We introduce a generalization of the original Coordinate Bethe Ansatz that allows to treat the case of open spin chains with non-diagonal boundary matrices. We illustrate it on two cases: the XXX and XXZ chains. Short review on a joint work with N. Crampe (L2C) and D. Simon (LPMA), see arXiv:1009.4119, arXiv:1105.4119 and arXiv:1106.3264.

We present in a unified and detailed way the nested Bethe ansatz for open spin chains based on Y(gl(n)), Y(gl(m|n)), (U) over cap (q)(gl(n)) or (U) over cap (q)(gl(m| n)) (super) algebras, with arbitrary representations (i.e., 'spins') on each site of the chain and diagonal boundary matrices (K(+)(u), K(-)(u)). The nested Bethe ansatz applies for a general K(-)(u), but a particular form of the K(+)(u) matrix. The construction extends and unifies the results already obtained for open spin chains based on the fundamental representation and for some particular super-spin chains. We give the eigenvalues, Bethe equations and the explicit form of the Bethe vectors for the corresponding models. The Bethe vectors are also expressed using a trace formula.

We identify generating functionals that satisfy dynamical exchange relations with the Lax matrices defining the face-type elliptic quantum algebra B-q,B-lambda ((g) over capl(2))(c), when the central charge takes the two possible values c =3D +/- 2. These generating functionals are constructed as quadratic trace-like objects in terms of the Lax matrices. The obtained structures are characterized as 'dynamical centers', i.e. the centrality property is deformed by dynamical shifts. For these values, the functionals define (genuine) abelian subalgebras of B-q,B-lambda ((g) over capl(2))(c).

We consider a one-dimensional model of a two-component Bose gas and study form factors of local operators in this model. For this aim, we use an approach based on the algebraic Bethe ansatz. We show that the form factors under consideration can be reduced to those of the monodromy matrix entries in a generalized GL(3)-invariant model. In this way we derive determinant representations for the form factors of local operators.

We study the vacuum state of spin chains where each site carries an arbitrary representation. We prove that the string hypothesis usually used to solve the Bethe ansatz equations is valid for representations characterized by rectangular Young tableaux. In these cases, we obtain the density of the centre of the strings for the vacuum. We work through different examples and, in particular, that for spin chains with a periodic array of impurities.

We study integrable models with gl(2 vertical bar 1) symmetry and solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for scalar products of Bethe vectors, when the Bethe parameters obey some relations weaker than the Bethe equations. This representation allows us to find the norms of on-shell Bethe vectors and obtain determinant formulas for form factors of the diagonal entries of the monodromy matrix.

We present a classification of W algebras and superalgebras arising in Abelian as well as non-Abelian Toda theories. Each model, obtained from a constrained WZW action, is related with an Sl(2) subalgebra (resp. OSp(1|2) superalgebra) of a simple Lie algebra (resp. superalgebra) Gscr. However, the determination of an U(1) Y factor, commuting with Sl(2) (resp. OSp(1|2)), appears, when it exists, particularly useful to characterize the corresponding W algebra. The (super) conformal spin contents of each W (super) algebra is performed. The class of all the superconformal algebras (i.e. with conformal spins s les 2) is easily obtained as a byproduct of our general results

We consider Casimir elements for the orthogonal and symplectic Lie algebras constructed with the use of the Brauer algebra. We calculate the images of these elements under the Harish-Chandra isomorphism and thus show that they (together with the Pfaffian-type element in the even orthogonal case) are algebraically independent generators of the centers of the corresponding universal enveloping algebras. (C) 2013 Elsevier Inc. All rights reserved.

We introduce a new Baxterisation for R-matrices that depend separately on two spectral parameters. The Baxterisation is based on a new algebra, close to but different from the braid group. We study representations of this new algebra on the vector space , when the generators act locally. The ones for are completely classified. We also introduce some representations for generic m: they allow us to recover the R-matrix of the multi-species generalization of the totally asymmetric simple exclusion process with different hopping rates.

We study integrable models solvable by the nested algebraic Bethe ansatz and possessing GL(3)-invariant R-matrix. Assuming that the monodromy matrix of the model can be expanded into series with respect to the inverse spectral parameter, we define zero modes of the monodromy matrix entries as the first nontrivial coefficients of this series. Using these zero modes we establish new relations between form factors of the elements of the monodromy matrix. We prove that all of them can be obtained from the form factor of a diagonal matrix element in special limits of Bethe parameters. As a result we obtain determinant representations for form factors of all the entries of the monodromy matrix. (C) 2015 The Authors. Published by Elsevier B.V.

We compute the Bethe equations of generalized Hubbard models, and study their thermodynamical limit. We argue how they can be connected to the ones found in the context of AdS/CFT correspondence, in particular with the so-called dressing phase problem. We also show how the models can be interpreted, in condensed matter physics, as integrable multi-leg Hubbard models.

Factoring out the spin 1 subalgebra of a W algebra leads to a new W structure which can be seen either as a rational finitely generated W algebra or as a polynomial non-linear W infin realization

We study the general L 0-regular gl (2) spin chain, i.e. a chain where the sites {i, i + L 0, i + 2L 0, ...} carry the same arbitrary representation (spin), gl(2). The basic example of such chain is obtained for L 0 = 2, where we recover the alternating spin chain. Firstly, we review different known results about the integrability and the spectrum. Secondly, we give an interpretation in terms of particles and conjecture a scattering matrix connecting such chains.

Recently the quantum Hamiltonian reduction was done in the case of general sl(2) embeddings into Lie algebras and superalgebras. In this paper we extend the results to the quantum Hamiltonian reduction of N=1 affine Lie superalgebras in the superspace formalism. We show that if we choose a gauge for the supersymmetry, and consider only certain equivalence classes of fields, then our quantum Hamiltonian reduction reduces to quantum Hamiltonian reduction of nonsupersymmetric Lie superalgebras. We construct explicitly the super energy-momentum tensor, as well as all generators of spin 1 (and 1/2); thus we construct explicitly all generators in the superconformal, quasi-superconformal and Z2times Z2 superconformal algebras

We show that a certain class of light-like Wilson loops exhibits a Yangian symmetry at one loop, or equivalently, in an Abelian theory. The Wilson loops we discuss are equivalent to one-loop MHV amplitudes in N = 4 super Yang-Mills theory in a certain kinematical regime. The fact that we find a Yangian symmetry constraining their functional form can be thought of as the effect of the original conformal symmetry associated to the scattering amplitudes in the N = 4 theory.