RIMS-1650 Integrable Systems: the R-Matrix Approach by Michael Semenov-Tian-Shansky December 2008 RESEARCH INSTITUTE for MATHEMA

RIMS-1650 Integrable Systems: the r-matrix Approach By Michael Semenov-Tian-Shansky December 2008 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan Integrable Systems: the r-matrix Approach Michael Semenov-Tian-Shansky Universite´ de Bourgogne UFR Sciences & Techniques 9 av- enue Alain Savary BP 47970 - 21078 Dijon Cedex FRANCE Current address: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 JAPAN 1991 Mathematics Subject Classification. Primary 37K30 Secondary 17B37 17B68 37K10 Key words and phrases. Lie groups, Lie algebras, Poisson geometry, Poisson Lie groups, Symplectic geometry, Integrable systems, classical Yang-Baxter equation, Virasoro algebra, Differential Galois Theory Abstract. These lectures cover the theory of classical r-matrices sat- isfying the classical Yang–Baxter equations and their basic applications in the theory of Integrable Systems as well as in geometry of Poisson Lie groups. Special attention is given to the factorization problems associated with classical r-matrices, the theory of dressing transformations, the geometric theory of difference zero curvature equations. We also discuss the relations between the Virasoro algebra and the Poisson Lie groups. Contents Introduction 4 Lecture 1. Preliminaries: Poisson Brackets, Poisson and Symplectic Manifolds, Symplectic Leaves, Reduction 7 1.1. Poisson Manifolds 7 1.2. Lie–Poisson Brackets 9 1.3. Poisson and Hamilton Reduction 12 1.4. Cotangent Bundle of a Lie Group. 18 Lecture 2. The r-Matrix Method and the Main Theorem 22 2.1. Introduction 22 2.2. Lie Dialgebras and Involutivity Theorem 23 2.3. Factorization Theorem 25 2.4. Factorization Theorem and Hamiltonian Reduction 26 2.5. Classical Yang-Baxter Identity and the General Theory of Classical r-Matrices 30 2.6. Lie Dialgebras and their Doubles. 32 Lecture 3. Classical r-matrices Related to Affine Lie Algebras 35 3.1. Loop Algebras and their Standard Decompositions 35 3.2. Factorization Theorem for Loop Algebras 38 3.3. Rational r-matrix and Multi-pole Lax Equations 39 Lecture 4. Poisson Lie Groups 45 4.1. Introduction 45 4.2. Main Definitions. Poisson Lie Groups and Lie Bialgebras 45 4.3. Yang–Baxter Identity and Tensor Formalism 51 4.4. Factorizable Lie Bialgebras 53 4.5. Examples of Lie Bialgebras 54 4.6. Sklyanin Brackets 61 4.7. Schouten bracket and the Gelfand–Dorfman Theorem 63 4.8. Jacobi Identity and Yang–Baxter Equation 65 4.9. Properties of Affine Poisson Structures on G 67 Lecture 5. Duality Theory for Poisson Lie Groups 69 5.1. The Drinfeld Double 69 5.2. Examples 70 5.3. Double of a Factorizable Lie Bialgebra 72 1 2 CONTENTS 5.4. Poisson Reduction. 74 5.5. Symplectic Double and Dressing Transformations 75 5.6. Factorizable Groups and their Duals. 78 5.7. Nonabelian Moment and Dressing Transformations 80 Lecture 6. Lax Equations on Lie groups 83 6.1. Introduction 83 6.2. Difference Equations with Periodic Coefficients and the Main Motivations 84 6.3. Lax Equations on Lie Groups 86 6.4. Free Dynamics 87 6.5. Twisted Double and General Reduction Theorem 88 6.6. Proof of Reduction Theorem. 91 6.7. Difference Lax Equations. 93 6.8. Poisson Lie Groups and Central Extensions 95 Lecture 7. Zero Curvature Equations and Current Algebras 99 7.1. Introduction 99 7.2. Current Algebras and Their Central Extensions 100 7.3. Double Loop Algebras 104 7.4. Monodromy Mapping 107 7.5. Local Conservation Laws 110 7.6. Dressing Transformations 116 7.7. Dressing Transformations as a Poisson Group Action 120 Lecture 8. Virasoro Algebra and Schroedinger Operators on the Circle122 8.1. Virasoro Algebra and its Coadjoint Orbits 122 8.2. Formal Diffeomorphisms of the Circle 125 8.3. Poisson Groups and Differential Galois Theory of the Schroedinger Equation on the Circle 129 Bibliography 139 CONTENTS 3 . Introduction The present notes are based on the lectures which the author was giving in 2006 and in 2008 at the Universitéde Bourgogne, Dijon, France during two special semesters on Integrable Systems. They were completed during his visit to the Kyoto Research Institute for Mathematical Science. The author expresses his sincere gratitude to RIMS for hospitality and excellent working conditions. His work was partly supported by the ANR program “GIMP” ANR-05-BLAN-0029-01 and the RFFI grant 05-01-00922. The notion of classical r-matrix introduced in late 1970’s by Sklyanin [S1], 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 [F]. It has proved to be an extremely useful tool which allows to understand geometric and algebraic aspects of the Theory of Inte- grable Systems from a single point of view and also lead to the discovery of a new kind of Poisson structures, the quadratic Poisson brackets on Poisson Lie groups. The r-matrix approach allows to take into account in a natural way several important features which are typical for various concrete examples of Integrable Systems: (i) Nonlinear integrable equations arise as compatibility conditions for an auxiliary system of linear equations. (ii) They are Hamiltonian with respect to a natural Poisson bracket. (iii) The integrals of motion are spectral invariants of the auxiliary linear problem. They are in involution with respect to the Poisson bracket associated withour system. (iv) The solution of these equations is reduced to some version of the Riemann–Hilbert problem. Somewhat earlier, a geometric approach to integrable equations of the Lax type has been proposed in the important works of Kostant [K] and Adler [A]; some key ingredients were already implicit in the earlier work of Zakharov and Shabat on dressing transformations. A link of these earlier ideas with the notion of the classical r-matrix was established in [STS1]. The r-matrix naturally unites all properties listed above and allows to deduce them from a single geometric theorem. Importantly, the notion of r-matrix relates the Hamiltonian structure of integrable equations with the Riemann problem (factorization problem) which provides the main technical tool of their explicit solution. 4 Introduction 5 Nonlinear integrable equations may be divided into several classes depending on the type of the associated linear problem: (a) Finite dimensional systems. (b) Infinite-dimensional systems with one or two spatial variables. (c) Discrete systems on one or two-dimensional lattices. In the first case the auxiliary linear problem is the eigenvalue problem for finite-dimensional matrices (possibly depending on an additional parameter). In case (b) the auxiliary linear operator is differential. Finally, in case (c) it is a difference operator. The formulation and the proof of the main theorem are particularly sim- ple for systems of the first two types. Integrable systems associated with difference operators require a separate study, since they are related to a rather complicated type of nonlinear Poisson structures. This case is particularly interesting, because it may be regarded as a semiclassical approximation in the Quantum Group theory. The associated notions of Poisson Lie groups and Poisson Hopf algebras was introduced by V.Drinfeld [D1], following the seminal work of Sklyanin and of the Leningrad school in general on the Quantum Inverse Scattering Method. In the context of difference systems on the lattice the role of classical r-matrices is in fact two-fold: On the one hand, it defines the factorization problem needed to solve nonlinear difference equations, and on the other hand, it determines a Poisson–Hopf structure which plays a major role in the study of lattice systems. These two aspects lead to two inequivalent abstract definitions. The first one, due to Drinfeld, is the notion of Lie bialgebra, associated with the Hopf properties. The second one, due to the present author, is associated with factorization aspect of classical r-matrices and is suited for the study of integrable systems. In these lectures we prefer to coin for this notion a new term “Lie dialgebra” suggested to the author by N.Reshetikhin. The mismatch between the two notions may be some- times misleading. In particular, while there are sharp classification results for Lie bialgebras, much less is known for the case of Lie dialgebras. The common ground is the important case of factorizable Lie bialgebras [RS1] which satisfy both sets of definitions. The method of classical r-matrix provides a way for the systematic con- struction of examples of integrable systems. Its starting point is the choice of an underlying Lie algebra. There are several important classes of Lie algebras which give rise to different kinds of interesting examples: 1. Finite-dimensional semisimple Lie algebras. The associated examples include open Toda lattices and other integrable systems which are integrable with the help of elementary functions (rational functions of time t or of exp t. 2. Loop algebras, or affine Lie algebras. They lead to systems which are integrable in abelian functions of t. Examples which may be obtained in this way include periodic Toda lattices and the majority 6 Introduction of classical examples from the XIX crntury textbooks on Analytical Mechanics. 3. Double loop algebras and their central extensions. This class of Lie algebras leads to integrable PDE’s with two independent variables which admit the so called zero curvature representation, such as the nonlinear Schroedinger equation, the sine-Gordon equation, etc. 4. Lie algebras of formal pseudo-differential operators. They may be used to get the Korteweg – de Vries equation and its higher analogs, although it is more practical to derive these equations from from the double loop algebras. 5. Lie algebras of vector fields on the line or on the circle and the associated loop algebras. These algebras (or, more precisely, their central extension, the Virasoro algebra) are again connected with the Korteweg – de Vries type equations.

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