Solutions of the Yang-Baxter equation

Kulish, P. P.; Sklyanin, E. K.

doi:10.1007/bf01091463

Cited by 400 publications

(450 citation statements)

References 15 publications

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“…This development was partly initiated and most strongly influenced by the contribution of the Leningrad school, which clarified the fundamental meaning of the Yang-Baxter equation for the understanding of exactly solvable one-dimensional systems [1,2,3,4]. It culminated in the invention of quantum groups [5] which are by now generally accepted as the mathematical framework of the theory.…”

Section: Introductionmentioning

confidence: 99%

Algebraic solution of the Hubbard model on the infinite interval

Murakami

Göhmann

1998

Nuclear Physics B

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We develop the quantum inverse scattering method for the one-dimensional Hubbard model on the infinite line at zero density. This enables us to diagonalize the Hamiltonian algebraically. The eigenstates can be classified as scattering states of particles, bound pairs of particles and bound states of pairs. We obtain the corresponding creation and annihilation operators and calculate the S-matrix. The Hamiltonian on the infinite line is invariant under the Yangian quantum group Y(su(2)). We show that the n-particle scattering states transform like n-fold tensor products of fundamental representations of Y(su(2)) and that the bound states are Yangian singlet. ‡

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Section: Introductionmentioning

confidence: 99%

Algebraic solution of the Hubbard model on the infinite interval

Murakami

Göhmann

1998

Nuclear Physics B

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“…The Yang-Baxter equation and its solutions play a key role in the theory of the completely integrable quantum models [1,2,3,4,5]. The general solution of the Yang-Baxter equation (R-matrix) is the operator R(u) acting in a tensor product V 1 ⊗ V 2 of two linear spaces.…”

Section: Introductionmentioning

confidence: 99%

Yang–Baxter -operators and parameter permutations

2007

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We present an uniform construction of the solution to the Yang-Baxter equation with the symmetry algebra sℓ(2) and its deformations: the q-deformation and the elliptic deformation or Sklyanin algebra. The R-operator acting in the tensor product of two representations of the symmetry algebra with arbitrary spins ℓ 1 and ℓ 2 is built in terms of products of three basic operators S 1 , S 2 , S 3 which are constructed explicitly. They have the simple meaning of representing elementary permutations of the symmetric group S 4 , the permutation group of the four parameters entering the RLL-relation.1

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“…Now, following [14], we give some definitions concerning supermatrices. For any n × n supermatrix F, one can define the Grassmann parities of rows and columns as p row (i) ≡ p(F i,1 ) and p col ( j) ≡ p (F 1, j ), respectively, where p(F i, j ) is the Grassmann parity of the matrix element F i, j .…”

Section: Periodic Tl Hierarchiesmentioning

confidence: 99%

“…We consider the 2m-periodic auxiliary linear problem 14) for the wave functions ψ j such that ψ j+2m = wψ j . One can check that (5.13) and (5.14) are equivalent to the following linear problem:…”

Section: The R-matrix Formalismmentioning

confidence: 99%

Generalized fermionic discrete Toda hierarchy

Gribanov

Kadyshevsky

Сорин

2004

Discrete Dynamics in Nature and Society

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We describe bi-Hamiltonian structure and Lax-pair formulation with the spectral parameter of the generalized fermionic Toda lattice hierarchy as well as its bosonic and fermionic symmetries for different (including periodic) boundary conditions. Its two reductions-N = 4 and N = 2 supersymmetric Toda lattice hierarchies-in different (including canonical) bases are investigated. Its r-matrix description, monodromy matrix, and spectral curves are discussed.

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Solutions of the Yang-Baxter equation

Cited by 400 publications

References 15 publications

Algebraic solution of the Hubbard model on the infinite interval

Algebraic solution of the Hubbard model on the infinite interval

Yang–Baxter -operators and parameter permutations

Generalized fermionic discrete Toda hierarchy

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