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

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

A minimum principle in mRNA editing ?

mRNA editing of sequences of many species is analyzed. The nature of the inserted nucleotides and the position of the insertion sites, once fixed the edited peptide chain, are explained by introducing a minimum principle in the framework of the crystal basis model of the genetic code introduced by the authors.Comment: revised extended version - includes analysis on more dat

Anyonic Realization of the Quantum Affine Lie Algebras

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.g

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.Comment: 6 figures, documen

A crystal base for the genetic code

The quantum enveloping algebra U_q(sl(2) \oplus sl(2)) in the limit q \to 0 is proposed as a symmetry algebra for the genetic code. In this approach the triplets of nucleotids or codons in the DNA chain are classified in crystal bases, tensor product of U_{q \to 0}(sl(2) \oplus sl(2)) representations. Such a construction might be compared to the baryon classification from quark building blocks in elementary particles physics, one of the main differences standing in the property of a crystal base to provide a natural order in the state constituents, this order being crucial in the codon. Then an operator ensuring the correspondence codon/amino-acid can be constructed out of the above algebra. It will be called the reading operator, and be such that two codons relative to the same (resp. different) amino-acid(s) acquire the same (resp. different) eigenvalue(s).Comment: LaTeX-2e document, package amsfonts, 11 page

Sum rules of 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 with experimental data shows that the prediction is satisfied within about 5 %.Comment: 7 page

Anyonic Realization of the Quantum Affine Lie Superalgebra U_q(A(M,N)^{(1)})

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

Anyonic Realization of the Quantum Affine Lie Algebra U_q(A_N)

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