arXiv subjects
Publications and source records attributed to L. Frappat.
Using the crystal basis model of the genetic code, a set of relations between the physical-chemical properties of the amino acids are derived and compared with the experimental data. A prevision for the not yet measured thermodynamical parameters of three amino acids is done.
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.
We construct universal twists connecting the centrally extended double Yangian DY(sl(2))_c with deformed double Yangians DY_r(sl(2))_c, thereby establishing the quasi-Hopf structures of the latter.
A self-contained description of algebraic structures, obtained by combinations of various limit procedures applied to vertex and face sl(2) elliptic quantum affine algebras, is given. New double Yangians structures of dynamical type are in particular defined. Connections between these structures are established. A number of them take the form of twist-like actions. These are conjectured to be evaluations of universal twists.
Scaling limits when q tends to 1 of the elliptic vertex algebras A_qp(sl(N)) are defined for any N, extending the previously known case of N=2. They realise deformed, centrally extended double Yangian structures DY_r(sl(N)). As in the quantum affine algebras U_q(sl(N)), and quantum elliptic affine algebras A_qp(sl(N)), these algebras contain subalgebras at critical values of the central charge c=-N-Mr (M integer, 2r=ln p/ln q), which become Abelian when c=-N or 2r=Nh for h integer. Poisson structures and quantum exchange relations are derived for their abstract generators.
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.
The elliptic algebra A_{q,p}(sl(N)_{c}) at the critical level c=-N has an extended center containing trace-like operators t(z). Families of Poisson structures, defining q-deformations of the W_N algebra, are constructed. The operators t(z) also close an exchange algebra when (-p^{1/2})^{NM} = q^{-c-N} for M \in Z. It becomes Abelian when in addition p=q^{Nh} where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed W_N algebras depending on the parity of h, characterizing the exchange structures at p =/ q^{Nh} as new W_{q,p}(sl(N)) algebras.
We present a direct construction of abstract generators for q-deformed W_N algebras. This procedure hinges upon a twisted trace formula for the elliptic algebra A_{q,p}(sl(N)_c) generalizing the previously known formulae for quantum groups.
We investigate the central extensions of the q-deformed (classical and quantum) Virasoro algebras constructed from the elliptic quantum algebra A_{q,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.
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).
We extend to the sl(N) case the results that we previously obtained on the construction of W_{q,p} algebras from the elliptic algebra A_{q,p}(\hat{sl}(2)_c). The elliptic algebra A_{q,p}(\hat{sl}(N)_c) at the critical level c=-N has an extended center containing trace-like operators t(z). Families of Poisson structures indexed by N(N-1)/2 integers, defining q-deformations of the W_N algebra, are constructed. The operators t(z) also close an exchange algebra when (-p^1/2)^{NM} = q^{-c-N} for M in Z. It becomes Abelian when in addition p=q^{Nh} where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed W_N algebras depending on the parity of h, characterizing the exchange structures at p \ne q^{Nh} as new W_{q,p}(sl(N)) algebras.
We define on the universal enveloping superalgebra of osp(1|2n) a nonstandard adjoint action, endowing it with a module structure. This allows, in particular, to construct a bosonic operator which anticommutes with all the fermionic generators and which appears to be the square root of a certain Casimir operator.
It is shown that the elliptic algebra ${\cal A}_{q,p}(\hat{sl}(2)_c)$ at the critical level c=-2 has a multidimensional center containing some trace-like operators t(z). A family of Poisson structures indexed by a non-negative integer and containing the q-deformed Virasoro algebra is constructed on this center. We show also that t(z) close an exchange algebra when p^m=q^{c+2} for m integer, they commute when in addition p=q^{2k} for k integer non-zero, and they belong to the center of ${\cal A}_{q,p}(\hat{sl}(2)_c)$ when k is odd. The Poisson structures obtained for t(z) in these classical limits contain the q-deformed Virasoro algebra, characterizing the structures at generic values of p, q and m as new ${\cal W}_{q,p}(sl(2))$ algebras.
We construct operators t(z) in the elliptic algebra introduced by Foda et al. ${\cal A}_{q,p}({\hat sl}(2)_c)$. They close an exchange algebra when p^m=q^{c+2} for m integer. In addition they commute when p=q^{2k} for k integer non-zero, and they belong to the center of ${\cal A}_{q,p}({\hat sl}(2)_c)$ when k is odd. The Poisson structures obtained for t(z) in these classical limits are identical to the q-deformed Virasoro Poisson algebra, characterizing the exchange algebras at generic values of p, q and m as new ${\cal W}_{q,p}(sl(2))$ algebras.
We present a classification of the possible quantum deformations of the supergroup $GL(1|1)$ and its Lie superalgebra $gl(1|1)$. In each case, the (super)commutation relations and the Hopf structures are explicitly computed. For each $R$ matrix, one finds two inequivalent coproducts whether one chooses an unbraided or a braided framework while the corresponding structures are isomorphic as algebras. In the braided case, one recovers the classical algebra $gl(1|1)$ for suitable limits of the deformation parameters but this is no longer true in the unbraided case.
It is shown that the elliptic algebra A_{p,q}(sl(2)_c) has a non trivial center at the critical level $c=-2$, generalizing the result of Reshetikhin and Semenov-Tian-Shansky for trigonometric algebras. A family of Poisson structures indexed by a non-negative integer $k$ is constructed on this center.
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 $ν$ of the anyons by q = exp(iπν). 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.
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.