Michel Semenov Tian Chanski - Academia.edu (original) (raw)

Papers by Michel Semenov Tian Chanski

WORLD SCIENTIFIC eBooks, May 21, 2018

arXiv (Cornell University), May 14, 1999

The q-deformed version of the Drinfeld-Sokolov reduction is extended to the case of the algebra o... more The q-deformed version of the Drinfeld-Sokolov reduction is extended to the case of the algebra of 'complex size matrices'; this construction generalizes earlier results of B.Khesin and F.Malikov on universal DS reduction and follows the pattern of recent studies of q-deformed DS reduction which were started by E.Frenkel, N.Reshetikhin and one of the authors.

Reviews in Mathematical Physics, Aug 23, 2018

The famous paper by L. D. Faddeev and B. S. Pavlov (1972) on automorphic wave equation explored a... more The famous paper by L. D. Faddeev and B. S. Pavlov (1972) on automorphic wave equation explored a highly romantic link between Scattering Theory (in the sense of Lax and Phillips) and Riemann hypothesis. An attempt to generalize this approach to general semisimple Lie groups leads to an interesting evolution system with multidimensional time explored by the author in 1976. In the present paper, we compare this system with a simpler one defined for zero curvature symmetric spaces and show that the Huygens principle for this system in the curved space holds if and only if it holds in the zero curvature limit.

Lecture Notes in Mathematics, 1984

HAL (Le Centre pour la Communication Scientifique Directe), Jul 1, 2017

International audienc

On February 26, 2017, after a long fight with cancer Ludwig Dmitrievich Faddeev passed away. For ... more On February 26, 2017, after a long fight with cancer Ludwig Dmitrievich Faddeev passed away. For the authors of this article Faddeev was a teacher and a constant source of scientific inspiration for many formative years.

Astérisque, 1994

L'accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisq... more L'accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ S. M. F.

Birkhäuser Basel eBooks, 2003

They were intended for participants with the background in Analysis and Operator Theory but witho... more They were intended for participants with the background in Analysis and Operator Theory but without special knowledge of Geometry and Lie Groups. The text below represents a sort of compromise: it is certainly impossible not to speak about Lie algebras and Lie groups at all; however, in order to make the main ideas reasonably clear, I tried to use only matrix algebras such as gl(n) and its natural subalgebras; Lie groups used are either GL(n) and its subgroups, or loop groups consisting of matrix-valued functions on the circle (possibly admitting an extension to parts of the Riemann sphere). I hope this makes the environment sufficiently easy to live in for an analyst. The main goal is to explain how the factorization problems (typically, the matrix Riemann problem) generate the entire small world of Integrable Systems along with the geometry of the phase space, Hamiltonian structure, Lax representations, integrals of motion and explicit solutions. The key tool will be the classical r-matrix (an object whose other guise is the well-known Hilbert transform). I do not give technical details, unless they may be exposed in a few lines; on the other hand, all motivations are given in full scale whenever possible. I hope that this choice agrees with the spirit of the Faro School and will help to bridge the gap between different branches of Mathematical Analysis.

Springer eBooks, Dec 10, 2007

The key concept discussed in these lectures is the relation between the Hamiltonians of a quantum... more The key concept discussed in these lectures is the relation between the Hamiltonians of a quantum integrable system and the Casimir elements in the underlying hidden symmetry algebra. (In typical applications the latter is either the universal enveloping algebra of an affine Lie algebra, or its q-deformation.) A similar relation also holds in the classical case. We discuss different guises of this very important relation and its implication for the description of the spectrum and the eigenfunctions of the quantum system. Parallels between the classical and the quantum cases are thoroughly discussed.

Communications in Mathematical Physics, Sep 1, 1986

A Lax pair for a new family of integrable systems on SO(4) is presented. The construction makes u... more A Lax pair for a new family of integrable systems on SO(4) is presented. The construction makes use of a twisted loop algebra of the G 2 Lie algebra. We also describe a general scheme producing integrable cases of the generalized rigid body motion in an external field which have a Lax representation with spectral parameter. Several other examples of multidimensional tops are discussed.

Physics Letters, Jul 1, 1988

The formation ofclassical r-matrices is used to construct families of compatible Poisson brackets... more The formation ofclassical r-matrices is used to construct families of compatible Poisson brackets for various nonlinear integrable systems connected with loop algebras. Poisson brackets are called compatible if their un-[X,~IR=~([RX, Y] + [X, RY]) (1) ear combination is again a Poisson bracket, i.e. satisfies the Jacobi identity. It is well known that satisfies the Jacobi identity. In this case (1) defines dynamical systems integrable via the inverse spectral a second Lie bracket in g called the R-bracket. Recall transform usually possess several compatible Pois-that the dual space to a Lie algebra g is equipped with son brackets and, moreover, the integrals of motion the natural Poisson bracket (the so-called Lie-Poisare in involution with respect to any of them. The 50fl bracket) specified by the condition that the properties of such compatible Poisson brackets and Poisson bracket of two linear functions on~cointhe hierarchies of conservation laws associated with cides with their Lie bracket in g. Hence in the presthem have been studied both axiomatically (the~-ent case these are two Poisson brackets on g*• The called "Magri-Lenard scheme") [1,2] and for var-Lie-Poisson bracket associated with the r-matrix R ious concrete examples [3-5].Our aim here is to will be also called the R-bracket. We have the folshow how these examples fit into the general pattern lowing basic statement: (1) Functions on g'~'invariant with respect to the based on the notion of the classical r-matrix [6,7]. In this approach families of compatible Poisson onginal Lie structure are in involution with respect to the R-bracket. brackets on the phase space are generated by linear (2) If H is an invariant function on g~,the asfamilies of Lie brackets (the so-called "Lie pencils") sociated hamiltonian equation on grelation to the on its dual. A systematic way to construct such Lie R-bracket has a generalized Lax form pencils is provided by the r-matrix method. The examples we give here are connected with the gener-dL alized AKNS systems (in particular, the Heisenberg~=-adM~L, M=~R(dH(L)). (2) ferromagnet and the nonlinear Schrodinger equation). Examples connected with the Virasoro algebra If~is self-dual, i.e. has a nondegenerate invariant which extend the results of ref. [81 are reported in inner product which allows to identify gwith g and a separate note [91.The impetus for the present study

Communications in Mathematical Physics, Jun 1, 1989

A "natural" Lax pair for the Kowalewski top is derived by using a general group-theoretic approac... more A "natural" Lax pair for the Kowalewski top is derived by using a general group-theoretic approach. This gives a new insight into the algebraic geometry of the top and leads to its complete solution via finite-band integration theory.

arXiv (Cornell University), Dec 31, 2019

Quantum Toda lattice may solved by means of the Representation Theory of semisimple Lie groups, o... more Quantum Toda lattice may solved by means of the Representation Theory of semisimple Lie groups, or alternatively by using the technique of the Quantum Inverse Scattering Method. A comparison of the two approaches, which is the purpose of the present review article, sheds a new light on Representation Theory and leads to a number of challenging questions.

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Communications in Mathematical Physics, Apr 1, 1998

We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-defo... more We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical W-algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL 2. The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL 2 , and this leads to a new elliptic classical rmatrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in [19]. We also consider a discrete analogue of this Poisson algebra. In the second part [31] the construction is generalized to the case of an arbitrary semisimple Lie algebra.

Contemporary mathematics, 1994

The Heisenberg double of a Hopf algebra may be regarded as a quantum analogue of the cotangent bu... more The Heisenberg double of a Hopf algebra may be regarded as a quantum analogue of the cotangent bundle of a Lie group. Quantum duality principle describes relations between a Hopf algebra, its dual, and their Heisenberg double in a way which extends both the theory of coadjoint orbits and the classical Fourier transform. We also describe the twisted Heisenberg double which is relevant for the study of nontrivial deformations of the quantized universal enveloping algebras.

arXiv (Cornell University), Feb 9, 1994

We compute the Poisson bracket relations for the monodromy matrix of the auxiliary linear problem... more We compute the Poisson bracket relations for the monodromy matrix of the auxiliary linear problem. If the basic Poisson brackets of the model contain derivatives, this computation leads to a peculiar kind of symmetry breaking which accounts for a 'spontaneous quantization' of the underlying global gauge group. A classification of possible patterns of symmetry breaking is outlined. * The present paper is a translation of an article originally published in Zapiski Nauchn.

Inventiones Mathematicae, Oct 1, 1981

The present article is the sequel to a previous paper by the same authors [1], Its aim is to give... more The present article is the sequel to a previous paper by the same authors [1], Its aim is to give an explicit solution of a factorization problem for groups of loops, and to establish a connection of Hamiltonian reduction methods with algebraic methods of Novikov and Krichever [2, 3] and of Mumford and van Moerbeke [4]. We also correct some erroneous statements in [1] concerning the factorization problem (see no. 2 below). To make our presentation more selfconsistent, we give an elementary proof of the reduction theorem in a slightly more general form as compared to [1]. This generality corresponds to that of [3, 4] where the same equations are treated in terms of finite-difference operators. An approach based on affine Lie algebras is also described by Adler and van Moerbeke [5]. However, the Hamiltonian reduction questions are not treated there. The authors are grateful to I.M. Krichever for valuable and stimulating discussions.

Springer eBooks, 1992

Fascinating links between Conformal Field Theory and Quantum groups discovered recently suggest t... more Fascinating links between Conformal Field Theory and Quantum groups discovered recently suggest that Quantum groups also have a direct bearing on the representation theory of Kac-Moody algebras. It is the purpose of the present note to trace down this hidden quantum group symmetry in the framework of Kae-Moody algebras. Our main result is that the monodromy of quantum Kac-Moody current

Encyclopaedia of mathematical sciences, 1994

Page 128. Chapter 2 Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Syst... more Page 128. Chapter 2 Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems AG Reyman, MA Semenov-Tian-Shansky Translated by the authors Introduction The present survey is devoted to a general ...

Inventiones Mathematicae, Feb 1, 1979

... o(gh)=zf(h-1)q)(g), heL:. 3 . Now we apply the reduction technics to get an algebra of commut... more ... o(gh)=zf(h-1)q)(g), heL:. 3 . Now we apply the reduction technics to get an algebra of commuting Hamiltonians described in [17]. We recall briefly some notions from [17]. Let a Lie algebra g be split into a linear sum of two subalgebras ...