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We construct $q$-deformations of quantum $\mathcal{W}_N$ algebras with elliptic structure functions. Their spin $k+1$ generators are built from $2k$ products of the Lax matrix generators of ${\mathcal{A}_{q,p}(\widehat{gl}(N)_c)}$). The closure of the algebras is insured by a critical surface condition relating the parameters $p,q$ and the central charge $c$. Further abelianity conditions are determined, either as $c=-N$ or as a second condition on $p,q,c$. When abelianity is achieved, a Poisson bracket can be defined, that we determine explicitly. One connects these structures with previously built classical $q$-deformed $\mathcal{W}_N$ algebras and quantum $\mathcal{W}_{q,p}(\mathfrak{sl}_N)$., Comment: 20 pages

We review some of the strategies that can be implemented to infer an $R$-matrix from the knowledge of its Hamiltonian. We apply them to the classification achieved in arXiv:1306.6303, on three state $U(1)$-invariant Hamiltonians solvable by CBA, focusing on models for which the $S$-matrix is not trivial. For the 19-vertex solutions, we recover the $R$-matrices of the well-known Zamolodchikov--Fateev and Izergin--Korepin models. We point out that the generalized Bariev Hamiltonian is related to both main and special branches studied by Martins in arXiv:1303.4010, that we prove to generate the same Hamiltonian. The 19-vertex SpR model still resists to the analysis, although we are able to state some no-go theorems on its $R$-matrix. For 17-vertex Hamiltonians, we produce a new $R$-matrix., Comment: 22 pages

We classify all Hamiltonians with rank 1 symmetry, acting on a periodic three-state spin chain, and solvable through (generalisation of) the coordinate Bethe ansatz (CBA). We obtain in this way four multi-parametric extensions of the known 19-vertex Hamiltonians (such as Zamolodchikov-Fateev, Izergin-Korepin, Bariev Hamiltonians). Apart from the 19-vertex Hamiltonians, there exists 17-vertex and 14-vertex Hamiltonians that cannot be viewed as subcases of the 19-vertex ones. In the case of 17-vertex Hamiltonian, we get a generalization of the genus 5 special branch found by Martins, plus three new ones. We get also two 14-vertex Hamiltonians. We solve all these Hamiltonians using CBA, and provide their spectrum, eigenfunctions and Bethe equations. A special attention is made to provide the specifications of our multi-parametric Hamiltonians that give back known Hamiltonians., Comment: 30 pages; web page: http://www.coulomb.univ-montp2.fr/3Ham

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., Comment: 30 pages

We compute the eigenfunctions, energies and Bethe equations for a class of generalized integrable Hubbard models based on gl(n|m)\oplus gl(2) superalgebras. The Bethe equations appear to be similar to the Hubbard model ones, up to a phase due to the integration of a subset of `simple' Bethe equations. We discuss relations with AdS/CFT correspondence, and with condensed matter physics., Comment: 36 pages

We propose a general framework that leads to one-dimensional XX and Hubbard models in full generality, based on the decomposition of an arbitrary vector space (possibly infinite dimensional) into a direct sum of two subspaces, the two corresponding orthogonal projectors allowing one to define a R-matrix of a universal XX model, and then of a Hubbard model using a Shastry type construction. The QISM approach ensures integrability of the models, the properties of the obtained R-matrices leading to local Hubbard-like Hamiltonians. In all cases, the energies, the symmetry algebras and the scattering matrices are explicitly determined. The computation of the Bethe Ansatz equations for some subsectors of the universal Hubbard theories are determined, while they are fully computed in the XX case. A perturbative calculation in the large coupling regime is also done for the universal Hubbard models., Comment: 33 pages

We construct the XX and Hubbard-like models based on unitary superalgebras gl(N|M) generalizing Shastry's and Maassarani's approach. We introduce the R-matrix of the gl(N|M) XX-type model; the one of the Hubbard-like model is defined by "coupling" two independent XX models. In both cases, we show that the R-matrices satisfy the Yang-Baxter equation. We derive the corresponding local Hamiltonian in the transfer matrix formalism and we determine its symmetries. A perturbative calculation "\`a la Klein and Seitz" is performed. Some explicit examples are worked out. We give a description of the two-particle scattering., Comment: Talk given by G. Feverati at the workshop "RAQIS'07 Recent Advances in Quantum Integrable Systems", 11-14 Sept. 2007, LAPTH, Annecy-le-Vieux, France

We consider the thermodynamical limit of a gl(N) spin chain with arbitrary representation at each site of the chain. We consider excitations (with holes and new strings) above the vacuum and compute their corrections in 1/L to the densities and the energy., Comment: 29 pages misprints in the example of sect 5.1 amended and a mistake in theorem 5.4 corrected

We consider the open spin-s XXZ quantum spin chain with N sites and general integrable boundary terms for generic values of the bulk anisotropy parameter, and for values of the boundary parameters which satisfy a certain constraint. We derive two sets of Bethe Ansatz equations, and find numerical evidence that together they give the complete set of $(2s+1)^{N}$ eigenvalues of the transfer matrix. For the case s=1, we explicitly determine the Hamiltonian, and find an expression for its eigenvalues in terms of Bethe roots., Comment: 23 pages -- Latex2e; misprints in appendix corrected

We construct XX- and Hubbard- like models based on unitary superalgebras gl(N|M) generalising Shastry's and Maassarani's approach of the algebraic case. We introduce the R-matrix of the gl(N|M) XX model and that of the Hubbard model defined by coupling two independent XX models. In both cases, we show that the R-matrices satisfy the Yang--Baxter equation, we derive the corresponding local Hamiltonian in the transfer matrix formalism and we determine the symmetry of the Hamiltonian. Explicit examples are worked out. In the cases of the gl(1|2) and gl(2|2) Hubbard models, a perturbative calculation at two loops a la Klein and Seitz is performed., Comment: 26 pages

From the defining exchange relations of the A_{q,p}(gl_{N}) elliptic quantum algebra, we construct subalgebras which can be characterized as q-deformed W_N algebras. The consistency conditions relating the parameters p,q,N and the central charge c are shown to be related to the singularity structure of the functional coefficients defining the exchange relations of specific vertex operators representations of A_{q,p}({gl_{N}) available when N=2., Comment: 23 pages

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., Comment: 48 pages; Bounds on parameters Mj corrected; References added; Examples added

We present an ``algebraic treatment'' of the analytical Bethe ansatz for open spin chains with soliton non preserving (SNP) boundary conditions. For this purpose, we introduce abstract monodromy and transfer matrices which provide an algebraic framework for the analytical Bethe ansatz. It allows us to deal with a generic gl(N) open SNP spin chain possessing on each site an arbitrary representation. As a result, we obtain the Bethe equations in their full generality. The classification of finite dimensional irreducible representations for the twisted Yangians are directly linked to the calculation of the transfer matrix eigenvalues., Comment: 17

We use the Jack symmetric functions as a basis of the Fock space, and study the action of the Virasoro generators $L_n$. We calculate explicitly the matrix elements of $L_n$ with respect to the Jack-basis. A combinatorial procedure which produces these matrix elements is conjectured. As a limiting case of the formula, we obtain a Pieri-type formula which represents a product of a power sum and a Jack symmetric function as a sum of Jack symmetric functions. Also, a similar expansion was found for the case when we differentiate the Jack symmetric functions with respect to power sums. As an application of our Jack-basis representation, a new diagrammatic interpretation is presented, why the singular vectors of the Virasoro algebra are proportional to the Jack symmetric functions with rectangular diagrams. We also propose a natural normalization of the singular vectors in the Verma module, and determine the coefficients which appear after bosonization in front of the Jack symmetric functions., Comment: 23 pages, references added

Two types of boundary conditions ("soliton preserving" and "soliton non-preserving") are investigated for the sl(n) and sl(m|n) open spin chains. The appropriate reflection equations are formulated and the corresponding solutions are classified. The symmetry and the Bethe Ansatz equations are derived for each case. The general treatment for non-diagonal reflection matrices associated to "soliton preserving" case is worked out. The connection between the "soliton non-preserving" boundary conditions and the twisted (super) Yangians is also discussed., Comment: 37 pages

We formulate the Bethe Ansatz equations for the open super spin chain based on the super Yangian of osp(M|2n) and with diagonal boundary conditions. We then study the bulk and boundary scattering of the osp(1|2n) open spin chain., Comment: references added

We present a classification of diagonal, antidiagonal and mixed reflection matrices related to Yangian and super-Yangian R matrices associated to the infinite series so(m), sp(n) and osp(m|n). We formulate the analytical Bethe Ansatz resolution for the so(m) and sp(n) open spin chains with boundary conditions described by the diagonal solutions., Comment: 36 pages ; references added ; typos corrected, precisions added

We extend to the sl(N) case the results that we previously obtained on the construction of W_{q,p} algebras from the elliptic algebra A_{q,p}(\hat{sl}(2)_c). The elliptic algebra A_{q,p}(\hat{sl}(N)_c) at the critical level c=-N has an extended center containing trace-like operators t(z). Families of Poisson structures indexed by N(N-1)/2 integers, defining q-deformations of the W_N algebra, are constructed. The operators t(z) also close an exchange algebra when (-p^1/2)^{NM} = q^{-c-N} for M in Z. It becomes Abelian when in addition p=q^{Nh} where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed W_N algebras depending on the parity of h, characterizing the exchange structures at p \ne q^{Nh} as new W_{q,p}(sl(N)) algebras., Comment: LaTeX2e Document - packages subeqn,amsfonts,amssymb - 30 pages

It is shown that the elliptic algebra A_{p,q}(sl(2)_c) has a non trivial center at the critical level $c=-2$, generalizing the result of Reshetikhin and Semenov-Tian-Shansky for trigonometric algebras. A family of Poisson structures indexed by a non-negative integer $k$ is constructed on this center., Comment: LaTeX 2.09 Document (should be run twice)

We give a realization of the quantum affine Lie superalgebras U_q(A(M,N))^(1) in terms of anyons defined on a one or two-dimensional lattice, the deformation parameter q being related to the statistical parameter $\nu$ of the anyons by q = exp(i\pi\nu). The construction uses anyons contructed from usual fermionic oscillators and deformed bosonic oscillators. As a byproduct, realization deformed in any sector of the quantum superalgebras U_q(A(M,N)) is obtained., Comment: 14p LaTeX Document (should be run twice)

We give a realization of the quantum affine Lie algebras $\uqa$ and $\uqc$ in terms of anyons defined on a one-dimensional chain (or on a two-dimensional lattice), the deformation parameter $q$ being related to the statistical parameter $\nu$ of the anyons by $q = e^{i\pi\nu}$. In the limit of the deformation parameter going to one we recover the Feingold-Frenkel fermionic construction of undeformed affine Lie algebras., Comment: 8p, LaTeX, subeqn.sty. Also available at http://lapphp0.in2p3.fr/preplapp/psth/anyon_ac.ps.gz

The main definitions and properties of Lie superalgebras are proposed a la facon de a short dictionary, the different items following the alphabetical order. The main topics deal with the structure of simple Lie superalgebras and their finite dimensional representations; rather naturally, a few pages are devoted to supersymmetry. This modest booklet has two ambitious goals: to be elementary and easy to use. The beginner is supposed to find out here the main concepts on superalgebras, while a more experimented theorist should recognize the necessary tools and informations for a specific use., Comment: 145p LaTeX Document, also available at http://lapphp0.in2p3.fr/preplapp/psth/DICTIONARY_SUPER.ps.gz

We give a realization of quantum affine Lie algebra $U_q(\hat A_{N-1})$ in terms of anyons defined on a two-dimensional lattice, the deformation parameter $q$ being related to the statistical parameter $\nu$ of the anyons by $q = e^{i\pi\nu}$. In the limit of the deformation parameter going to one we recover the Feingold-Frenkel fermionic construction of undeformed affine Lie algebra., Comment: 13p LaTeX Document (should be run twice)

We examine the two parameter deformed superalgebra $U_{qs}(sl(1|2))$ and use the results in the construction of quantum chain Hamiltonians. This study is done both in the framework of the Serre presentation and in the $R$-matrix scheme of Faddeev, Reshetikhin and Takhtajan (FRT). We show that there exists an infinite number of Casimir operators, indexed by integers $p > 1$ in the undeformed case and by $p \in Z$ in the deformed case, which obey quadratic relations. The construction of the dual superalgebra of functions on $SL_{qs}(1|2)$ is also given and higher tensor product representations are discussed. Finally, we construct quantum chain Hamiltonians based on the Casimir operators. In the deformed case we find two Hamiltonians which describe deformed $t-J$ models., Comment: 27 pages, LaTeX, one reference moved and one formula added

We perform an Hamiltonian reduction on a classical \cw(\cg, \ch) algebra, and prove that we get another \cw(\cg, \ch$'$) algebra, with $\ch\subset\ch'$. In the case $\cg=S\ell(n)$, the existence of a suitable gauge, called Generalized Horizontal Gauge, allows to relate in this way two \cw-algebras as soon as their corresponding \ch-algebras are related by inclusion., Comment: 11 p., Latex. There was a misprint on the last author

After some definitions, we review in the first part of this talk the construction and classification of classical $W$ (super)algebras symmetries of Toda theories. The second part deals with more recently obtained properties. At first, we show that chains of $W$ algebras can be obtained by imposing constraints on some $W$ generators: we call secondary reduction such a gauge procedure on $W$ algebras. Then we emphasize the role of the Kac-Moody part, when it exists, in a $W$ (super) algebra. Factorizing out this spin 1 subalgebra gives rise to a new $W$ structure which we interpret either as a rational finitely generated $W$ algebra, or as a polynomial non linear $W_\infty$ realization. (Plenary talk presented by P. SORBA at the $XXII^{th}$ International Conference on Differential Geometric Methods in Theoretical Physics. Ixtapa Mexico, September 1993.), Comment: 16 pages

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_\infty$ realization., Comment: 11 pages, LATEX, preprint ENSLAPP-AL-429/93 and NORDITA-93/47-P

In the same way the folding of the Dynkin diagram of A_{2n} (resp. A_{2n-1}) produces the B_n (resp. C_n) Dynkin diagram, the symmetry algebra W of a Toda model based on B_n (resp. C_n) can be seen as resulting from the folding of a W-algebra based on A_{2n} (resp. A_{2n-1}). More generally, W algebras related to the B-C-D algebra series can appear from W algebras related to the unitary ones. Such an approach is in particular well adapted to obtain fusion rules of W algebras based on non simply laced algebras from fusion rules corresponding to the A_n case. Anagously, super W algebras associated to orthosymplectic superalgebras are deduced from those relative to the unitary A(m,n) series., Comment: 27 pages, Latex

A classification of (super) $W$ algebras arising from non Abelian Toda and super Toda theories is presented. This classification is based on the $Sl(2)$ or $OSp(1|2)$ sub(super)algebras of the simple Lie (super)algebra underlying the model. This allows to compute the conformal spin content of each $W$ (super)algebra. {\em Based on two lectures given by L. Frappat and E. Ragoucy at the\\ XIX International Colloquium on Group Theoretical Methods in Physics\\ Salamanca (Spain), June 29 - July 4, 1992}, Comment: 10 pages in Latex

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) $\cg$. 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 $s\leq2$) is easily obtained as a byproduct of our general results., Comment: 66 pages (Latex), ENSLAPPP-A-391/92 Replaces previous unLatexable version, corrupted by mailer