Based on the NSF-CBMS Regional Conference lectures presented by Miwa in June 1993, this book surveys recent developments in the interplay between solvable lattice models in statistical mechanics and representation theory of quantum affine algebras. Because results in this subject were scattered in the literature, this book fills the need for a systematic account, focusing attention on fundamentals without assuming prior knowledge about lattice models or representation theory. After a brief account of basic principles in statistical mechanics, the authors discuss the standard subjects concerning solvable lattice models in statistical mechanics, the main examples being the spin $1/2$ XXZ chain and the six-vertex model. The book goes on to introduce the main objects of study, the corner transfer matrices and the vertex operators, and discusses some of their aspects from the viewpoint of physics. Once the physical motivations are in place, the authors return to the mathematics, covering the Frenkel-Jing bosonization of a certain module, formulas for the vertex operators using bosons, the role of representation theory, and correlation functions and form factors. The limit of the $XXX$ model is briefly discussed, and the book closes with a discussion of other types of models and related works.

Readership

Advanced graduate students and researchers.

Reviews

"The book provides a very clear exposition on the subject, and meanwhile gives an elementary introduction to representation theory ... serves very well as both introduction and reference."

-- Mathematical Reviews

"Very clearly written ... suitable for those who may not be expert either in quantum groups or in the theory of solvable lattice models."

-- Zentralblatt MATH

Table of Contents

• Background of the problem
• The spin $1/2$ XXZ model for $\Delta < -1$
• The six-vertex model in the anti-ferroelectric regime
• Solvability and symmetry
• Correlation functions--physical derivation
• Level one modules and bosonization
• Vertex operators
• Space of states--mathematical picture
• Traces of vertex operators
• Correlation functions and form factors
• The $XXX$ limit $q\rightarrow -1$
• Discussions
• List of formulas