Luc Frappat

A complete Bethe ansatz solution for the open spin-s?XXZ chain with general integrable boundary terms

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 of s = 1, we explicitly determine the Hamiltonian, and find an expression for its eigenvalues in terms of Bethe roots.

Classical and quantum sl(1‖2) superalgebras, Casimir operators and quantum chain Hamiltonians

We examine in this paper the two parameter deformed superalgebra Uqs(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≥2 in the undeformed case and by p∈Z in the deformed case, which obey quadratic relations. The construction of the dual superalgebra of functions of SLqs(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.

Super-Hubbard models and applications

We construct XX- and Hubbard- like models based on unitary superalgebras gl(N |M) generalising Shastrys and Maassaranis 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 loopsla Klein and Seitz is performed.

CENTRAL EXTENSIONS OF CLASSICAL AND QUANTUM Q-VIRASORO ALGEBRAS

We investigate the central extensions of the q-deformed (classical and quantum) Virasoro algebras constructed from the elliptic quantum algebra Aq,p(sl(N))c. After establishing the expressions of the cocycle conditions, we solve them, both in the classical and in the quantum case (for sl(2)). We find that the consistent central extensions are much more general that those found previously in the literature.

Symmetry and Codon Usage Correlations in the Genetic Code

The ratios of the codon usage in the quartets and sextets for the vertebrate series exhibit a correlated behaviour which fits naturally in the framework of the crystal basis model of the genetic code. Moreover the observed universal behaviour of these suitably normalized ratios can be easily explained.

Universal Hubbard models with arbitrary symmetry

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 an 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 R-matrices obtained 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 is carried out, while they are fully computed in the XX case. A perturbative calculation in the large coupling regime is also carried out for the universal Hubbard models.

Conspiracy in bacterial genomes

The rank ordered distribution of the codon usage frequencies for 109 eubacteria and 14 archaea is best fitted by a three-parameter function which is the sum of a constant, an exponential and a linear term in the rank n. The parameters depend (two parabolically) from the total GC content. The rank ordered distribution of the amino acids is fitted by a straight line. The Shannon entropy computed over all the codons is well fitted by a parabola in the GC content, while the partial entropies computed over subsets of the codons show peculiar different behavior. Moreover, the sum of the codon usage frequencies over particular sets, e.g. with C and A (respectively G and U) as ith nucleotide, show a clear linear dependence from the GC content, exhibiting a conspiracy effect.

Sum rules for codon usage probabilities

In the crystal basis model of the genetic code, it is deduced that the sum of usage probabilities of the codons with C and A in the third position for the quartets and/or sextets is independent of the biological species for vertebrates. A comparison between the theoretical predictions and the experimental data shows a satisfactory agreement.

A MODEL OF GENETIC CODE BASED ON UQ→0(sl(2)⊕sl(2))

A model of genetic code based on Uq→0(sl(2)⊕sl(2)) is presented. The ratios of the codon usage in the quartets and sextets for 7 biological species belonging to the vertebrate series, with a statistics of codons larger than 200.000, exhibit a correlated behaviour, which fits naturally in the model.

Universal Construction of \(\) Algebras

Abstract:We present a direct construction of the abstract generators for q-deformed algebras. New quantum algebraic structures of type are thus obtained. This procedure hinges upon a twisted trace formula for the elliptic algebra generalizing the previously known formulae for quantum groups. It represents the q-deformation of the construction of algebras from Lie algebras.