Algebraic analysis of solvable lattice models. Dedicated to Mikio Sato and Ludwig D. Faddeev.

*(English)*Zbl 0828.17018
Regional Conference Series in Mathematics. 85. Providence, RI: American Mathematical Society (AMS). xvi, 152 p. (1995).

“The aim of the present volume is to give a survey of the recent development on the interplay between solvable lattice models in statistical mechanics and representation theory of quantum affine algebras”. The Ising model is essentially the only such model for which the correlation functions are known explicitly. The main goal of these notes is to explain how the space of states and correlation functions for the six-vertex model (and its spin chain equivalent, the XXZ-model) can be expressed in terms of quantum affine algebras and their representations. Namely, the space of states for six-vertex model is \(H\otimes H^*\) (suitably completed tensor product), where \(H= V(\lambda_0) \otimes V(\lambda_1)\) and \(V(\lambda_0)\) and \(V(\lambda_1)\) are the two level 1 standard modules for \(U_q (\widehat {\text{sl}}_2)\). The correlation functions are then obtained by taking traces of suitable products of \(q\)-vertex operators, which are intertwining operators between tensor products of standard modules and finite-dimensional irreducible \(U_q (\widehat {\text{sl}}_2)\) modules (which depend on one or more complex parameters).

“Our expositions are organized as follows. In Chapter 0 we give a brief account of basic principles in statistical mechanics. We also touch upon the history of solvable models. The first three chapters are devoted to the standard subjects concerning solvable lattice models in statistical mechanics. Our main examples are the spin 1/2 XXZ chain and the six- vertex model. The setting for these models and their mutual equivalence are explained in Chapter 1 and Chapter 2, respectively. In Chapter 3, we discuss the integrability of the models. The role of the Yang-Baxter equation and the commuting transfer matrices are clarified. The rest of the chapter is devoted to the introduction of the quantum affine algebra \(U_q (\widehat {\text{sl}}_2)\) and the representation theoretical interpretation of the Yang-Baxter equations.

In Chapter 4 we introduce the main objects, the corner transfer matrices and the vertex operators. By a physical argument we then show how the correlation functions can be written as the trace of products of the vertex operators, and derive difference equations for them. Having these as physical motivations, we restart our mathematical discussions from the next chapters.

Chapter 5 is devoted to the Frenkel-Jing bosonization of the level 1 modules of \(U_q (\widehat {\text{sl}}_2)\). In Chapter 6 we derive the formulas for the vertex operators using bosons. In Chapter 7 we reformulate the physical setting in representation theoretical terms, such as the space of states, vacuum, translation, Hamiltonian and its eigenvectors. To derive the formulas for the correlation functions and the form factors we need to calculate the trace of the products of vertex operators. This computation is carried out in Chapter 8, and its application is given in Chapter 9. The limit of the XXZ model is briefly discussed in Chapter 10. We note that the formulas in Chapters 8-10 are presented here for the first time in such details. The last Chapter 11 is devoted to the discussion of the other types of models, and related works.”

These notes are very clearly written, and are suitable for those who may not be expert either in quantum groups or in the theory of solvable lattice models.

“Our expositions are organized as follows. In Chapter 0 we give a brief account of basic principles in statistical mechanics. We also touch upon the history of solvable models. The first three chapters are devoted to the standard subjects concerning solvable lattice models in statistical mechanics. Our main examples are the spin 1/2 XXZ chain and the six- vertex model. The setting for these models and their mutual equivalence are explained in Chapter 1 and Chapter 2, respectively. In Chapter 3, we discuss the integrability of the models. The role of the Yang-Baxter equation and the commuting transfer matrices are clarified. The rest of the chapter is devoted to the introduction of the quantum affine algebra \(U_q (\widehat {\text{sl}}_2)\) and the representation theoretical interpretation of the Yang-Baxter equations.

In Chapter 4 we introduce the main objects, the corner transfer matrices and the vertex operators. By a physical argument we then show how the correlation functions can be written as the trace of products of the vertex operators, and derive difference equations for them. Having these as physical motivations, we restart our mathematical discussions from the next chapters.

Chapter 5 is devoted to the Frenkel-Jing bosonization of the level 1 modules of \(U_q (\widehat {\text{sl}}_2)\). In Chapter 6 we derive the formulas for the vertex operators using bosons. In Chapter 7 we reformulate the physical setting in representation theoretical terms, such as the space of states, vacuum, translation, Hamiltonian and its eigenvectors. To derive the formulas for the correlation functions and the form factors we need to calculate the trace of the products of vertex operators. This computation is carried out in Chapter 8, and its application is given in Chapter 9. The limit of the XXZ model is briefly discussed in Chapter 10. We note that the formulas in Chapters 8-10 are presented here for the first time in such details. The last Chapter 11 is devoted to the discussion of the other types of models, and related works.”

These notes are very clearly written, and are suitable for those who may not be expert either in quantum groups or in the theory of solvable lattice models.

Reviewer: A.N.Pressley (London)

##### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

82-02 | Research exposition (monographs, survey articles) pertaining to statistical mechanics |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

82B23 | Exactly solvable models; Bethe ansatz |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

17B69 | Vertex operators; vertex operator algebras and related structures |