Classical integrable lattice models through quantum group related formalism

We effectively translate our earlier quantum constructions to the classical language and, using Yang-Baxterisation of the Faddeev-Reshetikhin-Takhtajan algebra, are able to construct the Lax operators and associated -matrices of classical integrable models. Thus, new as well as known lattice systems of different classes are generated, including new types of collective integrable models and canonical models with nonstandard matrices.Fachbereich 17-Mathematik/Informatik, GH-Universitän Kassel, Holländische Str. 36, 34109 Kassel, Germany. (Permanent address: Saha Institute of Nuclear Physics, AF/1 Bidhan Nagar, Calcutta 700 064, India.) Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 428–434, June, 1994.     

Conserved currents in quantum integrable models

The paper is devoted to the integrability of the quantum Klein-Gordon field theory with two exponential interactions. We find five different cases admitting integrability. For several fixed values of the coupling constant in these theories, conserved currents and noncommuting integrals of motion are found. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 34–44, October, 1997.     

On canonical variables in integrable models of magnets

Three integrable models: the deformed Heisenberg, Landau-Lifschits, and Ishimori magnets are written in terms of the stereographic projection. The Hamiltonians of these models are obtained and certain problems related to the existence of exact solutions are studied. In particular, stability of solitions is studied for the Heisenberg marnet, simplest stationary solutions are obtained for the Landau-Lifschits magnet, and Hamiltonians and topological charges are calculated for several known solutions of the Ishimori model. Bibliography: 33 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 117–143.     

Yang-Baxter algebra and generation of quantum integrable models

We discover an operator-deformed quantum algebra using the quantum Yang-Baxter equation with the trigonometric R-matrix. This novel Hopf algebra together with its q→1 limit seems the most general Yang-Baxter algebra underlying quantum integrable systems. We identify three different directions for applying this algebra in integrable systems depending on different sets of values of the deforming operators. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, and the associated Lax operators generate and classify them in a unified way. Variable values yield a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 470–485, June, 2007.     

The Harish-Chandra theorem for the quantum algebrau q (sl (3))

A basis of a quantum universal enveloping algebraU is constructed; the following theorem is proved with the help of this basis: For any nonzero element U, there exists a finite-dimensional representation such that(u) 0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45 No. 3, pp. 436–439, March, 1993.     

Construction of a new class of quantum integrable inhomogeneous models

Integrable inhomogenous or impurity models are usually constructed by either shifting the spectral parameter in the Lax operator or using another representation of the spin algebra. We propose a more involved general method for such construction in which the Lax operator contains generators of a novel quadratic algebra, a generalization of the known quantum algebra. In forming the monodromy matrix, we can replace any number of the local Lax operators with different realizations of the underlying algebra, which can result in spin chains with nonspin impurities causing changed coupling across the impurity sites, as well as with impurities in the form of bosonic operators. Following the same idea, we can also generate integrable inhomogeneous versions of the generalized lattice sine-Gordon model, nonlinear Schrödinger equation, Liouville model, relativistic and nonrelativistic Toda chains, etc.     

A categorification of the Casimir of quantum sl(2)

We categorify the Casimir element of the idempotented form of quantum sl(2).     

Three algebraic structures of quantum projective [sl(2, C)-invariant] field theory

Systematic studies are made of three algebraic structures of quantum projective [sl(2, )-invariant] field theory: the operator algebra Vert(sl(2, )), the infinite-dimensionalR matrixR _proj(u), and the deformationT () of the algebraT() of weighted-shift operators, which is associated with expansion of the renormalized pointwise product of vertex operator fields.State Geological and Prospecting Academy, Moscow; Department of Mathematics in the Research Institute of System Investigations (Information Technologies), Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 3, pp. 336–347, December, 1993.     

量子群U_q(sl(2 ) )的非负部分(英文)

设U≥ 0 是量子群Uq(sl(2 ) )的非负部分 .在本文中 ,我们确定了U≥ 0 的中心Z(U≥ 0 )和U≥ 0 的所有不可约表示     

The behaviour of cyclic variables in integrable systems

A general theorem on the behaviour of the angular variables of integrable dynamical systems as functions of time is established. Problems on the motion of the nodal line of a Kovalevskaya top and of a three- dimensional rigid body in a fluid are considered in integrable cases as examples. This range of topics is discussed for systems of a more general form obtained from completely integrable systems after changing the time.     

Composition factors of the kac module for the lie superalgebras sl(3/3) and sl(r/r)

The central results of this article are two theorems on G _ō and G primitivity on relative level 1 of the Kac module ?(?) for the Lie superal-gebra G = sl(m/n). These theorems are combined with Kac module homomorphisms in order to prove a conjecture by Hughes et al. [HKV] on the number of composition factors of the Kac module for 3- and r- fold atypical highest weights of sl(3/3) and sl(rZ/r) respectively. Our theorems on G _ō and G primitivity are general in nature and therefore the results also include infinite-dimensional representations.     

量子群Uq（sl（2））的非负部分

设U≥0是量子群Uq(sl(2))的非负部分,在本中,我们确定了U≥0的中心Z（U≥0）和U≥0有不可约表示。     

On the bosonization ofL-operators for the quantum affine algebraU q (sl 2)

Some relations between different objects associated with quantum affine algebras are reviewed. It is shown that the Frenkel-Jing bosonization of a new realization of the quantum affine algebra as well as bosonization of L-operators for this algebra can be obtained from Zamolodchikov-Faddeev algebras defined by the quantum R-matrix satisfying unitarity and crossing-symmetry conditions.On leave of absence from the ITP, Kiev 252143, Ukraine. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 64–77, July, 1995.     

Generic modules over the quantum groupU t (sl(2)) att a root of unit

LetA be a finite dimensionalk-algebra, an indecomposable (left)A-moduleM is called generic providedM is infinitek-dimensional but finite length as (right) End _A M-module. In this paper we given the explicit structure of generic modules and their endomorphism rings over the finite dimensional quantum groupU _t (sl(2)) att being a root of unit.