Genevieve Rollet

Construction of Dynamical Quadratic Algebras

We propose a dynamical extension of the quantum quadratic exchange algebras introduced by Freidel and Maillet. It admits two distinct fusion structures. A simple example is provided by the scalar Ruijsenaars-Schneider model. The semi-classical limit is also described.

Reflection k-matrices related to Temperley-Lieb R-matrices

We construct general solutions of the reflection equation associated with Temperley-Lieb R-matrices. We find their parameterization and obtain the Hamiltonians of the corresponding integrable spin systems.

Commuting quantum traces for quadratic algebras

Consistent tensor products on auxiliary spaces, hereafter denoted “fusion procedures,” and commuting transfer matrices are defined for general quadratic algebras, nondynamical and dynamical, inspired by results on reflection algebras. Applications of these procedures then yield integer-indexed families of commuting Hamiltonians.

The discrete Painlevé I equations: transcendental integrability and asymptotic solutions

The integrability of the discrete Painleve I equation (dP-I) is reviewed and its integrability studied. We establish the existence of a conserved quantity which is algebraic in the case of the autonomous dP-I equation and it is argued that the non-autonomous dP-I map has a non-algebraic invariant. Our analysis leads to, among other results the construction of asymptotic solutions with interesting structures.

Temperley-Lieb R-matrices from generalized Hadamard matrices

We construct new sets of rank n-representations of the Temperley-Lieb algebra TLN(q) that are characterized by two matrices with a generalized complex Hadamard property. We give partial classifications for the two matrices, in particular, in the case where they reduce to Fourier or Butson matrices.

Structures in BCN Ruijsenaars–Schneider models

We construct the classical r-matrix structure for the Lax formulation of BCN Ruijsenaars–Schneider systems proposed in Commun. Math. Phys., 115, 127 (1988). The r-matrix structure takes a quadratic form similar to the AN Ruijsenaars–Schneider Poisson bracket behavior, although the dynamical dependence is more complicated. Commuting Hamiltonians stemming from the BCN Ruijsenaars–Schneider Lax matrix are shown to be linear combinations of particular Koornwinder–van Diejen “external fields” Ruijsenaars–Schneider models, for specific values of the exponential one-body couplings. Uniqueness of such commuting Hamiltonians is established once the first of them and the general analytic structure are given.

Classification of the Solutions of Constant Rational Semi-Dynamical Reflection Equations

Abstract.We propose a classification of the solutions K to the semi-dynamical reflection equation with constant rational structure matrices associated to rational scalar Ruijsenaars-Schneider model. Four sets of solutions are identified and simple analytic transformations generate all solutions from these sets.Communicated by Petr Kulish

Classification of Non-Affine Non-Hecke Dynamical R-Matrices

A complete classification of non-affine dynamical quantum

C(2)N+1 Ruijsenaars–Schneider Models

R

On discrete evolutionary dynamics driven by quadratic interactions

-matrices obeying the