W-algebras and superalgebras from constrained WZW models: A group theoretical classification

We present a classification ofW algebras and superalgebras arising in Abelian as well as non Abelian Toda theories. Each model, obtained from a constrained WZW action, is related with anSl(2) subalgebra (resp.OSp(1/2) superalgebra) of a simple Lie algebra (resp. superalgebra)G. However, the determination of anU(1) _Y factor, commuting withSl(2) (resp.OSp(1/2)), appears, when it exists, particularly useful to characterize the correspondingW algebra. The (super) conformal spin contents of eachW (super) algebra is performed. The class of all the superconformal algebras (i.e. with conformal spinss<=2) is easily obtained as a byproduct of our general results.     

Extended super-Kač-Moody algebras and their super-derivation algebras

We study theN-extended super-Ka-Moody algebras, i.e. extensions of the Lie algebra of the loop group over the super-circleA _N . The extensions are characterized by 2-cocycles which are computed in terms of the cyclic cohomology of the Grassmann algebra withN generators. The graded algebra of super-derivations compatible with each extension is determined. The casesN=1,2,3 are examined in detail and their relation with the Ademollo et al. superconformal algebras is discussed. We examine the possibility of defining new superconformal algebras which, forN>1, generalize theN=1 Ramond-Neveu-Schwarz algebra.     

Exactly solvable models in atomic and molecular physics

We construct integrable generalized models in a systematic way exploring different representations of the gl(N)$\mathit{gl}(N)$ algebra. The models are then interpreted in the context of atomic and molecular physics, most of them related to different types of Bose–Einstein condensates. The spectrum of the models is given through the analytical Bethe ansatz method. We further extend these results to the case of the superalgebra gl(M|N)$\mathit{gl}(M|N)$, providing in this way models which also include fermions.     

On <Emphasis Type="Italic">osp</Emphasis>(<Emphasis Type="Italic">M</Emphasis>∣2<Emphasis Type="Italic">n</Emphasis>) Integrable Open Spin Chains

We consider open spin chains based on osp(M2n) Yangians and solve the reflection equations for some classes of reflection matrices, including the diagonal ones. Having then integrable open spin chains, we write the analytical Bethe Ansatz equations. More details and references can be found in D. Arnaudon et al.: Nucl. Phys B 668 (2003) 469 and 687 (2004) 257.     

Yangians and $$\mathcal{W} $$ -Algebras

We present a connection between -algebras and Yangians in the case of gl(N) algebras, as well as for twisted Yangians and super-Yangians. We illustrate this connection, which allows constructing an R-matrix for the -algebras and classifying their finite-dimensional irreducible representations, in the framework of the nonlinear Schrödinger equation in 1+1 dimensions.     

Super Yangian Y(osp(1|2)) and the Universal R-matrix of Its Quantum Double

We present the Drinfel'd realisation of the super Yangian Y(osp(1|2)), including the explicit expression for the coproduct. We show in particular that it is necessary to introduce supplementary Serre relations. The construction of its quantum double is carried out. This allows us to give the universal R-matrix of DY(osp(1|2)).Member of Institut Universitaire de France     

Strings fromN=2 gauged Wess-Zumino-Witten models

We present an algebraic approach to string theory. An embedding ofsl(2|1) in a super Lie algebra together with a grading on the Lie algebra determines a nilpotent subalgebra of the super Lie algebra. Chirally gauging this subalgebra in the corresponding Wess-Zumino-Witten model, breaks the affine symmetry of the Wess-Zumino-Witten model to some extension of theN=2 superconformal algebra. The extension is completely determined by thesl(2|1) embedding. The realization of the superconformal algebra is determined by the grading. For a particular choice of grading, one obtains in this way, after twisting, the BRST structure of a string theory. We classify all embeddings ofsl(2|1) into Lie super algebras and give a detailed account of the branching of the adjoint representation. This provides an exhaustive classification and characterization of both all extendedN=2 superconformal algebras and all string theories which can be obtained in this way.     

On the R-Matrix Realization of Yangians and their Representations

We study the Yangians associated with the simple Lie algebras of type B, C or D. The algebra can be regarded as a quotient of the extended Yangian whose defining relations are written in an R-matrix form. In this paper we are concerned with the algebraic structure and representations of the algebra . We prove an analog of the Poincaré–Birkhoff–Witt theorem for and show that the Yangian can be realized as a subalgebra of . Furthermore, we give an independent proof of the classification theorem for the finite-dimensional irreducible representations of which implies the corresponding theorem of Drinfeld for the Yangians . We also give explicit constructions for all fundamental representation of the Yangians. Communicated by Petr Kulish Dedicated to Daniel Arnaudon Submitted: November 22, 2005; Accepted: February 1, 2006     

Sugawara and Vertex Operator Constructions for Deformed Virasoro Algebras

From the defining exchange relations of the 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 available when N = 2. Communicated by Petr Kulish Submitted: January 13, 2006; Accepted: March 6, 2006 Dedicated to our friend Daniel Arnaudon