THE HIGHER ORDER q-DOLAN-GRADY RELATIONS AND QUANTUM INTEGRABLE SYSTEMS 6 1 0 2 n THÈSE a LMPT, Université François-Rabelais, Tours, France J 1 3 Thi Thao Vu ] h p - h THÈSE dirigée par : t a M. P. BASEILHAC C.R. CNRS, Université François-Rabelais de Tours m [ RAPPORTEURS : 3 v M. H. KONNO Professeur, Université des Sciences et Technologies Marines de Tokyo 3 M. P. TERWILLIGER Professeur, Université du Wisconsin-Madison 5 2 6 JURY : 0 . M. P. BASEILHAC C.R. CNRS, Université François-Rabelais de Tours 1 0 M. C. LECOUVEY Professeur, Université François-Rabelais de Tours 6 M. V. ROUBTSOV Professeur, Université d’Angers 1 M. H. SALEUR Chercheur CEA, Institut de Physique Théorique, CEA Saclay : v M. P. TERWILLIGER Professeur, Université du Wisconsin-Madison i X r a 1 2 Abstract In this thesis, the connection between recently introduced algebraic structures (tridiag- onal algebra, q-Onsager algebra, generalized q−Onsager algebras), related representation theory (tridiagonal pair, Leonard pair, orthogonal polynomials), some properties of these algebras and the analysis of related quantum integrable models on the lattice (the XXZ open spin chain at roots of unity) is first reviewed. Then, the main results of the thesis are described: (i) for the class of q−Onsager algebras associated with s(cid:99)l2 and ADE type simply-laced affine Lie algebras, higher order analogs of Lusztig’s relations are conjectured and proven in various cases, (ii) for the open XXZ spin chain at roots of unity, new ele- ments (that are divided polynomials of q−Onsager generators) are introduced and some of their properties are studied. These two elements together with the two basic elements of the q−Onsager algebra generate a new algebra, which can be understood as an analog of Lusztig’s quantum group for the q−Onsager algebra. Some perspectives are presented. Keywords : Tridiagonal algebra; Tridiagonal pair; q-Onsager algebra; Generalized q- Onsager algebra; XXZ open spin chain; root of unity. Notations Throughout this thesis, we use the following notations: 1. Let A,B denote generators, then [A,B] = AB−BA. [A,B] = qAB−q−1BA. q 2. Let n,m be integers, then (cid:20) n (cid:21) [n]q! (cid:89)n qn−q−n = , [n] ! = [l] , [n] = , [0] = 1 . m [m] ![n−m] ! q q q q−q−1 q q q q l=1 3. {x} denotes the integer part of x. Let j,m,n be integers, write j = m,n for j = m,m+1,...,n−1,n. 4. The Pauli matrices σ ,σ ,σ ,σ : 1 2 z ± (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 0 1 0 −i 1 0 σ = , σ = , σ = , 1 1 0 2 i 0 z 0 −1 (cid:18) (cid:19) (cid:18) (cid:19) 0 1 0 0 σ = , σ = . + 0 0 − 1 0 . 3 4 Contents Introduction 1 1 Mathematics: background 5 1.1 Tridiagonal algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 The q-Onsager algebra from different points of view . . . . . . . . . . . . . . 6 1.2.1 The q-Onsager algebra and Uq(s(cid:99)l2) . . . . . . . . . . . . . . . . . . . 8 1.2.2 The q-Onsager algebra and the U (sl )-loop algebra . . . . . . . . . 8 q 2 1.2.3 The q-Onsager algebra, the reflection equation and the algebra A . 9 q 1.3 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.1 Leonard pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.2 Tridiagonal pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.3 Connection with the theory of orthogonal polynomials . . . . . . . . 34 2 Mathematical Physics: background 41 2.1 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 Sklyanin’s formalism and the q−Onsager algebra . . . . . . . . . . . . . . . 44 2.3 Commuting quantities and the q−Dolan-Grady hierarchy. . . . . . . . . . . 48 2.4 The open XXZ spin chain and the q-Onsager algebra . . . . . . . . . . . . 49 3 Main Results 53 3.1 Higher order relations for the q-Onsager algebra . . . . . . . . . . . . . . . . 53 3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.1.2 Conjecture about the higher order relations of the q-Onsager algebra 55 3.1.3 Higher order relations and tridiagonal pairs . . . . . . . . . . . . . . 55 3.1.4 Recursion for generating the coefficients of the higher order q-Dolan- Grady relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.1.5 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Higher order relations for the generalized q−Onsager algebra . . . . . . . . 68 3.2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.2 Conjecture about the higher order relations of the generalized q- Onsager algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.3 Proof of the higher order relations for r ≤ 5 . . . . . . . . . . . . . . 70 3.2.4 Recursion relations of the coefficients of the higher order relations in generic case r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2.5 A two-variable polynomial generating function . . . . . . . . . . . . 74 3.3 The XXZ open spin chain at roots of unity . . . . . . . . . . . . . . . . . . 76 3.3.1 A background: the XXZ periodic spin chain at roots of unity . . . . 76 3.3.2 The case of the open XXZ spin chain . . . . . . . . . . . . . . . . . 78 3.3.3 Observations about the symmetries of the Hamiltonian . . . . . . . . 87 5 6 CONTENTS 4 Perspectives 89 4.1 A family of new integrable hierarchies . . . . . . . . . . . . . . . . . . . . . 89 4.2 Cyclic tridiagonal pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3 An analog of Lusztig quantum group for the q−Onsager algebra at roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5 Appendices 93 (m) (r,p) (r,p) 5.1 APPENDIX A: Coefficients η , M , N . . . . . . . . . . . . . . . . 93 k,j j j 5.2 APPENDIX B: A2r+2A∗r+1A, A2r+1A∗r+1A2, A2r+1A∗r+1 . . . . . . . . . . 95 (m) (r,p) 5.3 APPENDIX C: Coefficients η , M . . . . . . . . . . . . . . . . . . . . 97 k,j j 5.4 APPENDIX D: Algorithms for the q-Onsager algebra . . . . . . . . . . . . . 98 5.5 APPENDIX E: Algorithms for the generalized q-Onsager algebra . . . . . . 110 5.6 APPENDIX F: Powers of q−Onsager generators . . . . . . . . . . . . . . . 119 Introduction In the literature, the Onsager algebra and the Dolan-Grady relations first appeared in study of integrable systems (the XY, the Ising models,...) [O44, DG82, Dav91, Dav90]. Later on, they appeared in the context of mathematics in relation with certain subalgebras of s(cid:99)l2 [DR99]. From 2003, a q−deformed analog of the Dolan-Grady relations appeared in the context of mathematics as a special case of tridiagonal algebras [Terwilliger et al.]. Almost simultaneously, the q−Dolan-Grady relations appeared in the context of quantum integrablesystemsonthelatticeandcontinuum: theq−Onsageralgebrawasdefined(which q−Dolan-Gradyrelationsarethedefiningrelations)inrelationwiththequantumreflection equation,asanalgebrageneratingalargeclassofquantumintegrablesystemsonthelattice or continuum [Baseilhac et al.]. There is now a rather vast literature on the subject of tridiagonal algebras [Ter93III], the representation theory of tridiagonal pairs [ITT99] and Leonard pairs [Ter03], the q−Onsager algebra and its generalizations [Ter01, Bas0404], the connections with coideal subalgebrasofUq(s(cid:99)l2)[BB12]andwithanewinfinitedimensionalalgebracalledAq [BS09]. In the context of mathematical physics, these structures and the explicit analysis of some of their properties lead to several new exact non-perturbative results for the open XXZ spin chain [BK07], for the half-infinite XXZ spin chain [BB12, BK14], for the open affine Toda field theories [BB09, BF11]. From a general point of view, a new approach called ‘q−Onsagerapproach’hasemergedasanalternativetoexistingonesinquantumintegrable systems (Bethe ansatz [Bet31, FST80], separation of variable [Sk92], q−vertex operators [JM95, JKKKMW94]). Since 2007, this approach has been currently developed in different directions. Inthisthesis,weexploreonedirectionwhichoverlapsbetweenmathematicsandphysics. Namely, we investigate in detail some properties of the q−Onsager algebras (in particu- lar the existence and explicit construction of higher order relations between monomials of the fundamental generators) which will find application in the analysis of the open XXZ spin chain at roots of unity (characterization of the symmetry of the Hamiltonian at roots of unity). At the moment, the results of this thesis have been published in two articles [BV13, BV1312]. There is another article in preparation [BGSV15]. The manuscript of the thesis is divided into three main Chapters. Chapter 1. We summarize without proofs the relevant material on tridiagonal alge- bras, the q-Onsager algebra and some aspects of its representation theory: Leonard pairs, tridiagonal pairs and orthogonal polynomials. In the first part, tridiagonal algebras are defined by generators and relations. Several special cases corresponding to particular parameter sequences of the tridiagonal relations are mentioned such as the q-Serre relations or the Dolan-Grady relations. 1 2 CONTENTS The q-Onsager algebra is introduced in a second part. Its defining relations are the q−Dolan-Grady relations: these are ρ -deformed analogues of the q-Serre relations, and i correspond to a special parameter sequence of the tridiagonal algebra. In connection with the quantum affine algebra Uq(s(cid:99)l2) and the Uq(sl2)-loop algebra, we thus recall homo- morphisms from the q-Onsager algebra to these algebras. Finally, we recall the reflection equation algebra and indicate its relation with the q-Onsager algebra: the reflection equa- tion algebra is defined by generators which are entries of the solution of the “RKRK” equations for the Uq(s(cid:99)l2) R-matrix. The isomorphisms between the reflection equation algebra, the current algebra Oq(s(cid:99)l2), and the infinite dimensional algebra Aq generated by {W ,W ,G ,G˜ |k ∈ Z }arerecalled. Wealsorecalltheconstructionofacoaction −k k+1 k+1 k+1 + map for the q-Onsager algebra and the defining relations of the K-matrix as an intertwiner of irreducible finite dimensional representations of the q-Onsager algebra. Thus, a quotient of the q-Onsager algebra is isomorphic to a quotient of the reflection equation algebra. The last part recalls some aspects of the representation theory of tridiagonal algebras (including the case of the q−Onsager algebra), in particular the results of Terwilliger et al. about irreducible finite dimensional representations and the concept of tridiagonal pairs. For convenience, Leonard pairs (a subclass of the tridiagonal pairs) are introduced first. We recall the notion of Leonard pair, Leonard system as well as modification of a given Leonard pair in several ways. The relation between the Leonard pair and the tridiagonal algebra is also clarified. Namely, there exists a scalar sequence such that the Leonard pair satisfies the corresponding Askey-Wilson relations. Inversely, a pair of linear transforma- tions satisfying the Askey-Wilson relations allows to define a Leonard pair under certain conditions. One more important result is the classification of Leonard pairs, it is asserted that a sequence of scalars satisfying conditions (i)−(v) in Theorem (1.3.7) is necessary and sufficient to obtain a Leonard pair. In addition, we show that Leonard pairs arise naturally in relation with the Lie algebra sl and U (sl ). A more general object than the 2 q 2 concept of Leonard pair, namely the concept of tridiagonal pair, is also introduced and described in some details. We first recall the concept of a tridiagonal pair, of a tridiagonal system as well as properties of its (dual) eigenvalue sequence, the corresponding (dual) eigenspace sequence. It is asserted that the tridiagonal pair of q-Racah type satisfies the tridiagonalrelations,inverselyatridiagonalpaircanbeobtainedfromatridiagonalalgebra under several conditions. We also describe some special classes of tridiagonal pairs, such as Leonard pairs, tridiagonal pairs of the q-Serre type, mild tridiagonal pairs, sharp tridiag- onal pairs. Especially, the classification of the sharp tridiagonal pairs is clarified [INT10]. Last but not least, the relation between tridiagonal algebras and the theory of orthogo- nal polynomials is briefly described. We recall hypergeometric orthogonal polynomials, and describe the connection between the theory of Leonard pairs and the Askey-scheme of orthogonal polynomials [Ter0306]. Note that the extension to the theory of tridiagonal pairsleadstohypergeometricpolynomialsofseveralvariablesdefinedonadiscretesupport (Gasper-Rahman), as recently discovered in [BM15]. Chapter 2. We recall the known presentations of the Onsager algebra and how the so-called q−Onsager algebra appeared in the context of mathematical physics. We briefly recall the ‘q−Onsager approach’. First, we provide a historical background about the two different known presentations of the Onsager algebra: either the original presentation with generators A ,G [O44] or n m the presentation in terms of the Dolan-Grady relations [DG82]. We also recall the relation with the loop algebra of sl . Secondly, we recall how the q−Onsager algebra surprisingly 2 appeared in 2004 in the context of quantum integrable systems, through an analysis of so- INTRODUCTION 3 lutions of the reflection equation. In particular, we recall how the new infinite dimensional algebra A arises and how it is related with the q−Onsager algebra. It is explained how q its connections with the quantum loop algebra of sl2 and with Uq(s(cid:99)l2) naturally appear from the Yang-Baxter and reflection equation algebra formulation. Then, the so-called ‘q−Onsager approach’ is briefly recalled. Chapter 3. The three main results of the thesis are presented in some details. (i) and (ii): For the family of q−Onsager algebras (s(cid:99)l2 and ADE type), analogs of the higher order relations of Lusztig are conjectured and supporting evidence is presented in detail; (iii) The open XXZ spin chain is considered at roots of unity in the framework of the q−Onsager approach. A new algebra, an analog of Lusztig’s quantum group, naturally arises. For a class of finite dimensional representations, explicit generators and relations are described. With respect to the new algebra, symmetries of the Hamiltonian are explored. Recall the homomorphism from the q−Onsager algebra to the quantum affine Lie alge- bra Uq(s(cid:99)l2) [Bas0408, BB12]. Recall the homomorphism from the generalized q−Onsager algebras (higher rank generalizations of the q−Onsager algebra) to the quantum affine Lie algebra U (g) [BB09, Kol12]. By analogy with Lusztig’s higher order relations [Lusz93] q (cid:98) which arise for any quantum affine Lie algebra, it is thus expected that higher order rela- tions are satisfied. Successively, we obtained: 1. The higher order relations for the q-Onsager algebra (the s(cid:99)l2 case) [BV13] Let A,A∗ be the standard generators of the q−Onsager algebra. The r−th higher order relations for the q-Onsager algebra are conjectured. First, a generalization of the conjecture is proven for the case of tridiagonal pairs (i.e. certain irreducible finite dimensional representations on which A,A∗ act). Two-variable polynomials which determine the relations are given. Then, the special case of the q−Onsager algebra is considered in details. The conjecture is proven for r = 2,3. For r generic, the conjecture is studied recursively. A Maple software program is used to check the conjecture, which is confirmed for r ≤ 10. Also, for a special case, the higher order relations of Lusztig are recovered. 2. The higher order relations for the generalized q-Onsager algebra (the ADE serie) [BV1312] For each affine Lie algebra, a generalized q-Onsager algebra has been defined in [BB09]. Let A , i = 0,1,...,rank(g) be the standard generators of this algebra. i By analogy with the s(cid:99)l2 case, for any simply-laced affine Lie algebra analogues of Lusztig’s higher order relations are conjectured. The conjecture is proven for r ≤ 5. For r generic, the conjecture is studied recursively. A Maple software program is used to check the conjecture, which is confirmed for r ≤ 10. According to the parity of r, two new families of two-variable polynomials are proposed, which determine the structure of the higher order relations. Several independent checks are done, which support the conjecture. 3. The XXZ open spin chain at roots of unity [BGSV15] 4 CONTENTS Inspired by the fact that the XXZ periodic spin chain at roots of unity enjoys a sl loop algebra symmetry in certain sectors of the spectrum [DFM99], the aim is 2 to settle an algebraic framework for the analysis of the open XXZ spin chain at roots of unity within the q−Onsager approach. First, the two basic generators of the q−Onsager algebra are recalled, and their properties are studied for q a root of unity (spectrum, structure of the eigenspaces and action). They form a new object that we call a ‘cyclic tridiagonal pair’. Secondly, two new operators that are divided polynomials of the fundamental generators of the q-Onsager algebra are introduced. Westudysomeoftheirproperties(spectrum,structureoftheeigenspacesandaction). The relations satisfied by the four operators are described in details. They generate an explicit realization and first example of an analog of Lusztig quantum group for the q−Onsager algebra. Finally, we briefly discuss the conditions on the boundary parameters such that the Hamiltonian of the open XXZ spin chain commutes with some of the generators. In the end of this thesis, three families of open problems are presented in Chapter 4 and appendices are reported in Chapter 5.