Group Theory Structures Underlying Integrable Systems 1

Different group structures which underline the integrable systems are considered. In some cases, the quantization of the integrable system can be provided with substituting groups by their quantum counterparts. However, some other group structures keep non-deformed in the course of quantizing the integrable system although their treatment is to be changed. Manifest examples of the KP/Toda hierarchy and the Liouville theory are considered. 1. During last decades there was a great development of integrable theories, both classical and quantum. However, while the classical integrable systems mostly advanced as the theory of non-linear equations, their quantum counterparts were rather based on algebraic structures related to the R-matrix. Therefore, the quantizing procedure was not always immediate. However, the last progress in integrable theories allows one to introduce a unified framework equally applicable to the both classical and quantum integrable systems. This framework is based on using the group theory structures which underline integrable system. In fact, it was known for many years that the most elegant and effective approach to the classical integrable systems is to use the language of group theory [1]. However, it turns out that there are some different group structures underlying the same integrable system. Some of the groups act in the space of solutions to the integrable hierarchy, others can act just on the variables of equations (" in the space-time "). In order to quantize an integrable system, one needs just to replace the first type group structures by their quantum counterparts. It was demonstrated [2,3] that one can reformulate the classical non-linear equations in these group terms, i.e. the quantizing procedure becomes really immediate. On the other hand, the groups acting in the space-time still remain classical even for the quantum systems. In this short remark we would like to stress the difference between above mentioned group structures and to demonstrate how they can be applied. Indeed, the groups acting in the space of solutions