New formula for the eigenvectors of the Gaudin model in the sl(3) case

We propose new formulas for eigenvectors of the Gaudin model in the sl(3) case. The central point of the construction is the explicit form of some operator P, which is used for derivation of eigenvalues given by the formula

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.     

The argument shift method and the Gaudin model

We construct a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U ( ) of a semisimple Lie algebra . This family is parameterized by finite sequences μ, z ₁, ..., z _n , where μ ∈ ^* and z _i ∈ ℂ. The construction presented here generalizes the famous construction of the higher Gaudin Hamiltonians due to Feigin, Frenkel, and Reshetikhin. For n = 1, the corresponding commutative subalgebras in the Poisson algebra S( ) were obtained by Mishchenko and Fomenko with the help of the argument shift method. For commutative algebras of our family, we establish a connection between their representations in the tensor products of finite-dimensional -modules and the Gaudin model. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 40, No. 3, pp. 30–43, 2006 Original Russian Text Copyright ? by L. G. Rybnikov     

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.     

Integrable Systems Obtained by Puncture Fusion from Rational and Elliptic Gaudin Systems

Using the procedure for puncture fusion, we obtain new integrable systems with poles of orders higher than one in the Lax operator matrix and consider the Hamiltonians, symplectic structure, and symmetries of these systems. Using the Inozemtsev limit procedure, we find a Toda-like system in the elliptic case having nontrivial commutation relations between the phase-space variables.     

Projection method and a universal weight function for the quantum affine algebra

We calculate the projection of the product of the Drinfeld currents on the intersection of the different Borel subalgebras in the current realization of the quantum affine algebra . This projection yields a universal weight function and has the structure of nested Bethe vectors. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 286–303, February, 2007.     

The Yangian double of the Lie superalgebra A(m, n)

The Yangian double DY(A(m, n)) of the Lie superalgebra A(m, n) is described in terms of generators and defining relations. Normally ordered bases in the Yangian and its dual in the quantum double are introduced. We calculate the pairing between the elements of these bases and obtain a formula for the universal R-matrix of the Yangian double as well as a formula for the universal R-matrix (introduced by Drinfeld) of the Yangian.     

Bethe algebra and the algebra of functions on the space of differential operators of order two with polynomial kernel

We show that the algebra of functions on the scheme of monic linear second-order ordinary differential operators with prescribed n + 1 regular singular points, prescribed exponents at the singular points, and having the kernel consisting of polynomials only, is isomorphic to the Bethe algebra of the Gaudin model acting on the vector space Sing of singular vectors of weight Λ^(∞) in the tensor product of finite-dimensional polynomial -modules with highest weights .      

广义Bethe树上关于随机选择系统的一类极限定理

本文将随机选择系统的概念在广义 Bethe 树上进行了推广,同时研究了广义Bethe树上选择子序列的状态序偶出现频率的一类极限定理,它是 Bernoulli序列无规则性概念的进一步推广.     

RTT relation and long-range spin chain model

It is verified that the long-range interaction integrable chain models with Yangian symmetry can be obtained from RTT relation, which therefore make this kind of models merge into Yang-Baxter system. A general method for obtaining Hamiltonian from quantum determinant of transfer matrix satisfying RTT relation is given. Project supported by the National Natural Science Foundation of China.     

A quantum spin system with random interactions I

We study a quantum spin glass as a quantum spin system with random interactions and establish the existence of a family of evolution groups {τ_t(ω)}_ω∈/Ω of the spin system. The notion of ergodicity of a measure preserving group of automorphisms of the probability space Ω, is used to prove the almost sure independence of the Arveson spectrum Sp(τ(ω)) of τ_t(ε). As a consequence, for any family of (τ(ω),β) — KMS states {ρ(ω)}, the spectrum of the generator of the group of unitaries which implement τ(ω) in the GNS representation is also almost surely independent of ω.     

关于有理模和余理想子代数的性质

In this paper, for some used conceptions and notations, we see ［1］ and ［2］.§1. Rational Module and Its Exact Sequence In ［1］, Cai Chuanreng and Cheng Huixiang have proved that relative Hopf modules and rational modules are one by one corresponding. In ［2］, Zhang Liangyun has given the dual relationship between relative Hopf modules. Naturally, we have a question to ask: is the dual module of a rational module still a rational module? This answer is affirmative. Let H be a Hopf …     

The Kac Jordan superalgebra: automorphisms and maximal subalgebras

In this note the group of automorphisms of the Kac Jordan superalgebra is described and used to classify the maximal subalgebras.

     

The bipolar quantum drift-diffusion model

A fourth order parabolic system, the bipolar quantum drift-diffusion model in semiconductor simulation, with physically motivated Dirichlet-Neumann boundary condition is studied in this paper. By semidiscretization in time and compactness argument, the global existence and semiclassical limit are obtained, in which semiclassieal limit describes the relation between quantum and classical drift-diffusion models, Furthermore, in the case of constant doping, we prove the weak solution exponentially approaches its constant steady state as time increases to infinity.     

Maximal Abelian Subalgebras of the Hyperfinite Factor,Entropy and Ergodic Theory

Given a free ergodic action of a discrete abelian group G on a measure space (X, μ), the crossed product L ^∞ (X, μ)⋊ G contains two distinguished maximal abelian subalgebras. We discuss what kind of information about the action can be extracted from the positions of these two subalgebras inside the crossed product algebra. Received February 24, 2002, Accepted August 5, 2002     

量子群U_q(f(K))的局部有限子代数的稳定理想

利用量子群U=U_q(f(K))的表示理论及其局部有限子代数F(U)的子模结构,证明了U_q(f(K))的局部有限子代数F(U)的任一非零理想均可由若干个具有不同权的最高权向量的和生成.     

Affine flag varieties and quantum symmetric pairs,II. Multiplication formula

We establish a multiplication formula for a tridiagonal standard basis element in the idempotent version, i.e., the Lusztig form, of the coideal subalgebras of quantum affine ${\mathfrak{gl}}_n$ arising from the geometry of affine partial flag varieties of type C. We apply this formula to obtain the stabilization algebras ${\dot{\mathbf{K}}}_n^{\mathfrak{c}}$, ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{??}$, ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{??}$ and ${\dot{\mathbf{K}}}_{\eta }^{??}$, which are idempotented coideal subalgebras of quantum affine ${\mathfrak{gl}}_n$. The symmetry in the formula leads to an isomorphism of the idempotented coideal subalgebras ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{??}$ and ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{??}$ with compatible monomial, standard and canonical bases.