Infinite Dimensional Algebras and their Applications to Quantum Integrable Systems

Abstract

This special issue is centred around the workshop Infinite Dimensional Algebras and Quantum Integrable Systems II—IDAQUIS 2007, held at the University of Algarve, Faro, Portugal in July 2007. It was the second workshop in the IDAQUIS series following a previous meeting at the same location in 2003. The latest workshop gathered around forty experts in the field reviewing recent developments in the theory and applications ofintegrable systems in the form of invited lectures and in a number of contributions from the participants. All contributions contain significant newresults or provide a survey of the state of the art of thesubject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants are also included.The origins of the topic of this issue can be traced back a long way to the early investigations of completely integrable systems of classical mechanics in the fundamentalpapers by Euler, Lagrange, Jacobi, Liouville, Kowalevski and others. Bythe end of the nineteenth century all interesting examples seemed to have beenexhausted. A revival in the study of integrable systems began with thedevelopment of the classical inverse scattering method, or the theory ofsolitons. Later developments led to the basic geometrical ideas of thetheory, of which infinite dimensional algebras are a key ingredient. In a loose sense one may think that all integrable systems possesssome hidden symmetry. In the quantum version of these systems the representationtheory of these algebras may be exploited in the description of the structure of the Hilbert spaceof states.Modern examples of field theoretical systems such as conformal field theories,with the Liouville model being a prominent example, affine Toda fieldtheories and the AdS/CFT correspondence are based on algebraic structures like quantum groups, modular doubles, global conformal invariance, Hecke algebras, Kac-Moody algebras, Virasoro algebras etc. The exploitation of these mathematical structures inevitably leads to a deeper understanding of the physical systems.This issue provides some further progress in the investigations of the algebraic structures, such as Lie groups and Lie algebras, quantum groups, algebroids, etc,which have always played an important role in the development of the field. Quantum groups, for instance, have given an algebraic shape to the kinematicsof the quantum inverse scattering method and these ideas are developed further in thisissue.Some contributions focus on integrable systems with boundaries, which are interesting in their own right from a formal point of view as they exhibit some peculiarities which cannot be found within systems with periodic boundary conditions. The reflection equations and underlying quantum group covariant algebras allow for meaningful generalisations ofresults found in integrable scattering theories. Meanwhile the off shell structures have also been developed further and the first examples for form factor calculations, ultimatelyleading to correlation functions, are presented in this issue.Non-Hermitian Hamiltonian systems have already featured for some time in the contextof integrable models, as for instance in the form of affine Toda field theories with a complexcoupling constant or the Yang-Lee model. However, a systematic study of such types of models has only been initiated recently. It is now well understood that the reality of the spectrumof these models can be attributed either to the unbroken PT-symmetry of the entire system or to its pseudo(quasi)-Hermiticity. In reverse, one may take these concepts asstarting points for the construction of new types of models, such as integrableones which are the central topic of this special issue.We gratefully acknowledge the financial support provided by Clay Mathematics Institute, the Group of Mathematical Physics of the University of Lisbon, the Gulbenkian Foundation, the Luso-American Foundation and thePortuguese Foundation for Science and Technology, projects FACC/07/1/490and POCI/MAT/58452/2004. Finally, we would like to thank all the speakersand participants for creating an excellent atmosphere at the workshop. A special thank you goes to all contributors to this volume.We hope that this issue will provide a comprehensiveinsight into the theory of integrable systems for the newcomer to the field as well as to the expert.