Geneviève Rollet - Academia.edu (original) (raw)
Papers by Geneviève Rollet
Theory in Biosciences, Jul 21, 2016
After an introduction to the general topic of models for a given locus of a diploid population wh... more After an introduction to the general topic of models for a given locus of a diploid population whose quadratic dynamics is determined by a fitness landscape, we consider more specifically the models that can be treated using genetic (or train) algebras. In this case, any quadratic offspring interaction can produce any type of offspring and after the use of specific changes of basis, we study the evolution and possible stability of some examples. We also consider some examples that cannot be treated using the framework of genetic algebras. Among these are bistochastic matrices.
HAL (Le Centre pour la Communication Scientifique Directe), 2004
We propose a dynamical extension of the quantum quadratic exchange algebras introduced by Freidel... more 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.
Теоретическая и математическая физика, 2014
Построены новые наборы представлений ранга n алгебры Темперли-Либа T LN (q), характеризующихся дв... more Построены новые наборы представлений ранга n алгебры Темперли-Либа T LN (q), характеризующихся двумя матрицами, удовлетворяющими обобщенному комплексному свойству Адамара. Приведены частичные классификации для двух матриц, в частности в случае, когда они сводятся к матрицам Фурье или Батсона.
Теоретическая и математическая физика, 2014
Дана классификация неоператорных матриц K, являющихся решением уравнения Склянина для квантового ... more Дана классификация неоператорных матриц K, являющихся решением уравнения Склянина для квантового отражения для всех R-матриц, получаемых из недавно определенных общих представлений алгебры Темперли-Либа T LN (√ n) адамаровского типа ранга n. R-матрицы характеризуются универсальным набором алгебраических уравнений в конкретном каноническом базисе, который единственным образом определяется из мастер-матрицы, связанной с выбранной реализацией алгебры Темперли-Либа.
Theoretical and Mathematical Physics, Nov 1, 2011
The general solutions of the reflection equation associated with Temperley-Lieb Rmatrices are con... more The general solutions of the reflection equation associated with Temperley-Lieb Rmatrices are constructed. Their parametrization is defined and the Hamiltonians of corresponding integrable spin systems are given.
Theoretical and Mathematical Physics, Apr 1, 2014
We classify non-operatorial matrices K solving Skylanin's quantum reflection equation for all R-m... more We classify non-operatorial matrices K solving Skylanin's quantum reflection equation for all R-matrices obtained from the newly defined general rank-n Hadamard type representations of the Temperley-Lieb algebra T L N (√ n). They are characterized by a universal set of algebraic equations in a specific canonical basis uniquely defined from the "Master matrix" associated to the chosen realization of Temperley-Lieb algebra
Symmetry Integrability and Geometry-methods and Applications, Sep 28, 2012
A complete classification of non-affine dynamical quantum R-matrices obeying the Gl n (C)-Gervais... more A complete classification of non-affine dynamical quantum R-matrices obeying the Gl n (C)-Gervais-Neveu-Felder equation is obtained without assuming either Hecke or weak Hecke conditions. More general dynamical dependences are observed. It is shown that any solution is built upon elementary blocks, which individually satisfy the weak Hecke condition. Each solution is in particular characterized by an arbitrary partition {I(i), i ∈ {1,. .. , n}} of the set of indices {1,. .. , n} into classes, I(i) being the class of the index i, and an arbitrary family of signs (I) I∈{I(i), i∈{1,...,n}} on this partition. The weak Hecketype R-matrices exhibit the analytical behaviour R ij,ji = f (I(i) Λ I(i) − I(j) Λ I(j)), where f is a particular trigonometric or rational function, Λ I(i) = j∈I(i) λ j , and (λ i) i∈{1,...,n} denotes the family of dynamical coordinates.
Journal of Mathematical Physics, 2002
We construct the classical r-matrix structure for the Lax formulation of BC N Ruijsenaars-Schneid... more We construct the classical r-matrix structure for the Lax formulation of BC N Ruijsenaars-Schneider systems proposed in [18]. The r-matrix structure takes a quadratic form similar to the A N Ruijsenaars-Schneider Poisson bracket behavior, although the dynamical dependence is more complicated. Commuting Hamiltonians stemming from the BC N 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.
Theoretical and Mathematical Physics, Feb 1, 2014
New sets of rank n-representations of Temperley-Lieb algebra T L N (q) are constructed. They are ... more New sets of rank n-representations of Temperley-Lieb algebra T L N (q) are constructed. They are characterized by two matrices obeying a generalization of the complex Hadamard property. Partial classifications for the two matrices are given, in particular when they reduce to Fourier or Butson matrices.
Physics Letters, Mar 1, 1996
We compute the classical r-matrix for the relativistic generalization of the Calogero-Moser model... more We compute the classical r-matrix for the relativistic generalization of the Calogero-Moser model, or Ruijsenaars-Schneider model, at all values of the speed-of-light parameter λ. We connect it with the non-relativistic Calogero-Moser r-matrix (λ → −1) and the λ = 1 sine-Gordon soliton limit.
arXiv (Cornell University), Aug 22, 2000
(Title: Structures in BC_N Ruijsenaars-Schneider models)
Cette these se situe au carrefour de la mecanique statistique sur reseau et des systemes dynamiqu... more Cette these se situe au carrefour de la mecanique statistique sur reseau et des systemes dynamiques discrets. Le lien entre ces domaines apparait a travers l'existence de groupes infinis discrets de symetries des modeles de mecanique statistique que nous representons par des groupes de transformations birationnelles. Nous rappelons tout d'abord quelques notions sur les equations de yang-baxter et leurs generalisations ; leur lien a l'integralite quantique et la construction d'un groupe infini discret d'automorphismes de l'ensemble de leurs solutions. Nous presentons ensuite des methodes algebriques et numeriques d'etude des transformations birationnelles en tant que telles. Pour cette analyse, nous construisons de nombreux systemes dynamiques discrets de natures differentes: certains sont integrables et leurs orbites definissent des courbes elliptiques, d'autres, par iterations, decrivent des varietes de dimension plus grande que deux de facon reguliere, mais apparaissent aussi des situations intermediaires entre regularite et chacs pour des transformations dites presque integrables. Enfin, nous appliquons ces methodes a des modeles de mecanique statistique, d'une part pour analyser leurs diagrammes de phases, d'autre part pour rechercher des situations favorables a l'integrabilite pour des modeles de dimension trois et plus
arXiv (Cornell University), Jul 7, 2016
After an introduction to the general topic of models for a given locus of a diploid population wh... more After an introduction to the general topic of models for a given locus of a diploid population whose quadratic dynamics is determined by a fitness landscape, we consider more specifically the models that can be treated using genetic (or train) algebras. In this setup, any quadratic offspring interaction can produce any type of offspring and after the use of specific changes of basis, we study the evolution and possible stability of some examples. We also consider some examples that cannot be treated using the framework of genetic algebras. Among these are bistochastic matrices.
HAL (Le Centre pour la Communication Scientifique Directe), 2002
We define the notion of C (2) N +1 Ruijsenaars-Schneider models and construct their Lax formulati... more We define the notion of C (2) N +1 Ruijsenaars-Schneider models and construct their Lax formulation. They are obtained by a particular folding of the A 2N +1 systems. Their commuting Hamiltonians are linear combinations of Koornwinder-van Diejen "external fields" Ruijsenaars-Schneider models, for specific values of the exponential onebody couplings but with the most general 2 double-poles structure as opposed to the formerly studied BC N case. Extensions to the elliptic potentials are briefly discussed.
We propose a classification of the solutions K to the semi-dynamical reflection equa- tion with c... more We propose a classification of the solutions K to the semi-dynamical reflection equa- tion 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.
Compositio Mathematica
We construct a correspondence between the set of partitions of a finite set M and the set of pair... more We construct a correspondence between the set of partitions of a finite set M and the set of pairs of walks to the same vertex on a graph giving the Bratteli diagram of the partition algebra on M. This is the precise analogue of the correspondence between the set of permutations of a finite set and the set of pairs of Young tableaux of the same shape, called the Robinson–Schensted correspondence.
Letters in Mathematical Physics
We define the notion of C (2)N+1Ruijsenaars–Schneider models and construct their Lax formulation.... more We define the notion of C (2)N+1Ruijsenaars–Schneider models and construct their Lax formulation. They are obtained by a particular folding of the A 2N+1 systems. Their commuting Hamiltonians are linear combinations of Koornwinder–van Diejen ‘external fields’ Ruijsenaars–Schneider models, for specific values of the exponential one-body couplings but with the most general two double-poles structure as opposed to the formerly studied BC N case. Extensions to the elliptic potentials are briefly discussed.
Теоретическая и математическая физика, 2014
Дана классификация неоператорных матриц K, являющихся решением уравнения Склянина для квантового ... more Дана классификация неоператорных матриц K, являющихся решением уравнения Склянина для квантового отражения для всех R-матриц, получаемых из недавно определенных общих представлений алгебры Темперли-Либа T LN (√ n) адамаровского типа ранга n. R-матрицы характеризуются универсальным набором алгебраических уравнений в конкретном каноническом базисе, который единственным образом определяется из мастер-матрицы, связанной с выбранной реализацией алгебры Темперли-Либа.
Theory in Biosciences, Jul 21, 2016
After an introduction to the general topic of models for a given locus of a diploid population wh... more After an introduction to the general topic of models for a given locus of a diploid population whose quadratic dynamics is determined by a fitness landscape, we consider more specifically the models that can be treated using genetic (or train) algebras. In this case, any quadratic offspring interaction can produce any type of offspring and after the use of specific changes of basis, we study the evolution and possible stability of some examples. We also consider some examples that cannot be treated using the framework of genetic algebras. Among these are bistochastic matrices.
HAL (Le Centre pour la Communication Scientifique Directe), 2004
We propose a dynamical extension of the quantum quadratic exchange algebras introduced by Freidel... more 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.
Теоретическая и математическая физика, 2014
Построены новые наборы представлений ранга n алгебры Темперли-Либа T LN (q), характеризующихся дв... more Построены новые наборы представлений ранга n алгебры Темперли-Либа T LN (q), характеризующихся двумя матрицами, удовлетворяющими обобщенному комплексному свойству Адамара. Приведены частичные классификации для двух матриц, в частности в случае, когда они сводятся к матрицам Фурье или Батсона.
Теоретическая и математическая физика, 2014
Дана классификация неоператорных матриц K, являющихся решением уравнения Склянина для квантового ... more Дана классификация неоператорных матриц K, являющихся решением уравнения Склянина для квантового отражения для всех R-матриц, получаемых из недавно определенных общих представлений алгебры Темперли-Либа T LN (√ n) адамаровского типа ранга n. R-матрицы характеризуются универсальным набором алгебраических уравнений в конкретном каноническом базисе, который единственным образом определяется из мастер-матрицы, связанной с выбранной реализацией алгебры Темперли-Либа.
Theoretical and Mathematical Physics, Nov 1, 2011
The general solutions of the reflection equation associated with Temperley-Lieb Rmatrices are con... more The general solutions of the reflection equation associated with Temperley-Lieb Rmatrices are constructed. Their parametrization is defined and the Hamiltonians of corresponding integrable spin systems are given.
Theoretical and Mathematical Physics, Apr 1, 2014
We classify non-operatorial matrices K solving Skylanin's quantum reflection equation for all R-m... more We classify non-operatorial matrices K solving Skylanin's quantum reflection equation for all R-matrices obtained from the newly defined general rank-n Hadamard type representations of the Temperley-Lieb algebra T L N (√ n). They are characterized by a universal set of algebraic equations in a specific canonical basis uniquely defined from the "Master matrix" associated to the chosen realization of Temperley-Lieb algebra
Symmetry Integrability and Geometry-methods and Applications, Sep 28, 2012
A complete classification of non-affine dynamical quantum R-matrices obeying the Gl n (C)-Gervais... more A complete classification of non-affine dynamical quantum R-matrices obeying the Gl n (C)-Gervais-Neveu-Felder equation is obtained without assuming either Hecke or weak Hecke conditions. More general dynamical dependences are observed. It is shown that any solution is built upon elementary blocks, which individually satisfy the weak Hecke condition. Each solution is in particular characterized by an arbitrary partition {I(i), i ∈ {1,. .. , n}} of the set of indices {1,. .. , n} into classes, I(i) being the class of the index i, and an arbitrary family of signs (I) I∈{I(i), i∈{1,...,n}} on this partition. The weak Hecketype R-matrices exhibit the analytical behaviour R ij,ji = f (I(i) Λ I(i) − I(j) Λ I(j)), where f is a particular trigonometric or rational function, Λ I(i) = j∈I(i) λ j , and (λ i) i∈{1,...,n} denotes the family of dynamical coordinates.
Journal of Mathematical Physics, 2002
We construct the classical r-matrix structure for the Lax formulation of BC N Ruijsenaars-Schneid... more We construct the classical r-matrix structure for the Lax formulation of BC N Ruijsenaars-Schneider systems proposed in [18]. The r-matrix structure takes a quadratic form similar to the A N Ruijsenaars-Schneider Poisson bracket behavior, although the dynamical dependence is more complicated. Commuting Hamiltonians stemming from the BC N 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.
Theoretical and Mathematical Physics, Feb 1, 2014
New sets of rank n-representations of Temperley-Lieb algebra T L N (q) are constructed. They are ... more New sets of rank n-representations of Temperley-Lieb algebra T L N (q) are constructed. They are characterized by two matrices obeying a generalization of the complex Hadamard property. Partial classifications for the two matrices are given, in particular when they reduce to Fourier or Butson matrices.
Physics Letters, Mar 1, 1996
We compute the classical r-matrix for the relativistic generalization of the Calogero-Moser model... more We compute the classical r-matrix for the relativistic generalization of the Calogero-Moser model, or Ruijsenaars-Schneider model, at all values of the speed-of-light parameter λ. We connect it with the non-relativistic Calogero-Moser r-matrix (λ → −1) and the λ = 1 sine-Gordon soliton limit.
arXiv (Cornell University), Aug 22, 2000
(Title: Structures in BC_N Ruijsenaars-Schneider models)
Cette these se situe au carrefour de la mecanique statistique sur reseau et des systemes dynamiqu... more Cette these se situe au carrefour de la mecanique statistique sur reseau et des systemes dynamiques discrets. Le lien entre ces domaines apparait a travers l'existence de groupes infinis discrets de symetries des modeles de mecanique statistique que nous representons par des groupes de transformations birationnelles. Nous rappelons tout d'abord quelques notions sur les equations de yang-baxter et leurs generalisations ; leur lien a l'integralite quantique et la construction d'un groupe infini discret d'automorphismes de l'ensemble de leurs solutions. Nous presentons ensuite des methodes algebriques et numeriques d'etude des transformations birationnelles en tant que telles. Pour cette analyse, nous construisons de nombreux systemes dynamiques discrets de natures differentes: certains sont integrables et leurs orbites definissent des courbes elliptiques, d'autres, par iterations, decrivent des varietes de dimension plus grande que deux de facon reguliere, mais apparaissent aussi des situations intermediaires entre regularite et chacs pour des transformations dites presque integrables. Enfin, nous appliquons ces methodes a des modeles de mecanique statistique, d'une part pour analyser leurs diagrammes de phases, d'autre part pour rechercher des situations favorables a l'integrabilite pour des modeles de dimension trois et plus
arXiv (Cornell University), Jul 7, 2016
After an introduction to the general topic of models for a given locus of a diploid population wh... more After an introduction to the general topic of models for a given locus of a diploid population whose quadratic dynamics is determined by a fitness landscape, we consider more specifically the models that can be treated using genetic (or train) algebras. In this setup, any quadratic offspring interaction can produce any type of offspring and after the use of specific changes of basis, we study the evolution and possible stability of some examples. We also consider some examples that cannot be treated using the framework of genetic algebras. Among these are bistochastic matrices.
HAL (Le Centre pour la Communication Scientifique Directe), 2002
We define the notion of C (2) N +1 Ruijsenaars-Schneider models and construct their Lax formulati... more We define the notion of C (2) N +1 Ruijsenaars-Schneider models and construct their Lax formulation. They are obtained by a particular folding of the A 2N +1 systems. Their commuting Hamiltonians are linear combinations of Koornwinder-van Diejen "external fields" Ruijsenaars-Schneider models, for specific values of the exponential onebody couplings but with the most general 2 double-poles structure as opposed to the formerly studied BC N case. Extensions to the elliptic potentials are briefly discussed.
We propose a classification of the solutions K to the semi-dynamical reflection equa- tion with c... more We propose a classification of the solutions K to the semi-dynamical reflection equa- tion 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.
Compositio Mathematica
We construct a correspondence between the set of partitions of a finite set M and the set of pair... more We construct a correspondence between the set of partitions of a finite set M and the set of pairs of walks to the same vertex on a graph giving the Bratteli diagram of the partition algebra on M. This is the precise analogue of the correspondence between the set of permutations of a finite set and the set of pairs of Young tableaux of the same shape, called the Robinson–Schensted correspondence.
Letters in Mathematical Physics
We define the notion of C (2)N+1Ruijsenaars–Schneider models and construct their Lax formulation.... more We define the notion of C (2)N+1Ruijsenaars–Schneider models and construct their Lax formulation. They are obtained by a particular folding of the A 2N+1 systems. Their commuting Hamiltonians are linear combinations of Koornwinder–van Diejen ‘external fields’ Ruijsenaars–Schneider models, for specific values of the exponential one-body couplings but with the most general two double-poles structure as opposed to the formerly studied BC N case. Extensions to the elliptic potentials are briefly discussed.
Теоретическая и математическая физика, 2014
Дана классификация неоператорных матриц K, являющихся решением уравнения Склянина для квантового ... more Дана классификация неоператорных матриц K, являющихся решением уравнения Склянина для квантового отражения для всех R-матриц, получаемых из недавно определенных общих представлений алгебры Темперли-Либа T LN (√ n) адамаровского типа ранга n. R-матрицы характеризуются универсальным набором алгебраических уравнений в конкретном каноническом базисе, который единственным образом определяется из мастер-матрицы, связанной с выбранной реализацией алгебры Темперли-Либа.