Spin chain related to the quantum superalgebra slq(1¦1)

We consider an integrable system with R-matrix related to the algebra sl _q(1 | 1). The Hamiltonian of the system is constructed, and its spectrum is found by means of the algebraic Bethe ansatz. The symmetry algebra of the chain is written out. The partition function of the model on the lattice with domain wall boundary conditions is calculated. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 146–162.     

Scalar products in models with the GL(3) trigonometric R-matrix: General case

We study quantum integrable models with the GL(3) trigonometric R-matrix solvable by the nested algebraic Bethe ansatz and obtain an explicit representation for a scalar product of generic Bethe vectors in terms of a sum over partitions of Bethe parameters. This representation generalizes the known formula for scalar products in models with the GL(3)-invariant R-matrix.     

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.     

Dynamic systems related to the Cremmer-GervaisR-matrix

The generalized Cremmer-Gervais R-matrix, which is a twist of the standard sl_q (3) R-matrix, depends on two additional parameters. We discuss the properties of this R-matrix and construct two associated dynamic systems: the q-oscillator that is covariant with respect to the corresponding quantum group and an integrable spin chain with a non-Hermitian Hamiltonian. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116. No.1, pp. 101–112, July, 1998     

Hidden quantumR-matrix in the discrete-time classical Heisenberg magnet

We construct local M-operators for an integrable discrete-time version of the classical Heisenberg magnet by convoluting the twisted quantum trigonometric 4×4 R-matrix with certain vectors in its “quantum” space. Components of the vectors are identified with τ-functions of the model. Hirota's bilinear formalism is extensively used. The construction generalizes the known representation of M-operators in continuous-time models in terms of Lax operators and the classical τ-matrix. This paper was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 2, pp. 179–204, November, 2000.     

From Lie algebras of vector fields to algebraic group actions

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → Der_K(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.     

Outer automorphisms ofsl(2), integrable systems, and mappings

Outer automorphisms of infinite-dimensional representations of the Lie algebra sl(2) are used to construct Lax matrices for integrable Hamiltonian systems and discrete integrable mappings. The known results are reproduced, and new integrable systems are constructed. Classical r-matrices, corresponding to the Lax representation with the spectral parameter are dynamic. This scheme is advantageous because quantum systems naturally arise in the framework of the classical r-matrix Lax representation and the corresponding quantum mechanical problem admits a variable separation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 2, pp. 205–216, February, 1999.     

IntegrableN-dimensional systems on the Hopf algebra andq-deformations

We construct the class of integrable classical and quantum systems on the Hopf algebras describing n interacting particles. We obtain the general structure of an integrable Hamiltonian system for the Hopf algebra A(g) of a simple Lie algebra g and prove that the integrals of motion depend only on linear combinations of k coordinates of the phase space, 2·ind g≤k≤g·ind g, whereind g andg are the respective index and Coxeter number of the Lie algebra g. The standard procedure of q-deformation results in the quantum integrable system. We apply this general scheme to the algebras sl(2), sl(3), and o(3, 1). An exact solution for the quantum analogue of the N-dimensional Hamiltonian system on the Hopf algebra A(sl(2)) is constructed using the method of noncommutative integration of linear differential equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 373–390, September, 2000     

Zamolodchikov-Faddeev algebras for Yangian doubles at level 1

The representation theory of centrally extended Yangian doubles is investigated. The intertwining operators are constructed for infinite dimensional representations of , which are deformed analogs of the highest weight representations of the affine algebra at level 1. We give bosonized expressions for the intertwining operators and verify that they generate an algebra isomorphic to the Zamolodchikov-Faddeev algebra for the SU(2)-invariant Thirring model. From them, we compose L-operators by Miki’s method and verify that they coincide with L-operators constructed from the universal R-matrix. The matrix elements of the product of these operators are calculated explicitly and are shown to satisfy the quantum (deformed) Knizhnik-Zamolodchikov equation associated with the universal R-matrix for . This paper was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 25–45. January, 1997.     

Quantum differential operators on the quantum plane

The universal enveloping algebra of a Lie algebra acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or quantum group) is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring. We present this situation for the quantum group of sl₂.     

Factorization of the universalR-matrix forU_q \left( {\hat sl_2 } \right)^1

The factorization of the universal R-matrix corresponding to the so-called Drinfeld Hopf structure is described in the example of the quantum affine algebra . As a result of the factorization procedure, we deduce certain differential equations on the factors of the universal R-matrix that allow uniquely constructing these factors in the integral form. This article was written at the request of the Editorial Board Translated from Teoreticheskava i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 179–214, August, 2000     

Scalar products in models with a GL(3) trigonometric R-matrix: Highest coefficient

We study quantum integrable models with a GL(3) trigonometric R-matrix solvable by the nested algebraic Bethe ansatz. Scalar products of Bethe vectors in such models can be expressed in terms of bilinear combinations of the highest coefficients. We show that there exist two different highest coefficients in the models with a GL(3) trigonometric R-matrix. We obtain various representations for the highest coefficients in terms of sums over partitions. We also prove several important properties of the highest coefficients, which are necessary for evaluating the scalar products.     

R-Matrix quantization of the elliptic Ruijsenaars-Schneider model

It is shown that the classical L-operator algebra of the elliptic Ruijsenaars-Schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions. It is governed by two dynamic τ and -matrices satisfying a closed system of equations. The corresponding quantum R- and -matrices are found as solutions to quantum analogues of these equations. We present the quantum L-operator algebra and show that the system of equations for R and arises as the compatibility condition for this algebra. It turns out that the R-matrix is twist-equivalent to the Felder elliptic R^F-matrix, with playing the role of the twist. The simplest representation of the quantum L-operator algebra corresponding to the elliptic Ruijsenaars-Schneider model is obtained. The connection of the quantum L-operator algebra to the fundamental relation R^B LL=LLR^B with the Belavin elliptic R-matrix is established. As a by-product of our construction, we find a new N-parameter elliptic solution to the classical Yang-Baxter equation. This paper was written at the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 2, pp. 182–217, May, 1997.     

Coefficient Matrices of a Quantum Discrete Auxiliary Linear Problem

In the present paper, we describe general properties of quantum matrices that are coefficient matrices of an auxiliary problem for quantum discrete three-dimensional integrable models. Our goal is to prove a universal functional equation for the quantum determinant in the case of a finite-dimensional representation of a local Weyl algebra. Bibliography: 4 titles.     

Factorization of the r-matrix for the quantum algebra <Emphasis Type="Italic">Uq</Emphasis>(<Emphasis Type="Italic">sl</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>)

We propose a method for construction of the general solution of the Yang–Baxter equation with the U _q (sℓ _n ) symmetry algebra. This method is based on the factorization property of the corresponding L-operator. We present a closed-form expression for the universal R-matrix in the form of a difference operator acting on the space of functions of n(n − 1) variables. Bibliography: 16 titles.     

The algebraic structure of discrete zero curvature equations associated with integrable couplings and application to enlarged Volterra systems

An algebraic structure of discrete zero curvature equations is established for integrable coupling systems associated with semi-direct sums of Lie algebras. As an application example of this algebraic structure, a τ-symmetry algebra for the Volterra lattice integrable couplings is engendered from this theory.     

The cartan-weyl basis for the yangian doubleDY (sl 3)

Using the analogues of Cartan-Weyl elements, we construct a realization of the Yangian Y (sl₃). This enables us to derive the explicit comultiplication formulas in Y (sl₃), the quantum double DY (sl₃) of the Yangian, and the universal R-matrix. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 3, pp. 423–438, June, 1997.     

Tensor factorization and Spin construction for Kac-Moody algebras

In this paper we discuss the “Factorization phenomenon” which occurs when a representation of a Lie algebra is restricted to a subalgebra, and the result factors into a tensor product of smaller representations of the subalgebra. We analyze this phenomenon for symmetrizable Kac-Moody algebras (including finite-dimensional, semi-simple Lie algebras). We present a few factorization results for a general embedding of a symmetrizable Kac-Moody algebra into another and provide an algebraic explanation for such a phenomenon using Spin construction. We also give some application of these results for semi-simple, finite-dimensional Lie algebras.We extend the notion of Spin functor from finite-dimensional to symmetrizable Kac-Moody algebras, which requires a very delicate treatment. We introduce a certain category of orthogonal g-representations for which, surprisingly, the Spin functor gives a g-representation in Bernstein-Gelfand-Gelfand category O. Also, for an integrable representation, Spin produces an integrable representation. We give the formula for the character of Spin representation for the above category and work out the factorization results for an embedding of a finite-dimensional, semi-simple Lie algebra into its untwisted affine Lie algebra. Finally, we discuss the classification of those representations for which Spin is irreducible.