Spectrum and Bethe ansatz equations for the U_ {q}(gl(N)) closed and open spin chains in any representation

Abstract

We consider the N-site U_{q}(gl(N)) integrable spin chain with periodic and open diagonal soliton-preserving boundary conditions. By employing analytical Bethe ansatz techniques we are able to determine the spectrum and the corresponding Bethe ansatz equations for the general case, where each site of the spin chain is associated to any representation of U_{q}(gl(N)). In the case of open spin chain, we study finite dimensional representations of the quantum reflection algebra, and prove in full generality that the pseudo-vacuum is a highest weight of the monodromy matrix. For these two types of spin chain, we study the (generalized) "algebraic" fusion procedures, which amount to construct the quantum contraction and the Sklyanin determinant for the affine U_{q}(gl(N)) and quantum reflection algebras. We also determine the symmetry algebra of these two types of spin chains, including general K and K^+ diagonal matrices for the open case. The case of open spin chains with soliton non-preserving boundary conditions is also presented in the framework of quantum twisted Yangians. The symmetry algebra of this spin chains is studied. We also give an exhaustive classification of the invertible matricial solutions to the corresponding reflection equation.

arXiv comment48 pages; Bounds on parameters Mj corrected; References added; Examples added

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• Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra I
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• Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables
• Nested Bethe ansatz for `all' open spin chains with diagonal boundary conditions
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• Bethe Ansatz for the Universal Weight Function
• Bethe Ansatz and Bethe Vectors Scalar Products
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• The master T-operator for vertex models with trigonometric $R$-matrices as classical tau-function
• On Separation of Variables for Reflection Algebras
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• Thermodynamics, density profiles and correlation functions of the inhomogeneous one-dimensional spinor Bose gas
• Dynamic Lattice Supersymmetry in gl(n|m) Spin Chains
• (Quantum) twisted Yangians: symmetry, Baxterisation and centralizers
• New reflection matrices for the U_q(gl(m|n)) case
• Contracted and expanded integrable structures
• Nested algebraic Bethe ansatz for open spin chains with even twisted Yangian symmetry
• Generating Series for Nested Bethe Vectors
• Transmission matrices in gl(N) & U_q(gl(N)) quantum spin chains
• Central extension of the reflection equations and an analog of Miki's formula
• Generalized Landau-Lifshitz models on the interval
• Momentum reconstruction and contact of the one-dimensional Bose-Fermi mixture
• Scattering matrix for a general gl(2) spin chain
• Representations of twisted q-Yangians