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.     

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.     

Quantum Groups and the Genetic Code

The quantum algebra in the limit q 0 is proposed as a symmetry algebra for the genetic code. The nucleotide triplets (codons) in the DNA chain are classified in crystal bases. This construction can be compared to the baryon classification from quarks in particle physics, one main difference being the natural order in the state constituents provided by the crystal base. An operator ensuring the correspondence between codons and amino acids is constructed from the above algebra: two codons corresponding to the same or different amino acids acquire the same or different eigenvalues respectively. Then a set of relations between the physicochemical properties of the amino acids are derived and compared with the experimental data. Correlations of the codon usage for quartets and sextets are determined, fitting naturally in the framework of this model.     

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     

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     

Higher Sugawara Operators for the Quantum Affine Algebras of Type A

We give explicit formulas for the elements of the center of the completed quantum affine algebra in type A at the critical level that are associated with the fundamental representations. We calculate the images of these elements under a Harish-Chandra-type homomorphism. These images coincide with those in the free field realization of the quantum affine algebra and reproduce generators of the q-deformed classical \({{\mathcal W}}\)-algebra of Frenkel and Reshetikhin.