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L. Frappat

Publications and source records attributed to L. Frappat.

At least 19 recordsLinked to original sources
arXiv2025

Quadratic Poisson brackets for the Camassa--Holm peakons

We establish quadratic Poisson brackets for the generalized Camassa--Holm peakon structure introduced in \cite{AFR23}. The calculation is based on the halving of the spectral parameter dependent $r$-matrix used to define the linear Poisson structure of this model. This quadratic structure, together with the linear one, establish the bi-Hamiltonian structure of the generalized Camassa--Holm peakon model. When the deformation parameter tends to $\pm2$, the spectral parameter dependence drops out, and we recover the linear and quadratic Poisson structure of the Camassa--Holm peakon model. When the spectral parameter tends to the fixed points of the involution defining the halving, we recover the Ragnisco--Bruschi deformation of the Camassa--Holm peakon model, thereby establishing a new quadratic Poisson structure thereof.

nlin.SI↗
arXiv2023

$\lambda$-Griffiths polynomials: Bispectrality and biorthogonality

We introduce a generalization of bivariate Griffiths polynomials depending on an additional parameter $\lambda$. These $\lambda$-Griffiths polynomials are bivariate, bispectral and biorthogonal. For two specific values of the parameter $\lambda$, they become orthogonal. One of the value is related to the usual bivariate Griffiths polynomials, while the second value produces new orthogonal bivariate polynomials.

math-ph↗
arXiv2023

A new integrable structure associated to the Camassa-Holm peakons

We provide a closed Poisson algebra involving the Ragnisco--Bruschi generalization of peakon dynamics in the Camassa--Holm shallow-water equation. This algebra is generated by three independent matrices. From this presentation, we propose a one-parameter integrable extension of their structure. It leads to a new $N$-body peakon solution to the Camassa--Holm shallow-water equation depending on two parameters. We present two explicit constructions of a (non-dynamical) $r$-matrix formulation for this new Poisson algebra. The first one relies on a tensorization of the $N$-dimensional auxiliary space by a 4-dimensional space. We identify a family of Poisson commuting quantities in this framework, including the original ones. This leads us to constructing a second formulation identified as a spectral parameter representation.

nlin.SI↗
arXiv2022

Integrable quadratic structures in peakon models

We propose realizations of the Poisson structures for the Lax representations of three integrable $n$-body peakon equations, Camassa--Holm, Degasperis--Procesi and Novikov. The Poisson structures derived from the integrability structures of the continuous equations yield quadratic forms for the $r$-matrix representation, with the Toda molecule classical $r$-matrix playing a prominent role. We look for a linear form for the $r$-matrix representation. Aside from the Camassa--Holm case, where the structure is already known, the two other cases do not allow such a presentation, with the noticeable exception of the Novikov model at $n=2$. Generalized Hamiltonians obtained from the canonical Sklyanin trace formula for quadratic structures are derived in the three cases.

nlin.SI↗
arXiv2020

Algebraic structure of classical integrability for complex sine-Gordon

The algebraic structure underlying the classical $r$-matrix formulation of the complex sine-Gordon model is fully elucidated. It is characterized by two matrices $a$ and $s$, components of the $r$ matrix as $r=a-s$. They obey a modified classical reflection/Yang--Baxter set of equations, further deformed by non-abelian dynamical shift terms along the dual Lie algebra $su(2)^*$. The sign shift pattern of this deformation has the signature of the twisted boundary dynamical algebra. Issues related to the quantization of this algebraic structure and the formulation of quantum complex sine-Gordon on those lines are introduced and discussed.

nlin.SI↗
arXiv2019

Elliptic deformation of $\mathcal{W}_N$-algebras

We construct $q$-deformations of quantum $\mathcal{W}_N$ algebras with elliptic structure functions. Their spin $k+1$ generators are built from $2k$ products of the Lax matrix generators of ${\mathcal{A}_{q,p}(\widehat{gl}(N)_c)}$). The closure of the algebras is insured by a critical surface condition relating the parameters $p,q$ and the central charge $c$. Further abelianity conditions are determined, either as $c=-N$ or as a second condition on $p,q,c$. When abelianity is achieved, a Poisson bracket can be defined, that we determine explicitly. One connects these structures with previously built classical $q$-deformed $\mathcal{W}_N$ algebras and quantum $\mathcal{W}_{q,p}(\mathfrak{sl}_N)$.

math.QA↗
arXiv2017

Dynamical centers for the elliptic quantum algebra ${\mathcal{B}}_{q,λ}(\hat{gl}_2)_c$

We identify generating functionals that satisfy dynamical exchange relations with the Lax matrices defining the face-type elliptic quantum algebra ${\mathcal{B}}_{q,λ}(\hat{gl}_{2})_c$, when the central charge takes the two possible values $c =\pm 2$. These generating functionals are constructed as quadratic trace-like objects in terms of the Lax matrices. The obtained structures are characterized as "dynamical centers", i.e. the centrality property is deformed by dynamical shifts. For these values, the functionals define (genuine) abelian subalgebras of ${\mathcal{B}}_{q,λ}(\hat{gl}_{2})_c$.

math-ph↗
arXiv2016

3-state Hamiltonians associated to solvable 33-vertex models

Using the nested coordinate Bethe ansatz, we study 33-vertex models, where only one global charge with degenerate eigenvalues exists and each site possesses three internal degrees of freedom. In the context of Markovian processes, they correspond to diffusing particles with two possible internal states which may be exchanged during the diffusion (transmutation). The first step of the nested coordinate Bethe ansatz is performed providing the eigenvalues in terms of rapidities. We give the constraints ensuring the consistency of the computations. These rapidities also satisfy Bethe equations involving $4\times 4$ R-matrices, solutions of the Yang--Baxter equation which implies new constraints on the models. We solve them allowing us to list all the solvable 33-vertex models.

math-ph↗
arXiv2016

Deformed Virasoro algebras from elliptic quantum algebras

We revisit the construction of deformed Virasoro algebras from elliptic quantum algebras of vertex type, generalizing the bilinear trace procedure proposed in the 90's. It allows us to make contact with the vertex operator techniques that were introduced separately at the same period. As a by-product, the method pinpoints two critical values of the central charge for which the center of the algebra is extended, as well as (in the $gl(2)$ case) a Liouville formula.

math-ph↗
arXiv2015

A new braid-like algebra for Baxterisation

We introduce a new Baxterisation for R-matrices that depend separately on two spectral parameters. The Baxterisation is based on a new algebra, close to but different from the braid group. This allows us to recover the R-matrix of the multi-species generalization of the totally asymmetric simple exclusion process with different hopping rates.

math-ph↗
arXiv2014

R-matrices of three-state Hamiltonians solvable by Coordinate Bethe Ansatz

We review some of the strategies that can be implemented to infer an $R$-matrix from the knowledge of its Hamiltonian. We apply them to the classification achieved in arXiv:1306.6303, on three state $U(1)$-invariant Hamiltonians solvable by CBA, focusing on models for which the $S$-matrix is not trivial. For the 19-vertex solutions, we recover the $R$-matrices of the well-known Zamolodchikov--Fateev and Izergin--Korepin models. We point out that the generalized Bariev Hamiltonian is related to both main and special branches studied by Martins in arXiv:1303.4010, that we prove to generate the same Hamiltonian. The 19-vertex SpR model still resists to the analysis, although we are able to state some no-go theorems on its $R$-matrix. For 17-vertex Hamiltonians, we produce a new $R$-matrix.

math-ph↗
arXiv2013

Classification of three-state Hamiltonians solvable by Coordinate Bethe Ansatz

We classify all Hamiltonians with rank 1 symmetry, acting on a periodic three-state spin chain, and solvable through (generalisation of) the coordinate Bethe ansatz (CBA). We obtain in this way four multi-parametric extensions of the known 19-vertex Hamiltonians (such as Zamolodchikov-Fateev, Izergin-Korepin, Bariev Hamiltonians). Apart from the 19-vertex Hamiltonians, there exists 17-vertex and 14-vertex Hamiltonians that cannot be viewed as subcases of the 19-vertex ones. In the case of 17-vertex Hamiltonian, we get a generalization of the genus 5 special branch found by Martins, plus three new ones. We get also two 14-vertex Hamiltonians. We solve all these Hamiltonians using CBA, and provide their spectrum, eigenfunctions and Bethe equations. A special attention is made to provide the specifications of our multi-parametric Hamiltonians that give back known Hamiltonians.

math-ph↗
arXiv2010

Including a phase in the Bethe equations of the Hubbard model

We compute the Bethe equations of generalized Hubbard models, and study their thermodynamical limit. We argue how they can be connected to the ones found in the context of AdS/CFT correspondence, in particular with the so-called dressing phase problem. We also show how the models can be interpreted, in condensed matter physics, as integrable multi-leg Hubbard models.

hep-th↗
arXiv2009

Bethe equations for generalized Hubbard models

We compute the eigenfunctions, energies and Bethe equations for a class of generalized integrable Hubbard models based on gl(n|m)\oplus gl(2) superalgebras. The Bethe equations appear to be similar to the Hubbard model ones, up to a phase due to the integration of a subset of `simple' Bethe equations. We discuss relations with AdS/CFT correspondence, and with condensed matter physics.

hep-th↗
arXiv2009

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

math-ph↗
arXiv2009

Bethe Ansatz for the Universal Weight Function

We consider universal off-shell Bethe vectors given in terms of Drinfeld realization of the algebra $U_q(\widehat{gl}_N)$ [arXiv:math/0610517,arXiv:0711.2819]. We investigate ordering properties of the product of the transfer matrix and these vectors. We derive that these vectors are eigenvectors of the transfer matrix if their Bethe parameters satisfy the universal Bethe equations [arXiv:math-ph/0512037].

math.QA↗