Recent Developments in Integrable Systemsand Related Topics of Mathematical Physics
Our book is a selection of works presented at the Conference of Mathematical Physics “Kezenoi-Am 2016”. The Organising and Programme Committee of the conference tried to create a programme which embraces the variety of research directions inspired by the modern developments in Mathematical Physics and the theory of Integrable Systems. The authors of the included papers are well known mathematicians from several research groups in Europe and Russia. We hope that the book will attract the attention to these areas of research, and will be interesting
both to experts and young researchers. The Conference of Mathematical Physics “Kezenoi-Am 2016” is the first in the
series of conferences which are being held partly in the city of Grozny, and partly at a beautiful mountain lake “Kezenoi-Am” of the Chechen Republic, Russia. These conferences are generously supported by the Chechen State University (CheSU), which is driven by the goal to support mathematical culture in Chechen republic. It is important to note that the Chairman of the Organizing Committee, Rector of the CheSU, Prof. Zaurbek Saidov, encourages the idea that the organization of international conferences with the participation of world recognized researches is the
optimal way to motivate and attract students to research. During the participants’ welcoming, he stressed out that he considers our conference to be a quite important step towards this direction. We are especially grateful to him for his support and help in the organization of this series of successful international conferences. We are also grateful to the Vice-rector of CheSU, Prof. Zaur Kindarov for his support and hospitality. Dr. Dmitry Grinev played an essential role in the organization of these conferences. We would like to thank him for his activity and patience throughout this procedure. Finally, we acknowledge that this work was carried out within the framework of the State Programme of the Ministry of Education and Science of the Russian Federation, project 1.12873.2018/12.1.
Victor M. Buchstaber, Sotiris Konstantinou-Rizos, Alexander V. Mikhailov
Yaroslavl, Russia June 2018
In general, quantum matrix algebras are associated with a couple of compatible braidings. A particular example of such an algebra is the so-called Reflection Equation algebra In this paper we analyze its specific properties, which distinguish it from other quantum matrix algebras (in first turn, from the RTT one). Thus, we exhibit a specific form of the Cayley-Hamilton identity for its generating matrix, which in a limit turns into the Cayley-Hamilton identity for the generating matrix of the enveloping algebra U(gl(m)). Also, we consider some specific properties of the braided Yangians, recently introduced by the authors. In particular, we exhibit an analog of the Cayley-Hamilton identityfor the generating matrix of such a braided Yangian. Besides, by passing to a limit of this braided Yangian, we get a Lie algebra similar to that entering the construction of the rational Gaudin model. In its enveloping algebra we construct a Bethe subalgebra by the method due to D.Talalaev.