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.     

Quantum volterra model and the universalR-matrix

We prove that an integrable system solved by the quantum inverse scattering method can be described by a purely algebraic object (universal R-matrix) and a proper algebraic representation. For the quantum Volterra model, we construct the L-operator and the fundamental R-matrix from the universal R-matrix for the quantum affine algebra U_q(ŝl₂) and the q-oscillator representation for it. Thus, there is an equivalence between an integrable system with the symmetry algebra A and the representation of this algebra. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 3, pp. 384–396, December, 1997.     

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     

Galois groups of quantum group actions and regularity of fixed-point algebras

It is shown that, for a minimal and integrable action of a locally compact quantum group on a factor, the group of automorphisms of the factor leaving the fixed-point algebra pointwise invariant is identified with the intrinsic group of the dual quantum group. It is proven also that, for such an action, the regularity of the fixed-point algebra is equivalent to the cocommutativity of the quantum group.

     

量子群中的Coxeter变换的“阶”

设g是有限维复单李代数。本文考虑量子群U_q(g)中两个特殊的自同构及它们作用在U_q(g)上及其可积U_q(g)-模上的性态。     

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.     

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.     

Bethe Vectors for Orthogonal Integrable Models

Theoretical and Mathematical Physics - We consider quantum integrable models associated with the $$\mathfrak{so}_3$$ algebra and describe Bethe vectors of these models in terms of the current...     

Action of Clifford Algebra on the Space of Sequences of Transfer Operators

We deduce from a determinant identity on quantum transfer matrices of generalized quantum integrable spin chain model their generating functions. We construct the isomorphism of Clifford algebra modules of sequences of transfer matrices and the boson space of symmetric functions. As an application, tau-functions of transfer matrices immediately arise from classical tau-functions of symmetric functions.     

Categorification of integrable representations of quantum groups

We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac–Moody algebra. As a byproduct, we obtain a geometric realization of Lusztig's canonical bases of these representations as well as a new positivity result. The main ingredient in the underlying geometric construction is a class of micro-local perverse sheaves on quiver varieties.     

I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type

We introduce a notion of I-factorial quantum torsor, which consists of an integrable ergodic action of a locally compact quantum group on a type I-factor such that also the crossed product is a type I-factor. We show that any such I-factorial quantum torsor is at the same time a I-factorial quantum torsor for the dual locally compact quantum group, in such a way that the construction is involutive. As a motivating example, we show that quantized compact semisimple Lie groups, when amplified via a crossed product construction with the function algebra on the associated weight lattice, admit I-factorial quantum torsors, and give an explicit realization of the dual quantum torsor in terms of a deformed Heisenberg algebra for the Borel part of a quantized universal enveloping algebra.     

“Twisted” rational <Emphasis Type="Italic">r</Emphasis>-matrices and the algebraic Bethe ansatz: Applications to generalized Gaudin models,Bose–Hubbard dimers,and Jaynes–Cummings–Dicke-type models

We construct quantum integrable systems associated with the Lie algebra gl(n) and non-skew-symmetric “shifted and twisted” rational r-matrices. The obtained models include Gaudin-type models with and without an external magnetic field, n-level (n?1)-mode Jaynes–Cummings–Dicke-type models in the Λ-configuration, a vector generalization of Bose–Hubbard dimers, etc. We diagonalize quantum Hamiltonians of the constructed integrable models using a nested Bethe ansatz.     

Maximally Superintegrable Gaudin Magnet: A Unified Approach

A classical integrable Hamiltonian system is defined by an Abelian subalgebra (of suitable dimension) of a Poisson algebra, while a quantum integrable Hamiltonian system is defined by an Abelian subalgebra (of suitable dimension) of a Jordan–Lie algebra of Hermitian operators. We propose a method for obtaining large Abelian subalgebras inside the tensor product of free tensor algebras, and we show that there exist canonical morphisms from these algebras to Poisson algebras and Jordan–Lie algebras of operators. We can thus prove the integrability of some particular Hamiltonian systems simultaneously at both the classical and the quantum level. We propose a particular case of the rational Gaudin magnet as an example.     

本刊英文版Vol.30(2014),No.6论文摘要（英文）

<正>Categorification of Integrable Representations of Quantum Groups Hao ZHENG Abstract We eategorify the highest weight integrable representations and their tensor prod—ucts of a symmetric quantum Kac—Moody algebra.As a byproduct,we obtain a geometric realization of Lusztig's canonical bases of these representations as well as a new positivity result.The main ingredient in the underlying geometric construction is a class of micro-local perverse     

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.     

Gaudin models with irregular singularities

We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from B. Feigin et al. (1994) [17] to the case of non-highest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P¹ with regular as well as irregular singularities at finitely many points. We construct eigenvectors of these Hamiltonians, using Wakimoto modules of critical level, and show that their spectra on finite-dimensional representations are given by opers with trivial monodromy. We also comment on the connection between the generalized Gaudin models and the geometric Langlands correspondence with ramification.     

量子群的基变换与范畴同构

令Ｍ是Ｚ［ｖ］的由ｖ－１和奇素数ｐ生成的理想，Ｕ是Ａ＝Ｚ［ｖ］Ｍ上相伴于对称Ｃａｒｔａｎ矩阵的量子群， Ａ－Γ是环同态， Ｕг＝ＵＡΓ［Ｕг］是Ｕг的量子坐标代数，本文建立了量子坐标代数的基变换：即在相关约束条件下有Г－Ｈｏｐｆ同构 Ａ［Ｕ］ＡГ≌Г［Ｕг］．我们证明了有限秩 Ａ自由 １型可积 Ｕ模范畴和有限秩 Ａ自由 Ａ［Ｕ］余模范畴是同构的．特别，当 Г是域时，局部有限 １型 Ｕг模范畴和Г［Ｕг］余模范畴是同构的．最后，我们还证明了在［１］中定义的诱导函子和Ｂ．Ｐａｒｓｈａｌｌ与王建磐博士在［２］中研究的诱导函子的一致性．     

Representation of the Zamolodchikov-Faddeev algebra

For quantum completely integrable models with an infinite number of degrees of freedom, such as vector nonlinear Schrödinger equations on the line, isotropic and anisotropic generalized Heisenberg ferromagnets, operators are constructed which satisfy the permutation relations of Zamolodchikov's algebra.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 109, pp. 83–92, 1981.     

Factorization of the Loop Algebra and Integrable Toplike Systems

With any Lie algebra of Laurent series with coefficients in a semisimple Lie algebra and its decomposition into a sum of the subalgebra consisting of the Taylor series and a complementary subalgebra, we associate a hierarchy of integrable Hamiltonian nonlinear ODEs. In the case of the so(3) Lie algebra, our scheme covers all classical integrable cases in the Kirchhoff problem of the motion of a rigid body in an ideal fluid. Moreover, the construction allows generating integrable deformations for known integrable models.