kosmann-schwarzbach y., grammaticos b., tamizhmani k.m. (eds.) integrability of nonlinear systems (springer lnp 638, 2004)(342s)

Y Kosmann-Schwarzbach B Grammaticos K.M Tamizhmani (Eds.) Integrability of Nonlinear Systems 13 Editors Yvette Kosmann-Schwarzbach Centre de Mathématiques École Polytechnique Palaiseau 91128 Palaiseau, France K M Tamizhmani Department of Mathematics Pondicherry University Kalapet Pondicherry 605 014, India Basil Grammaticos GMPIB, Universit´ Paris VII e Tour 24-14, 5e étage, case 7021 place Jussieu 75251 Paris, France Y Kosmann-Schwarzbach, B Grammaticos, K M Tamizhmani (eds.), Integrability of Nonlinear Systems, Lect Notes Phys 638 (Springer-Verlag Berlin Heidelberg 2004), DOI 10.1007/b94605 Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de ISSN 0075-8450 ISBN 3-540-20630-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright All 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each volume As a rule, no reprints of individual contributions can be supplied Manuscript Submission The manuscript in its final and approved version must be submitted in ready to print form The corresponding electronic source files are also required for the production process, in particular the online version Technical assistance in compiling the final manuscript can be provided by the publisher‘s production editor(s), especially with regard to the publisher’s A own LTEX macro package which has been specially designed for this series LNP Homepage (springerlink.com) On the LNP homepage you will find: −The LNP online archive It contains the full texts (PDF) of all volumes published since 2000 Abstracts, table of contents and prefaces are accessible free of charge to everyone Information about the availability of printed volumes can be obtained −The subscription information The online archive is free of charge to all subscribers of the printed volumes −The editorial contacts, with respect to both scientific and technical matters −The author’s / editor’s instructions Preface This second edition of Integrability of Nonlinear Systems is both streamlined and revised The eight courses that compose this volume present a comprehensive survey of the various aspects of integrable dynamical systems Another expository article in the first edition dealt with chaos: for this reason, as well as for technical reasons, it is not reprinted here Several texts have been revised and others have been corrected or have had their bibliography brought up to date The present edition will be a valuable tool for graduate students and researchers The first edition of this book, which appeared in 1997 as Lecture Notes in Physics 495, was the development of the lectures delivered at the International School on Nonlinear Systems which was held in Pondicherry (India) in January 1996, organized by CIMPA-Centre International de Math´matiques Pures et e Appliqu´es/International Center for Pure and Applied Mathematics and Pone dicherry University In February 2003, another International School was held in Pondicherry, sponsored by CIMPA, UNESCO and the Pondicherry Government, dealing with Discrete Integrable Systems The lectures of that school are now being edited as a volume in the Lecture Notes in Physics series by B Grammaticos, Y Kosmann-Schwarzbach and Thamizharasi Tamizhmani, and will constitute a companion volume to the essays presented here We are very grateful to the scientific editors of Springer-Verlag, Prof Wolf Beiglbăck and Dr Christian Caron, who invited us to prepare a new edition o We acknowledge with thanks the renewed editorial advice of Dr Bertram E Schwarzbach, and we thank Miss Sandra Thoms for her expert help in the production of the book Paris, September 2003 The Editors Contents Introduction The Editors Analytic Methods Painlev´ Analysis e τ -functions, Bilinear and Trilinear Forms Lie-Algebraic and Group-Theoretical Methods Bihamiltonian Structures 2 3 Nonlinear Waves, Solitons, and IST M.J Ablowitz Fundamentals of Waves IST for Nonlinear Equations in 1+1 Dimensions Scattering and the Inverse Scattering Transform 11 IST for 2+1 Equations 19 Remarks on Related Problems 24 Integrability – and How to Detect It B Grammaticos, A Ramani General Introduction: Who Cares about Integrability? Historical Presentation: From Newton to Kruskal Towards a Working Definition of Integrability 3.1 Complete Integrability 3.2 Partial and Constrained Integrability Integrability and How to Detect It 4.1 Fixed and Movable Singularities 4.2 The Ablowitz-Ramani-Segur Algorithm Implementing Singularity Analysis: From Painlev´ to ARS e and Beyond Applications to Finite and Infinite Dimensional Systems 6.1 Integrable Differential Systems 6.2 Integrable Two-Dimensional Hamiltonian Systems 6.3 Infinite-Dimensional Systems Integrable Discrete Systems Do Exist! Singularity Confinement: The Discrete Painlev´ Property e 31 31 33 41 43 47 48 49 50 54 66 66 68 71 73 77 VIII Contents Applying the Confinement Method: Discrete Painlev´ Equations and Other Systems e 9.1 The Discrete Painlev´ Equations e 9.2 Multidimensional Lattices and Their Similarity Reductions 9.3 Linearizable Mappings 10 Discrete/Continuous Systems: Blending Confinement with Singularity Analysis 10.1 Integrodifferential Equations of the Benjamin-Ono Type 10.2 Multidimensional Discrete/Continuous Systems 10.3 Delay-Differential Equations 11 Conclusion 79 79 86 87 87 89 90 90 90 Introduction to the Hirota Bilinear Method J Hietarinta Why the Bilinear Form? From Nonlinear to Bilinear 2.1 Bilinearization of the KdV Equation 2.2 Another Example: The Sasa-Satsuma Equation 2.3 Comments Constructing Multi-soliton Solutions 3.1 The Vacuum, and the One-Soliton Solution 3.2 The Two-Soliton Solution 3.3 Multi-soliton Solutions Searching for Integrable Evolution Equations 4.1 KdV 4.2 mKdV and sG 4.3 nlS 95 95 95 96 97 98 99 99 100 101 101 102 102 103 Lie Bialgebras, Poisson Lie Groups, and Dressing Transformations Y Kosmann-Schwarzbach Introduction Lie Bialgebras 1.1 An Example: sl(2, C) 1.2 Lie-Algebra Cohomology 1.3 Definition of Lie Bialgebras 1.4 The Coadjoint Representation 1.5 The Dual of a Lie Bialgebra 1.6 The Double of a Lie Bialgebra Manin Triples 1.7 Examples 1.8 Bibliographical Note Classical Yang-Baxter Equation and r-Matrices 2.1 When Does δr Define a Lie-Bialgebra Structure on g? 2.2 The Classical Yang-Baxter Equation 2.3 Tensor Notation 107 107 110 110 111 113 114 114 115 116 119 119 119 122 126 Contents IX 2.4 R-Matrices and Double Lie Algebras 2.5 The Double of a Lie Bialgebra Is a Factorizable Lie Bialgebra 2.6 Bibliographical Note Poisson Manifolds The Dual of a Lie Algebra Lax Equations 3.1 Poisson Manifolds 3.2 The Dual of a Lie Algebra 3.3 The First Russian Formula 3.4 The Traces of Powers of Lax Matrices Are in Involution 3.5 Symplectic Leaves and Coadjoint Orbits 3.6 Double Lie Algebras and Lax Equations 3.7 Solution by Factorization 3.8 Bibliographical Note Poisson Lie Groups 4.1 Multiplicative Tensor Fields on Lie Groups 4.2 Poisson Lie Groups and Lie Bialgebras 4.3 The Second Russian Formula (Quadratic Brackets) 4.4 Examples 4.5 The Dual of a Poisson Lie Group 4.6 The Double of a Poisson Lie Group 4.7 Poisson Actions 4.8 Momentum Mapping 4.9 Dressing Transformations 4.10 Bibliographical Note Appendix The ‘Big Bracket’ and Its Applications Appendix The Poisson Calculus and Its Applications 128 131 132 133 133 135 136 138 139 142 145 146 146 147 149 151 151 153 155 155 158 158 162 162 165 Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: A Review and Extensions of Tests for the Painlev´ Property e M.D Kruskal, N Joshi, R Halburd Introduction Nonlinear-Regular-Singular Analysis 2.1 The Painlev´ Property e 2.2 The α-Method 2.3 The Painlev´ Test e 2.4 Necessary versus Sufficient Conditions for the Painlev´ Property e 2.5 A Direct Proof of the Painlev´ Property for ODEs e 2.6 Rigorous Results for PDEs Nonlinear-Irregular-Singular Point Analysis 3.1 The Chazy Equation 3.2 The Bureau Equation Coalescence Limints 175 175 180 182 185 187 191 192 194 195 196 199 201 X Contents Eight Lectures on Integrable Systems F Magri, P Casati, G Falqui, M Pedroni 1st Lecture: Bihamiltonian Manifolds 2nd Lecture: Marsden–Ratiu Reduction 3rd Lecture: Generalized Casimir Functions 4th Lecture: Gel’fand–Dickey Manifolds 5th Lecture: Gel’fand–Dickey Equations 6th Lecture: KP Equations 7th Lecture: Poisson–Nijenhuis Manifolds 8th Lecture: The Calogero System 209 210 215 220 225 229 235 241 247 Bilinear Formalism in Soliton Theory J Satsuma Introduction Hirota’s Method Algebraic Inentities Extensions 4.1 q-Discrete Toda Equation 4.2 Trilinear Formalism 4.3 Ultra-discrete Systems Concluding Remarks 251 251 252 255 259 259 260 263 266 Quantum and Classical Integrable Systems M.A Semenov-Tian-Shansky Introduction Generalities 2.1 Basic Theorem: Linear Case 2.2 Two Examples Quadratic Case 3.1 Abstract Case: Poisson Lie Groups and Factorizable Lie Bialgebras 3.2 Duality Theory for Poisson Lie Groups and Twisted Spectral Invariants 3.3 Sklyanin Bracket on G (z) Quantization 4.1 Linear Case 4.2 Quadratic Case Quasitriangular Hopf Algebras 269 269 271 273 283 291 293 296 304 305 305 319 Quantum and Classical Integrable Systems 4.2 319 Quadratic Case Quasitriangular Hopf Algebras For an expert in Quantum Integrability, the Gaudin model is certainly a sort of limiting special case The real thing starts with the quantization of quadratic Poisson bracket relations (3.7) This is a much more complicated problem which eventually requires the whole machinery of Quantum Groups (and has led to their discovery) The substitute for the Poisson bracket relations (3.30) is the famous relation R(u v −1 )L1 (u)L2 (v)R(u v −1 )−1 = L2 (v)L1 (u), (4.22) where R(u) is the quantum R-matrix satisfying the quantum Yang-Baxter identity R12 (u)R13 (u v)R23 (v) = R23 (v)R13 (u v)R12 (u) (4.23) To bring a quantum mechanical system into Lax form one has to arrange quantum observables into a Lax matrix L(u) (which is a rational function of u) and to find an appropriate R-matrix satisfying (4.22), (4.23) The first examples of quantum Lax operators were constructed by trial and error method; in combination with the Bethe Ansatz technique this has led to the explicit solution of important problems ( [18, 53], Faddeev [14, 15]) The algebraic concept which brings order to the subject is that of quasitriangular Hopf algebra [12] The main examples of quasitriangular Hopf algebras arise as q-deformations of universal enveloping algebras associated with Manin triples Remarkably, the general pattern represented by Theorems 1, survives q-deformation The standard way to describe quantum deformations of simple finite-dimensional or affine Lie algebras is by means of generators and relations generalizing the classical Chevalley–Serre relations [12, 30] We shall recall this definition below in Sect 4.2.2 in the finite-dimensional case, and in Sect 4.2.3 for the quantized universal enveloping algebra of the loop algebra L (sl2 ) A dual approach, due to [17], is to construct quantum universal enveloping algebras as deformations of coordinate rings on Lie groups (regarded as linear algebraic groups) Of course, the construction of a quantum deformation of the Poisson algebra F (G) was one of the first results of the quantum group theory and is, in fact, a direct generalization of the Baxter commutation relations RT T = T T R A nontrivial fact, first observed by Faddeev, Reshetikhin and Takhtajan, is that the dual Hopf algebra (usually described as a q-deformation of the universal enveloping algebra) may also be regarded as a deformation of a Poisson algebra of functions on the dual group More generally, the FRT construction is related to the quantum duality principle, which we shall now briefly discuss Let (g, g∗ ) be a factorizable Lie bialgebra, G, G∗ the corresponding dual Poisson groups, F (G) , F (G∗ ) the associated Poisson-Hopf algebras of functions on G, G∗ , and Fq (G) , Fq (G∗ ) their quantum deformations (For simplicity we choose the deformation parameter q to be the same for both algebras.) The 320 M.A Semenov-Tian-Shansky quantum duality principle asserts that these algebras are dual to each other as Hopf algebras More precisely, there exists a nondegenerate bilinear pairing Fq (G) ⊗ Fq (G∗ ) → C [[q]] which sets the algebras Fq (G) , Fq (G∗ ) into duality as Hopf algebras Hence, in particular, we have, up to an appropriate completion, Fq (G∗ ) Uq (g) Fq (G) Uq (g∗ ) In the dual way, we have also For factorizable Lie bialgebras the quantum deformations Fq (G) , Fq (G∗ ) may easily be constructed once we know the corresponding quantum R-matrices An equivalence of this formulation to the definition of Drinfeld and Jimbo is not immediate and requires the full theory of universal R-matrices Namely, starting with Drinfeld’s definition of a quasitriangular Hopf algebra we may construct ‘quantum Lax operators’ whose matrix coefficients generate the quantized algebras of functions Fq (G) , Fq (G∗ ) Using explicit formulæ for universal Rmatrices one can, in principle, express these generators in terms of the DrinfeldJimbo generators In the context of integrable models, the FRT formulation has several important advantages: it allows stating the quantum counterpart of the main commutativity theorem as well as a transparent correspondence between classical and quantum integrable systems The analogue of the FRT realization in the affine case is nontrivial, the key point being the correct treatment of the central element which corresponds to the central extension; it was described by [41] 4.2.1 Factorizable Hopf Algebras and the Faddeev-ReshetikhinTakhtajan Realization of Quantized Universal Enveloping Algebras Definition Let A be a Hopf algebra with coproduct ∆ and antipode S; let ∆ be the opposite coproduct in A; A is called quasitriangular if ∆ (x) = R∆(x)R−1 (4.24) for all x ∈ A and for some distinguished invertible element R ∈ A ⊗ A (the universal R-matrix) and, moreover, (∆ ⊗ id) R = R13 R23 , (id ⊗ ∆) R = R13 R12 (4.25) Identities (4.25) imply that R satisfies the Yang-Baxter identity R12 R13 R23 = R23 R13 R12 (4.26) Quantum and Classical Integrable Systems 321 (We use the standard tensor notation to denote different copies of the spaces concerned.) Let A0 be the dual Hopf algebra equipped with the opposite coproduct (in other words, its coproduct is dual to the opposite product in A) Put R+ = R, R− = σ(R−1 ) (here σ is the permutation map in A ⊗ A, σ(x ⊗ y) = y ⊗ x) and define the mappings R± : A0 −→ A : f −→ f ⊗ id, R± ; axioms (4.25) imply that R± are Hopf algebra homomorphisms Define the action A0 ⊗ A → A by R+ (fi ) x S(R− (fi ), where ∆0 f = (1) f ·x= (2) i (1) fi (2) ⊗ fi (4.27) i Definition A is called factorizable if A is a free A0 -module generated by ∈ A (Let us denote the corresponding linear isomorphism A0 → A by F for future reference.) There are several important examples of factorizable Hopf algebras: – Let A be an arbitrary Hopf algebra; then its Drinfeld double D (A) is factorizable [42] – Let g be a finite-dimensional semisimple Lie algebra; then A = Uq (g) is factorizable ˆ g – Let g be an affine Lie algebra; then A = Uq (ˆ) is factorizable (Observe that the last two cases are actually special cases of the first one; up to now, the double remains the principal (if not the only) source of factorizable Hopf algebras.) Let g be a simple Lie algebra Let A = Uq (g) be the corresponding quantized enveloping algebra, and let A0 be the dual of A with the opposite coproduct Let R be the universal R-matrix of A Let V, W be finite-dimensional irreducible representations of Uq (g) Let L ∈ A ⊗ A0 be the canonical element Set LV = (ρV ⊗ id)L, V R±W = (ρV ⊗ ρW )R± (4.28) We may call LV ∈ End V ⊗ A0 the universal quantum Lax operator (with auxiliary space V ) Property (4.24) immediately implies that V L2 L1 = R+ W L1 L2 W V V W V R+ W −1 (4.29) Proposition 31 The associative algebra Fq (G) generated by the matrix coefficients of LV satisfying the commutation relations (4.29) and with the matrix coproduct ∆LV = LV ⊗ LV 322 M.A Semenov-Tian-Shansky is a quantization of the Poisson-Hopf algebra F (G) with the Poisson bracket (3.6), (3.8).5 The dual algebra Fq (G∗ ) is described in the following way Let again L ∈ A ⊗ A0 be the canonical element Put ± TV = TV = ρV ⊗ R± L ∈ End V ⊗ A, + TV (id ⊗ (4.30) − S) TV (4.31) ± Proposition 32 (i) The matrix coefficients of TV satisfy the following commutation relations: ±2 ±1 ±1 ±2 TW TV = R V W TV TW −2 +1 +1 −2 TW TV = R V W TV TW R −1 RV W V W −1 , (4.32) (ii) The associative algebra Fq (G∗ ) generated by the matrix coefficients of T ±V satisfying the commutation relations (4.32) and with the matrix coproduct ∆T ±V = T ±V ⊗ T ±V is a quantization of the Poisson-Hopf algebra F (G∗ ) with Poisson bracket (3.9), (3.13) (iii) TV = (id ⊗ F )LV The matrix coefficients of TV satisfy the following commutation relations: V R+ W −1 V 1 V TW R+ W TV = TV R− W −1 V TW R− W (4.33) (iv) The associative algebra generated by the matrix coefficients of TV satisfying the commutation relations (4.32) is a quantization of the Poisson-Hopf algebra F (G∗ ) with Poisson bracket (3.13) (v) The pairing ±1 V TV , L2 = R+ W W sets the algebras Fq (G), Fq (G∗ ) into duality as Hopf algebras Formulae (4.29, 4.32, 4.33) are the exact quantum analogues of the Poisson bracket relations (3.8, 3.9, 3.13), respectively 4.2.2 Quantum Commutativity Theorem and Quantum Casimirs In complete analogy with the linear case the appropriate algebra of observables which is associated with quantum Lax equations is not the quasitriangular Hopf algebra A but rather its dual A0 ; the Hamiltonians arise from the Casimir As usual, ∆L = L ⊗ L is the condensed notation for ∆Lij = k Lik ⊗ Lkj Quantum and Classical Integrable Systems 323 elements of A We may summarize this picture in the following heuristic correspondence principle Quadratic Classical Case Quadratic Quantum case F(G) A0 = Uq (g∗ ) Fq (G) ∗ F (G ) A = Uq (g) Fq (G∗ ) Classical r-matrices Quantum R-matrices Casimir functions in F (G∗ ) Casimir operators Z ⊂ Uq (g) Symplectic leaves in G Irreducible representations of A0 Symplectic leaves in G∗ Irreducible representations of A According to this correspondence principle, in order to get a ‘quantum Lax system’ associated with a given factorizable Hopf algebra A we may proceed as follows: – Choose a representation (π, V) of A0 – Choose a Casimir element ζ ∈ Z (A) and compute its inverse image F −1 (ζ) ∈ A0 with respect to the factorization map F : A0 → A – Put Hζ = π F −1 (ζ) Moreover, we expect that the eigenvectors of the quantum Hamiltonian Hζ can be expressed as matrix coefficients of appropriate representations of A In order to describe quantum Casimirs explicitly we have to take into account the effect of twisting We shall start with their description in the finite-dimensional case The results stated below are the exact quantum counterparts of those described in Sect 3.2.2 Let g be a finite-dimensional simple Lie algebra, A = Uq (g) the quantized universal enveloping algebra of g We shall recall its standard definition in terms of the Chevalley generators [12, 30] Let P be the set of simple roots of g For αi ∈ P we set qi = q αi ,αi We denote the Cartan matrix of g by Aij −1 Definition Uq (g) is a free associative algebra with generators ki , ki , ei , fi , i ∈ P , and relations −1 −1 ki · ki = ki ki = 1, [ai , ej ] = [ei , fj ] = [ki , kj ] = 0, A qi ij ej , −2 k − ki δij i −2 qi − qi −Aij [ai , fj ] = qi ej , ; moreover, we assume the following relations (q-deformed Serre relations): 1−Aij (−1)n n=0 1−Aij (−1)n n=0 (Here m n q − Aij n − Aij n 1−Aij −n qi ej en = 0, i 1−Aij −n qi fj fin = ei fi are the q-binomial coefficients.) 324 M.A Semenov-Tian-Shansky Theorem [12] Uq (g) is a quasitriangular Hopf algebra As usual, we denote by R ∈ Uq (g) ⊗ Uq (g) its universal R-matrix and put RV W = (ρV ⊗ ρW ) R We recall the following well known corollary of definition Let A be the ‘Cartan subgroup’ generated by the elements ki , i ∈ P, C [A] its group algebra We denote by Uq (n± ), Uq (b± ) ⊂ Uq (g) the subalgebras generated by ei , fi (respectively, by ki , ei and by ki , fi ) Proposition 33 (i) C [A], Uq (n± ), Uq (b± ) are Hopf subalgebras in Uq (g) (ii) Uq (n± ) is a two-sided Hopf ideal in Uq (b± ) and Uq (b± )/Uq (n± ) C [A] Let π± : Uq (b± ) → C [A] be the canonical projection Consider the embedding i : C [A] → C [A] ⊗ C [A] : h −→ (id ⊗ S) ∆h (Here ∆ is the coproduct in the commutative and cocommutative Hopf algebra C [A] and S is its antipode.) Proposition 34 The dual of Uq (g) may be identified with the subalgebra A0 = {x ∈ Uq (b+ ) ⊗ Uq (b− ); π+ ⊗ π− (x) ∈ i (C [A])} Lemma 10 For any h ∈ A, h ⊗ h R = R h ⊗ h The algebra A admits a (trivial) family of deformations Ah = Uq (g)h , h ∈ A, defined by the following prescription: ei −→ hei , fi −→ hfi , ki −→ hki (4.34) It is instructive to look at the commutation relations of Ah in the FRT rea± lization Let V, W ∈ Rep A; we define the ‘generating matrices’ TV ∈ A ⊗ ± End V, TW ∈ A ⊗ End W , as in (4.31) ± ± Proposition 35 (i) The deformation (4.34) maps TV into ρV (h∓1 )TV ; the commutation relations in the deformed algebra Ah amount to ±2 ±1 ±1 ±2 RV W TW TV = R V W TV TV −2 +1 V W +1 −2 V W −1 , TW T V = Rh T V T W R −1 , (4.35) V where Rh W = ρV ⊗ ρW (Rh ) and Rh = h ⊗ h−1 Rh−1 ⊗ h + − −1 (ii) Put Th = ρV h2 TV TV ; the ‘operator-valued matrix’ Th satisfies the following commutation relations V R+ W −1 V 1 V TW R+ W TV = TV R− W −1 V TW R− W (4.36) ∗ Let LV ∈ A ⊗ End V be the ‘universal Lax operator’ introduced in (4.28) Fix h ∈ A and put h lV = tr V ρV (h2 )LV (4.37) h Elements lV ∈ A0 are usually called (twisted) transfer matrices The following theorem is one of the key results of QISM (as applied to the finite-dimensional case) Quantum and Classical Integrable Systems 325 Theorem 10 For any representations V, W and for any h ∈ A, h h h h l V lW = lW lV (4.38) The pairwise commutativity of transfer matrices is a direct corollary of the commutation relations (4.29); twisting by elements h ∈ A is compatible with these relations, since (h ⊗ h) R = R (h ⊗ h) We want to establish a relation between the transfer matrices and the Casimirs of Uq (g)h (where h ∈ A may be different from h) To account for twisting we must slightly modify the factorization map Let A × A → A be the natural action of the ‘Cartan subgroup’ A on A by right translations, and let A × A0 → A0 be the contragredient action The algebra A = Uq (g) is factorizable; let F : A0 → A be the isomorphism induced by the action (4.27); for h ∈ A put F h = F ◦ h Let us compute the image of the universal Lax operator LV ∈ End V ⊗ A0 under the ‘factorization mapping’ id ⊗ F h It is easy to see that h± 1± id ⊗ F h (LV ) := TV = id ⊗ h±1 TV Put h+ h− th = trV TV TV V −1 = trV ρV (h)2 TV ; (4.39) h clearly, we have th = F (lV ) V Theorem 11 (i) [17] Suppose that h = q −ρ , where 2ρ ∈ h is the sum of the positive roots of g Then all coefficients of th are central in A = Uq (g) V (ii) The center of A is generated by th (with V ranging over all irreducible V s finite-dimensional representations of Uq (g)).(iii) For any s ∈ A we have lV = sh−1 h F (tV ) Theorem 11 is an exact analogue of Theorem for finite-dimensional factorizable quantum groups Its use is twofold: first, it provides us with commuting quantum Hamiltonians; second, the eigenfunctions of these Hamiltonians may be constructed as appropriate matrix coefficients (‘correlation functions’) of irreducible representations of Uq (g) Again, to apply the theorem to more interesting and realistic examples we must generalize it to the affine case 4.2.3 Quantized Affine Lie Algebras Two important classes of infinitedimensional quasitriangular Hopf algebras are quantum affine Lie algebras and the full Yangians (that is, the doubles of the Yangians defined in [12]) For concreteness I shall consider only the first class To avoid technical difficulties we shall consider the simplest nontrivial case, that of the affine sl2 The standard definition of Uq (sl2 ) is by means of generators and relations: 326 M.A Semenov-Tian-Shansky Definition Uq (sl2 ) is a free associative algebra over C q, q −1 with genera−1 tors ei , fi , Ki , Ki , i = 0, 1, which satisfy the following relations: Ki Kj = Kj Ki , Ki ej = q Aij ej Ki , Ki fj = q −Aij fj Ki , −1 [ei , fj ] = δij q − q −1 Ki − Ki , (4.40) where Aij is the Cartan matrix of the affine sl2 , A00 = A11 = −A01 = −A10 = In addition to (4.40) the following q-Serre relations are imposed: e3 ej − + q + q −1 i fi3 fj − 1+q+q −1 e2 ej ei − ei ej e2 − ej e3 = 0, i i i fi2 fj fi − fi fj fi2 − fj fi3 (4.41) = The Hopf structure in Uq (sl2 ) is defined by ∆Ki = Ki ⊗ Ki , ∆ei = ei ⊗ + Ki ⊗ ei , ∆fi = fi ⊗ + Ki ⊗ ei (4.42) The element K0 K1 is central in Uq (sl2 ); the quotient of Uq (sl2 ) over the relation K0 K1 = q k , k ∈ Z, is called the level k quantized universal enveloping algebra Uq (sl2 )k An alternative realization of Uq (sl2 )k is based on the explicit use of the quantum R-matrix; in agreement with the quantum duality principle, this realization may be regarded as an explicit quantization of the Poisson algebra of functions Fq (G∗ ), where G∗ is the Poisson dual of the loop group L(SL2 ) The relevant Poisson structure on G∗ has to be twisted so as to take into account the central charge k; moreover, in complete analogy with the non-deformed case, the center of Uq (sl2 )k is nontrivial only for the critical value of the central charge kcrtit = −2 (The critical value k = −2 is a specialization, for g = sl2 , of the general formula k = −h∨ ; thus the value of kcrit is not affected by q-deformation.) Introduce the R-matrix   0 −1 z (q−q )   0  1−z −1 (4.43) R(z) =  q−zq−1 q−zq−1  1−z   q−q −1 q−zq q−zq −1 0 0 ± Let Uq (gl2 )k be an associative algebra with generators lij [n] , i, j = 1, 2, n ∈ ±Z+ In order to describe the commutation relations in Uq (gl2 )k we introduce Quantum and Classical Integrable Systems 327 ± the generating series L± (z) = lij (z) , ± lij (z) = ∞ ± lij [±n] z ±n ; (4.44) n=0 ± + − we assume, moreover, that lij [0] are upper (lower) triangular: lij [0] = lji [0] = for i < j The defining relations in Uq (gl2 )k are + − − + lii [0] lii [0] = lii [0] lii [0] = 1, i = 1, 2, z z L± (z) L± (w) = L± (w) L± (z) R , R 2 w w z −k z k q q R L+ (z) L− (w) = L− (w) L+ (z) R 2 w w (4.45) (we again use the standard tensor notation) Relations (4.45) are understood z as relations between formal power series in w Observe the obvious parallels between (4.45) and (4.35): the shift in the argument of R reflects the effects of the central extension; in the finite-dimensional case this extension is of course trivial and the shift may be eliminated by a change of variables The commutation relations (4.45) are the exact quantum analogue of the Poisson bracket relations (3.26) with p = q k Lemma 11 The coefficients of the formal power series ± q det(L± (z)) := l11 zq −1 ± l12 ± ± ± l22 (z) − l21 (z) l11 (z) (z) are central in Uq (gl2 )k Theorem 12 The quotient of Uq (gl2 )k over the relations q det(L± (z)) = is isomorphic to Uq (sl2 )k The accurate proof of this theorem (and of its generalization to arbitrary quantized affine Lie algebras) is far from trivial (see [9]); it relies on still another realization of quantized affine Lie algebras, the so-called Drinfeld’s new realization [13]; the important point for us is that the realization described in Theorem 12 is adapted to the explicit description of the Casimir elements Let us set L (z) = L+ q −k/2 z L− zq k/2 −1 (4.46) (this is the so-called full quantum current) Theorem 13 [41] (i) The quantum current (4.46) satisfies the commutation relations R q −k/2 z z L1 (z)R−1 ( ) L2 (w) = w w L2 (w)R( (4.47) q k/2 z z )L1 (z)R w w −1 328 M.A Semenov-Tian-Shansky (ii) Suppose that k = −2; then the coefficients of the formal series t(z) = tr q 2ρ L (z) (4.48) are central elements in the algebra Uq (sl2 )k As usual, ρ stands for half the sum of the positive roots In the present setting q The critical value k = −2 is a special case of we have simply q 2ρ = q −1 the general formula k = −h∨ (the dual Coxeter number); thus the critical value remains the same as in the non-deformed case The definition of the quantum current and formulæ (4.47,4.48) are in fact quite general and apply to arbitrary quantized loop algebras There are many reasons to expect that the center of the quotient algebra A−h∨ = A/(c + h∨ ) for the critical value of the central charge is generated by (4.48) (with V ranging over all irreducible finite-dimensional representations of Uq (ˆ)) (cf [8,23]) There g are two main subtle points: Our construction involved the 2-dimensional ‘evaluation representation’ Uq (sl2 ) → End C((z)) In the general case we must describe finite-dimensional representations V (z) of Uq (g) with spectral parameter; this requires additional technical efforts In the sl2 case it was possible to use the generating series (4.44) as an exact alternative to the Drinfeld-Jimbo definition; in the general case the analogues of relations (4.45) are still valid but there may be others; anyway, the generating series L±(z) may be defined as the image of the canonical V element: L±(z) = id ⊗ ρV (z) ◦ (R± ⊗ id)L; V the definition of the quantum current and Theorem 13 are still valid We may now use Theorem 13 to relate the elements of the center to quantum Hamiltonians Our main pattern remains the same For concreteness let us again assume that A = Uq (sl2 ) Let us take its dual A0 as the algebra of observables Let L ∈ A ⊗ A0 be the canonical element For any finite-dimensional sl2 -module V there is a natural representation of A in V ((z)) (evaluation representation) Let W be another finite-dimensional sl2 -module Set LV (z) = (ρV (z) ⊗id)L, LW (z) = (ρW (z) ⊗ id)L, RV W (z) = (ρV (z) ⊗ ρW (z) )R We may call LV (z) ∈ End V ((z)) ⊗ A0 the universal quantum Lax operator (with auxiliary space V ) Fix a finitedimensional representation π of A0 ; one can show that (id ⊗ π) LV (z) is rational in z (Moreover, [55] showed how to use this dependence on z to classify finitedimensional representations of A0 ) Property (4.24) immediately implies that LW (w)LV (v) = RV W (vw−1 )LV (v)LW (w) RV W (vw−1 ) 1 −1 (4.49) Quantum and Classical Integrable Systems 329 Moreover, RV W (z) satisfies the Yang-Baxter identity (4.23) Let A be a Cartan subgroup of G Fix h ∈ A and put h lV (v) = tr V ρV (h2 )LV (v), h h h h h then lV (v)lW (w) = lW (w)lV (v) Elements lV (v) are called (twisted) transfer matrices It is convenient to extend A by adjoining to it group-like elements which ˆ ˆ ˆ correspond to the extended Cartan subalgebra a = a ⊕ Cd in g Let A = A × C∗ be the corresponding ‘extended Cartan subgroup’ in A generated by elements ˆ h = q λ , λ ∈ h, t = q kd Let A × A → A be its natural action on A by right ˆ × A0 → A0 the contragredient action The algebra A = Uq (ˆ) is g translations, A factorizable; let F : A0 → A be the isomorphism induced by the action (4.27); ˆ for h ∈ H put F h = F ◦ h, Lh (z)± = V Lh V (z) = id ⊗ R± ◦ h LV (z) ∈ End V Lh V + (z) (id ⊗ S) Lh V z ±1 − ˆ (z) , h ∈ H ⊗ A, (4.50) It is easy to see that Lh (z)± = id ⊗ h±1 L1 (z)± V V and hence h TV (z) = (id ⊗ h) TV (z) (id ⊗ h) ; moreover, if h = s−1 t, where s ∈ H and t = q kd , we have also h t TV (z) = (ρV (s) ⊗ id) TV (z) (ρV (s) ⊗ id) h t s Put th = tr V TV (z) = tr V ρV (s)2 TV (z); clearly, we have th (z) = F t (lV (z)) Let V V ∨ 2ρ be the sum of the positive roots of g = sl2 , h = its dual Coxeter number (our notation again hints at the general case) ∨ Theorem 14 (i) [41] Suppose that h = q −ρ q −h d Then all coefficients of th (z) V −1 s are central in U (ii) For any s ∈ H we have lV (z) = F sh (th (z)) V Thus the duality between Hamiltonians and Casimir operators holds for quantum affine algebras as well This allows us to anticipate connections between the generalized Bethe Ansatz, the representation theory of quantum affine algebras at the critical level and the q-KZ equation [24, 54] The results bearing on these connections are already abundant [8, 23, 56], although they are still not in their final form As we have already observed in the classical context, the Hopf structure on A0 is perfectly suited to the study of lattice systems Let N ∆(N ) : A0 −→ A0 330 M.A Semenov-Tian-Shansky be the iterated coproduct map; algebra ⊗N A0 may be interpreted as the algebra of observables associated with a ‘multiparticle system’ Set ˆV th (z) = ∆(N ) th (z); V ˆV the Laurent coefficients of th (z) provide a commuting family of Hamiltonians in ⊗N A0 Let in : A0 → ⊗N A0 be the natural embedding, in : x → 1⊗ ⊗x⊗ ⊗1; set Ln = (id ⊗ in ) LV Then V x ˆV th (z) = tr V Ln V ρV (h) (4.51) n Formula (4.51) has a natural interpretation in terms of lattice systems: Ln may V be regarded as ‘local’ Lax operators attached to the points of a periodic lattice Γ = Z/N Z; commuting Hamiltonians for the big system arise from the x monodromy matrix MV = Ln associated with the lattice Finally, the twist V h ∈ H defines a quasiperiodic boundary condition on the lattice The study of the lattice system again breaks into two parts: (a) Find the joint spectrum of ˆV (z) (b) Reconstruct the Heisenberg operators corresponding to ‘local’ obl servables and compute their correlation functions This is the Quantum Inverse Problem (profound results on it are due to [54]) Acknowledgements The present lectures were prepared for the CIMPA Winter School on Nonlinear Systems which was held in Pondicherry, India, in January 1996 I am deeply grateful to the Organizing Committee of the School and to the 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Basil Grammaticos GMPIB, Universit´ Paris VII e Tour 24-14, 5e étage, case 7021 place Jussieu 75251 Paris, France Y Kosmann-Schwarzbach, B Grammaticos, K M Tamizhmani (eds.), Integrability of Nonlinear. .. theory of nonlinear systems, and in particular of integrable systems, is related to several very active fields of theoretical physics For instance, the role played in the theory of integrable systems. .. critical points of the solutions are poles, the equation passes the test It contributes to the determination of the integrability or nonintegrability of nonlinear equations, defined in terms of their