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## Bethe subalgebras in Braided Yangians and Gaudin-type models

In Gurevich and Saponov (J Geom Phys 138:124–143, 2019) the notion of braided Yangians of Reflection Equation type was introduced. Each of these algebras is associated with an involutive or Hecke symmetry R. In these algebras quantum analogs of certain symmetric polynomials (elementary symmetric ones, power sums) were defined. In the present paper we show that these quantum symmetric polynomials commute with each other and consequently generate a commutative Bethe subalgebra. As an application, we get some Gaudin-type models.

By generalized Yangians, we mean Yangian-like algebras of two different classes. One class comprises the previously introduced so-called braided Yangians. Braided Yangians have properties similar to those of the reflection equation algebra. Generalized Yangians of the second class, RT T -type Yangians, are defined by the same formulas as the usual Yangians but with other quantum R-matrices. If such an R-matrix is the simplest trigonometric R-matrix, then the corresponding RT T -type Yangian is called a q-Yangian. We claim that each generalized Yangian is a deformation of the commutative algebra Sym(gl(m)[t−1]) if the corresponding R-matrix is a deformation of the flip operator. We give the explicit form of the corresponding Poisson brackets.

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

We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skewinvertible involutive and Hecke symmetries. By using the Cayley–Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904–911, 2008).

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.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.