.
This equation is called Lax equation. Notice that the above derivative is taken when we con-sider the composition of L with the Hamiltonian flow of XH ∈Γ(T M).
The equation described above can be easily solved. Actually, if we consider the initial value problem
dL dt =
P, L with L(0) = L0,
the solution is given by
L(t ) = д(t)L0д(t )−1,
where д : (−ϵ, ϵ) → GL(r, R) is determined by the initial value problem dд
in fact, we have the following L(t ) = д(t)L0д(t )−1 ⇐⇒ dL dt = dд dtL0д(t ) −1+ д(t)L 0dд −1 dt , now we can rewrite the second expression on the right side above as follows
dL dt = dд dtд(t ) −1L(t ) + L(t)д(t)dд−1 dt , from these we can use
д(t )д(t )−1 = 1 ⇐⇒ dд dtд(t ) −1+ д(t)dд−1 dt = 0, hence we obtain dL dt = dд dtд(t ) −1L(t ) − L(t )dд dtд(t ) −1= P, L,
where P (t) = dдdtд(t )−1, with д(0) = 1. The above computations shows us that if we have a Lax
pair for a Hamiltonian system, we can always solve the initial value problem ˙L = [P, L], by solving P (t) = dдdtд(t )−1, and the solution has the form L(t) = д(t)L
0д(t )−1.
The main point which makes issues related to the existence of a Lax pair important and inter-esting in the study of Hamiltonian systems is the following. Suppose we have a Lax pair (L, P) for Hamiltonian system (M, ω, H ), if we consider a smooth function F : gl(r, R) → R which is invariant by the adjoint action, i.e.
F (дXд−1) = F (X ), for all д ∈ GL(r, R), and X ∈ gl(r, R),
and take the composition I = F ◦ L ∈ C∞(M), we obtain a function which is constant over the
Hamiltonian flow of XH ∈Γ(T M). In fact, we have
I (t ) = F (L(t)) = F (д(t)L0д(t )−1) = F (L0) = constant,
from where we obtain
H, I
M = XH(I ) = 0.
It follows that a Lax pair is an useful tool in the study of integrability since we can get quan-tities in involution by the procedure described above.
Lax pairs are not unique in general. In fact, besides of changes in the size of the matrix valued functions we can consider the natural action of the gauge group3 G (M × Rr) on such a pair
(L, P ) defined by
3Here we have G (M × Rr) = a ∈ Aut(M × Rr)
pr1◦a = idM
and the identification G (M × Rr)
L → дLд−1, and P → дPд−1+dд
dtд−1,
where д : (M, ω) → GL(r, R) is a smooth function. From the above action foreL = дLд−1, we can write deL dt = dд dtLд−1+ д dL dtд−1+ дL dд−1 dt , since dL dt = [P, L] andeL = дLд−1, we obtain deL dt = dд dtд −1 e L + дP, Lд−1+ e Lдdд−1 dt . Notice that дP, Lд−1 = дPд−1 e L −LдPдe −1. Sincedдdtд−1+ дdд −1 dt = 0, we have deL dt = дPд−1+dд dtд −1, e L =⇒ dedt +L e L,P = 0,e whereP = дPдe −1+ dд dtд−1.
Remark 2.3.1. In the last expression ofP above we used the following notatione
e
P (x) = д(x)P (x)д(x)−1+dд
dt (x)д(x)
−1,
with dдdt(x) = д∗(XH(x)), for every x ∈ M.
Let us illustrate how the ideas described so far can be applied in concrete cases
Example 2.3.1. (Harmonic Oscillator) A basic example to illustrate the previous discussion is provided by the Harmonic Oscillator. Consider the Hamiltonian system (R2, dp ∧ dq, H ),
where the Hamiltonian function is given by
H (q,p) = 1
2(p2+ C2q2). A straightforward calculation shows us that
dH + ιXH(dp ∧ dq) = 0 ⇐⇒ XH = p∂q −C2∂p.
From this we obtain the following equations of motion dq
dt =p and dp
dt =−C2q.
We have a Lax pair L, P : (R2, dp ∧ dq) → gl(2, R) for the Hamiltonian system (R2, dp ∧ dq, H )
L = p Cq Cq −p and P = 1 2 0 −C C 0 .
In fact, a straightforward calculation shows us that dL dt + L, P = 0 ⇐⇒ (dq dt = p, dp dt = −C2q.
Here it is important to observe that
H (q,p) = 12det(L) = 14Tr(L2).
Furthermore, we have L2= 2H (q,p)1 and we can check that
Tr(L2n) = 2n+1H (q,p)n and Tr(L2n+1) = 0.
Since the algebra of invariant functions by the adjoint action is generated by X → Tr(Xk), for every X ∈ gl(2, R),
the previous comments yield a complete description of the quantities in involution provided by the Lax pair, i.e. smooth functions of the form
Ik(q,p) = Tr(Lk).
Once integrability is a trivial issue in this case, the above calculations provide a simple illustra-tion of interesting properties of the Lax matrices in the study of Hamiltonian systems, further discussions and nontrivial examples can be found in [12].
To find a Lax pair for a Hamiltonian system is not a simple task, and the existence of such a pair does not necessarily ensure integrability. An interesting fact which we will describe below is that the integrability condition ensures the existence of a Lax pair.
Actually, if we have an integrable system (M, ω, H ), we can consider the equation of motion after a canonical transformation (qi, pi) → (ψi, Fi) as follows
dψi dt = H,ψ i M = ∂H∂F i = Ci and dFi dt = H, F i M = 0,
now we define the Lie algebra generated by {Ai, Bi | i = 1, . . . ,n} with the following bracket relations
Ai, Aj = 0, Ai, Bj = 2δijBj, Bi, Bj = 0.
It follows from Ado’s theorem that this Lie algebra can be realized as a matrix Lie algebra. From these we can define the Lax pair by
L = n X i=1 FiAi+ 2FiψiBi and P = − n X i=1 ∂H ∂FiBi.
A straightforward calculation shows us that dL dt + L, P = 0 ⇐⇒ X i dFi dt Ai +h2 dFi dt ψi + 2Fi dψi dt − dH dFiiBi = 0,
from where we see the equivalence between the equation of motion associated to XH ∈Γ(T M)
and the Lax equation.
Now, suppose that for a Hamiltonian system (M, ω, H ) we have a Lax pair (L, P) : (M, ω) → gl(r, R) such that L can be diagonalized, namely
L = U ΛU−1,
where Λ = diag(λ1, . . . , λr). We can check that the functions defined by λk are conserved
quantities, i.e. H, λk M = 0, ∀k = 1, . . . ,r. Let us introduce some notations, consider {Eij}as being the canonical basis for gl(r, R). With respect to this basis we can write
L =X
ij
LijEij.
Since the components Lij of L are functions defined on (M, ω), we can evaluate the Poisson
bracket Lij, Lkl M and gather the result in the following way. We set
L1 = L ⊗ 1 =X ij Lij(Eij ⊗1) and L2 = 1 ⊗ L =X ij Lij(1 ⊗ Eij), and we define L1, L2 M by L1, L2 M = X ij,kl Lij, Lkl MEij ⊗Ekl.
From the last comments for an integrable system (M, ω, H ) we have the following result [12, p. 14]
Proposition 2.3.1. The involution property of the eigenvalues of L is equivalent to the existence of a matrix-valued function r12 on the phase space (M, ω) such that:
L
1, L2 M = r12, L1 − r21, L2,
where the matrix-valued functions r12and r21are, respectively, defined by
r12 =X
ij,kl
rij,klEij ⊗Ekl and r21 =X
ij,kl
rij,klEkl ⊗Eij,
In the context of the above proposition the Jacobi identity on the Poisson bracket provides the following constraint on the matrix r
L
1, r12, r13+ r12, r23+ r32, r13+ L2, r13 M−L3, r12 M+ cyc. perm. = 0,
here “cyc. perm.” means cyclic permutations of tensor indices 1, 2, 3, see [12, p. 15] for more details about the above equation.
The main feature of the last equation is that if r is constant the Jacobi identity is satisfied if
r, r := r12, r13+ r12, r23+ r32, r13= 0.
When r is antisymmetric, r12 = −r21, the above equation r,r = 0 is called the classical
Yang-Baxter equation (CYBE).
The CYBE first appeared explicitly in the literature on integrable Hamiltonian systems, but it is a special case of the Schouten bracket in diffential geometry, introduced in 1940’s, see [31, p. 50] for more details. It is worth to point out that besides of the approach via integrable systems, solutions of the CYBE are also interesting for the study of quantum groups and related topics, see [31].
2.4
Collective Hamiltonians
In order to study Hamiltonian systems in coadjoint orbits it will be useful to set some basic facts about the Lie-Poisson structure of the dual space g∗ of the Lie algebra associated to compact
and connected Lie groups, see [77], [164, ex. 1.1.3], or [12, p. 522-525].
Definition 2.4.1. Let M be a smooth manifold and let C∞(M) denote the algebra of real-valued smooth functions on M. Consider a given bracket operation denoted by
·, · M: C∞(M) × C∞(M) → C∞(M).
The pair (M, {·, ·}M) is called Poisson manifold if the R-vector spaceC∞(M) with the bracket {·, ·}M
defines a Lie algebra and
H, f д M = дH, f M + f H,д M, ∀f ,д, H ∈ C∞(M).
Let g be a Lie algebra of a compact and connected Lie group G 4. We have a Poisson bracket
{·, ·}g∗on the manifold g∗ defined as follows. Given F1, F2 ∈C∞(g∗) and ξ ∈ g∗, we have
F1, F2
g∗(ξ ) = −ξ, (dF1)ξ, (dF2)ξ
,
4Here it is worthwhile to point out that given a vector space V there exists a correspondence between Lie
algebra structures on V and linear Poisson structures on V∗, see [141, p. 367] for more details about this
where we use the identification T∗
ξg∗ g, and from this (dF1)ξ, (dF2)ξ ∈ g.
Will be convenient to denote by ∇F (ξ ) the element of g which satisfies the pairing (dF )ξ(η) = η, ∇F (ξ ) ,
for every F ∈ C∞(g∗), ξ ∈ g∗ and η ∈ T
ξg∗. From these we can rewrite the previous expression of {·, ·}g∗ as follows
F
1, F2 g∗(ξ ) = −ξ, ∇F1(ξ ), ∇F2(ξ ) .
With the above bracket, the pair (g, {·, ·}g∗) is a Poisson manifold.
Definition 2.4.2. Let (M, {·, ·}M) be a Poisson manifold. A smooth functionC ∈ C∞(M), is called
Casimir function if satisfies
C, F
M = 0,
for every F ∈ C∞(M).
Example 2.4.1. Consider the Poisson manifold (g∗, {·, ·}g∗), described previously. Suppose
that C ∈ C∞(g∗) is a Casimir function. Then
C, F g∗ = 0,
for every F ∈ C∞(g∗). If F = l
X ∈C∞(g∗), where
lX(ξ ) = ξ, X ,
∀ξ ∈ g∗, a straightforward calculation shows us that
∇lX(ξ ) = X ∀ξ ∈ g∗. From these we have
0 = C,lX g∗(ξ ) = −ξ, ∇C(ξ ), X = −(dC)ξ(ad
∗(X )ξ ).
Since g = Lie(G) and G is connected, it follows that the Casimir functions of (g, {·, ·}g∗) are
Ad∗-invariant functions.
Notice that the above equation is also true if we take X = ∇F (ξ ) in the right side, i.e. 0 = (dC)ξ(ad∗(∇F (ξ ))ξ ) = −ξ, ∇C(ξ ), ∇F (ξ ) = C, F g∗(ξ ),
for some F ∈ C∞(g∗), and C ∈ C∞(g∗) Ad∗-invariant function.
It follows that the Casimir functions of (g, {·, ·}g∗) are exactly the Ad∗-invariant functions. For
a more general discussion about Casimir functions with respect to the Lie-Poisson bracket see [116, p. 463].
We are interested in studying Hamiltonian systems defined by the following kind of function Definition 2.4.3. Let (M, ω,G, Φ) be a Hamiltonian G-space. Given a smooth function F ∈ C∞(g∗), a collective Hamiltonian is defined by the pullback H = Φ∗(F ) ∈ C∞(M).
Now we will provide a expression for the Hamiltonian vector field XH ∈Γ(T M), associated to a collective Hamiltonian H = Φ∗(F ) ∈ C∞(M), for more details see [77, p. 241].
We notice that by fixing a basis {Xi}for g, and denoting by {Xi∗}its dual, we have
Φ =X i ΦiX∗ i, and DΦ = X i dhΦ, XiiXi∗,
where each component function Φi = hΦ, X
ii, satisfies the equation dhΦ, Xii+ ιδτ (Xi)ω = 0.
We recall that δτ denotes the infinitesimal action associated to the Hamiltonian action τ : G → Diff(M). Therefore, given H = Φ∗(F ) ∈ C∞(M), we have
dH = dF ◦ DΦ = dF (X
i
dhΦ, XiiXi∗).
From the previous equations for the components of Φ it follows that
dF (X i dhΦ, XiiXi∗) = − X i hXi∗, (∇F ) ◦ Φiιδτ (Xi)ω, thus we obtain XH = δτ ((∇F ) ◦ Φ).
The above description yields the following proposition
Proposition 2.4.4. Let (M, ω,G, Φ) be a Hamiltonian G-space and H = Φ∗(F ) ∈ C∞(M) a collective Hamiltonian. Given p ∈ M, the trajectory of XH ∈ Γ(T M) through the point p ∈ M, is
given by
φt(p) = τ (exp(t∇F (Φ(p))))p.
Proof. The proof follows from the above expression for XH.
Remark 2.4.1. Here we notice that the above expression φt(p) = τ (exp(t∇F (Φ(p))))p denotes a
curve which satisfies
d dt
The above curve is not necessarily the flow of XΦ∗(F ), it is in fact the Hamiltonian flow of the vector
field δτ (∇F (Φ(p))) ∈ Γ(T M) through the point p ∈ M. As we will see below the curve obtained in Proposition 2.4.4 will be the Hamiltonian flow of Φ∗(F ) when F ∈ C∞(g∗)Ad∗, i.e. when F is
Ad∗-invariant.
Let us briefly describe how we can find the Hamiltonian flow associated to a collective Hamiltonian Φ∗(F ) ∈ C∞(M). At first we take a trivialization of the tangent bundleTG of G by right invariant
vector fields. From this we consider the following vector field v ∈ Γ(TG) for a fixed point p ∈ M v : G → TG, such that vд= (Rд)∗(∇F (Φ(τ (д)p))),
where Rд: G → G denotes the right translation. Now we consider the following smooth map
induced by the action τ : G → Diff(M)
Ap: G → M, such that Ap(д) = τ (д)p.
A straightforward calculation shows us the following fact δτ (X )τ (д)p = (DAp)д((Rд)∗X ).
Therefore, if we take the solution of the initial value problem dд
dt =vд(t ), with д(0) = e, we can define a curve φt(p) = τ (д(t))p which satisfies
d
dtφt(p) = (DAp)д(t )( dд
dt) = (DAp)д(t )((Rд(t ))∗(∇F (Φ(τ (д(t ))p)))).
The above expression can be rewritten as d
dtφt(p) = δτ (∇F (Φ(τ (д(t))p)))τ (д(t ))p = δτ (∇F (Φ(φt(p))))φt(p),
i.e. we have a solution for the initial value problem d
dtφt(p) = δτ (∇F (Φ(φt(p))))φt(p) = XΦ∗(F )(φt(p)), with φ0(p) = p.
Now we observe the following fact. If F ∈ C∞(g∗)Ad∗, it follows that
∇F (Ad∗(д)ξ ) = Ad(д)∇F (ξ ), for every ξ ∈ g∗and д ∈ G. Thus we obtain
vд = (Rд)∗(∇F (Φ(τ (д)p))) = (Lд)∗(∇F (Φ(p))),
dд
dt =vд(t ), with д(0) = e,
becomes exactly the equation of left invariant vector fields through the identity element. Therefore, in this last case, we have д(t) = exp(t∇F (Φ(p))) and the Hamiltonian flow associated to the col-lective Hamiltonian Φ∗(F ) is exactly the curve described in Proposition 2.4.4. Further discussions
about the Hamiltonian flow of collective Hamiltonians can be found in [77, p. 241-242].
Let us illustrate the above ideas by means of an example which is the setting which we are interested in
Example 2.4.2. Consider now the Hamiltonian G-space (O(λ), ωO (λ), G, Φ). If we take a
col-lective Hamiltonian H = Φ∗(F ) ∈ C∞(O (λ)), from the above proposition we have
XH = ad∗((∇F ) ◦ Φ).
Since Φ in this case is just the inclusion map, we have the following expression for the trajec-tory of XH through the point ξ ∈ O(λ)
φt(ξ ) = Ad∗(exp(t∇F (Φ(ξ ))))ξ .
It follows that the dynamic defined by H = Φ∗(F ) ∈ C∞(O (λ)) can be understood through the
equation which defines the left invariant vector field associated to ∇F (ξ ) ∈ g.
2.5
Thimm’s trick
Now we will describe how to obtain quantities in involution when we consider Hamiltonian systems defined by collective Hamiltonians.
As we have seen the Hamiltonian vector field associated to a collective Hamiltonian Φ∗(F ) ∈
C∞(M), is given by
XF ◦Φ= δτ ((∇F ) ◦ Φ).
If we consider other collective Hamiltonian Φ∗(I ) ∈ C∞(M), for some I ∈ C∞(g∗), we have
F ◦ Φ, I ◦ Φ
M(p) = ω(δτ (∇F (Φ(p))p, δτ (∇I (Φ(p))p)
where ∇F (Φ(p)), ∇I (Φ(p)) ∈ g, for every p ∈ M. Let us introduce a result which will be important for us
Proposition 2.5.1. Let (M, ω,G, Φ) be a Hamiltonian G-space. Given p ∈ M and ξ = Φ(p) ∈ g∗. Let ι : G · p ,→ M be the natural inclusion map, then
ι∗ω = Φ∗ω O (ξ ),
Proof. See [141, p. 497]. From the above result and the previous comments we obtain5
F ◦ Φ, I ◦ Φ M(p) = F, I g∗(Φ(p)),
for every p ∈ M and F, I ∈ C∞(g∗).
Now we will use the above results in order to describe Thimm’s trick [158]. Let (M, ω,G, Φ) be a Hamiltonian G-space as before, if we take a closed and connected subgroup K ⊂ G, we have a natural Hamiltonian action of K on (M, ω) induced by restriction, it follows that we have a Hamiltonian K-space (M, ω, K, ΦK), where the moment map
ΦK: (M, ω) → k∗ = Lie(K)∗,
is given by
ΦK = πK ◦Φ,
where πK: g∗ → k∗is the projection induced by the inclusion k ,→ g. If we take two collective Hamiltonians Φ∗(F ), Φ∗
K(I ) ∈ C∞(M), we obtain F ◦ Φ, I ◦ Φ K M = F ◦ Φ, I ◦ πK ◦Φ M, hence we have F ◦ Φ, I ◦ Φ K M = F, I ◦ πK g∗◦Φ.
From the last equality we have the following proposition
Proposition 2.5.2. Let (M, ω,G, Φ) be a Hamiltonian G-space, and let K ⊂ G be a closed and connected subgroup. If we consider the Hamiltonian system (M, ω, Φ∗
K(I )), then all collective
Hamiltonians obtained from the Casimir functions of (g∗, {·, ·}
g∗) and (k∗, {·, ·}k∗) are quantities in
involution for the system (M, ω, Φ∗ K(I )).
Proof. This result is a consequence of the ideas developed in [158], [77] see also [76, prop. 3.1]. In fact, from the above comments, for Φ∗(F ), Φ∗
K(I ) ∈ C∞(M) we have
F ◦ Φ, I ◦ Φ
K M(p) = F, I ◦ πK g∗(Φ(p)) = 0,
if F ∈ C∞(g∗) is a Casimir. Similarly, for Φ∗
K(F ), Φ∗K(I ) ∈ C∞(M), we have
F ◦ Φ
K, I ◦ ΦK M(p) = F, I k∗(ΦK(p)) = 0,
5Given a map between Poisson manifolds Φ: (M, {·, ·}M) → (N , {·, ·}N), we say that Φ is a Poisson map if
