Related papers: On Bethe vectors in the sl_{N+1} Gaudin model
We show that the Bethe vectors are non-zero vectors in the sl_{r+1} Gaudin model. Moreover, we show that the norm of a Bethe vector is equal to the Hessian of the corresponding master function at the corresponding non-degenerate critical…
Consider a tensor product of finite-dimensional irreducible gl_{N+1}-modules and its decomposition into irreducible modules. The gl_{N+1} Gaudin model assigns to each multiplicity space of that decomposition a commutative (Bethe) algebra of…
We present a method to construct a basis of singular and non-singular common eigenvectors for Gaudin Hamiltonians in a tensor product module of the Lie algebra SL(2). The subset of singular vectors is completely described by analogy with…
We consider the Gaudin model associated to a point z in C^n with pairwise distinct coordinates and to the subspace of singular vectors of a given weight in the tensor product of irreducible finite-dimensional sl_2-representations, [G]. The…
We discuss the Bethe ansatz in the Gaudin model on the tensor product of finite-dimensional $sl_2$-modules over the field $F_p$ with $p$ elements, where $p$ is a prime number. We define the Bethe ansatz equations and show that if…
There is a correspondence between highest weight vectors in the tensor product of finite-dimensional irreducible sl(N+1)-modules marked by distinct complex numbers, on the one hand, and elements of the intersection of the Schubert varieties…
We derive explicit formulas for solutions of the Bethe Ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of…
We show that the Shapovalov norm of a Bethe vector in the Gaudin model is equal to the Hessian of the logarithm of the corresponding master function at the corresponding isolated critical point. We show that different Bethe vectors are…
We propose a new method of diagonalization of hamiltonians of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on Wakimoto modules over affine algebras at the critical level. We construct eigenvectors of these…
We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra $gl_N$. The models are defined on a tensor product irreducible $gl_N$-modules. For each model, there exist $N$ one-parameter families of commuting…
We prove that Bethe vectors generically form a base in a tensor product of irreducible heighest weight $gl_2$-modules or $U_q(gl_2)$-modules. We apply this result to difference equations with regular singular points. We show that if such an…
We examine the commuting elements $\theta_i=\sum_{j\neq i} \frac{s_{ij}}{z_i-z_j}$, $z_i\neq z_j$, $s_{ij}$ the transposition swapping $i$ and $j$, and we study their actions on irreducible $S_n$ representations. By applying Schur-Weyl…
The eigenvectors of the osp(1|2) invariant Gaudin hamiltonians are found using explicitly constructed creation operators. Commutation relations between the creation operators and the generators of the loop superalgebra are calculated. The…
Heine and Stieltjes in their studies of linear second-order differential equations with polynomial coefficients having a polynomial solution of a preassigned degree, discovered that the roots of such a solution are the coordinates of a…
The first goal of this paper is to give a precise and simple definition for off-shell Bethe vectors in a generic $g$-invariant integrable model for $g=gl_n$, $o_{2n+1}$, $sp_{2n}$ and $o_{2n}$. We prove from our definition that the…
Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$ generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of…
We compute scalar products of off-shell Bethe vectors in models with $o_{2n+1}$ symmetry. The scalar products are expressed as a sum over partitions of the Bethe parameter sets, the building blocks being the so-called highest coefficients.…
The Gaudin model based on the sl_2-invariant r-matrix with an extra Jordanian term depending on the spectral parameters is considered. The appropriate creation operators defining the Bethe states of the system are constructed through a…
In the derivation of the generating function of the Gaudin Hamiltonians with boundary terms, we follow the same approach used previously in the rational case, which in turn was based on Sklyanin's method in the periodic case. Our derivation…
Gaudin model based on the orthosymplectic Lie superalgebra osp(1|2) is studied. The eigenvectors of the osp(1|2) invariant Gaudin hamiltonians are constructed by algebraic Bethe Ansatz. Corresponding creation operators are defined by a…