Quantum Fields on Star Graphs with Bound States at the Vertex

We investigate the propagation of a massless scalar field on a star graph, modeling the junction of $n$ quantum wires. The vertex of the graph is represented by a point-like impurity (defect), characterized by a one-body scattering matrix. The general case of off-critical scattering matrix with bound and/or antibound states is considered. We demonstrate that the contribution of these states to the scalar field is fixed by causality (local commutativity), which is the key point of our investigation. Two different regimes of the theory emerge at this stage. If bound sates are absent, the energy is conserved and the theory admits unitary time evolution. The behavior changes if bound states are present, because each such state generates a kind of damped harmonic oscillator in the spectrum of the field. These oscillators lead to the breakdown of time translation invariance. We study in both regimes the electromagnetic conductance of the Luttinger liquid on the quantum wire junction. We derive an explicit expression for the conductance in terms of the scattering matrix and show that antibound and bound states have a different impact, giving raise to oscillations with exponentially damped and growing amplitudes respectively.Comment: LaTex 1+29 pages, 2 figures: Expanded version with new title and abstract; clarifying comments, fig.2 and references added; final version to appear in J. Math. Phy

A Minimum Principle in Codon-Anticodon Interaction

Imposing a minimum principle in the framework of the so called crystal basis model of the genetic code, we determine the structure of the minimum set of anticodons which allows the translational-transcription for animal mitochondrial code. The results are in very good agreement with the observed anticodons.Comment: 13 pages, 6 Tables, to appear in Biosystem

Symmetry and Minimum Principle at the Basis of the Genetic Code

The importance of the notion of symmetry in physics is well established: could it also be the case for the genetic code? In this spirit, a model for the Genetic Code based on continuous symmetries and entitled the "Crystal Basis Model" has been proposed a few years ago. The present paper is a review of the model, of some of its first applications as well as of its recent developments. Indeed, after a motivated presentation of our mathematical model, we illustrate its pertinence by applying it for the elaboration and verification of sum rules for codon usage probabilities, as well as for establishing relations and some predictions between physical-chemical properties of amino-acids. Then, defining in this context a "bio-spin" structure for the nucleotides and codons, the interaction between a couple of codon-anticodon can simply be represented by a (bio) spin-spin potential. This approach will constitute the second part of the paper where, imposing the minimum energy principle, an analysis of the evolution of the genetic code can be performed with good agreement with the generally accepted scheme. A more precise study of this interaction model provides informations on codon bias, consistent with data.Comment: To appear in BIOMAT 2016, 326 - 362, 201

Yangian realisations from finite W algebras

We construct an algebra homomorphism between the Yangian Y(sl(n)) and the finite W-algebras W(sl(np),n.sl(p)) for any p. We show how this result can be applied to determine properties of the finite dimensional representations of such W-algebras.Comment: 26 pages, Latex2

Non-Polynomial Realizations of W-Algebras

Relaxing first-class constraint conditions in the usual Drinfeld-Sokolov Hamiltonian reduction leads, after symmetry fixing, to realizations of W algebras expressed in terms of all the J-current components. General results are given for G a non exceptional simple (finite and affine) algebra. Such calculations directly provide the commutant, in the (closure of) G enveloping algebra, of the nilpotent subalgebra $G_-$, where the subscript refers to the chosen gradation in G. In the affine case, explicit expressions are presented for the Virasoro, $W_3$, and Bershadsky algebras at the quantum level.Comment: 33 pages, LaTeX file, minor LaTex error correcte

Dictionary on Lie Superalgebras

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.g

W-realization of Lie algebras: application to so(4,2) and Poincare algebras

The property of some finite W-algebras to appear as the commutant of a particular subalgebra in a simple Lie algebra G is exploited for the obtention of new G-realizations from a "canonical" differential one. The method is applied to the conformal algebra so(4,2) and therefore yields also results for its Poincare subalgebra. Unitary irreducible representations of these algebras are recognized in this approach, which is naturally compared -or associated- to the induced representation technic.Comment: LaTeX, 18 page

Crystalizing the genetic code

New developments are presented in the framework of the model introduced by the authors in refs. [1,2] and in which nucleotides as well as codons are classified in crystal bases of the quantum group U_q(sl(2)+sl(2)) in the limit q -> 0. An operator which gives the correspondence between the amino-acids and the codons is now obtained for any known genetic code. The free energy released by base pairing of dinucleotides as well as the relative hydrophilicity and hydrophobicity of the dinucleosides are also computed. For the vertebrate series, a universal behaviour in the ratios of codon usage frequencies is put in evidence and is shown to fit nicely in our model. Then a first attempt to represent the mutations relative to the deletion of a pyrimidine by action of a suitable crystal spinor operator is proposed. Finally recent theoretical descriptions are reviewed and compared with our model