Claude ROGER - Academia.edu (original) (raw)

Papers by Claude ROGER

International audienceIn d = 2 with variables (x, t), the superalgebraic trick of adding a supple... more International audienceIn d = 2 with variables (x, t), the superalgebraic trick of adding a supplementary odd variable allows the construction of a " square root of time " , an operator D satisfying D 2 = ∂/∂t in superspace of dimension (2|1). We already used that trick to obtain a Miura transform in dimension(2|1) for non stationary Schrödinger type operators[5]. We shall discuss here the construction of an algebra of pseudodifferential symbols in dimension (2|1); that algebra generalizes the one for d = 1, used in construction of hierarchies from isospectral deformations of stationary Schrödinger type operators.En dimension d=2 en les variables (x,t), l'astuce superalgébrique d'ajouter une variable impaire supplémentaire permet la construction d'une racine carrée du temps, un opérateur D vérifiant D^2=∂/∂t dans le superespace de dimension (2|1). Nous avons utilisé cet artifice pour construire une transformée de Miura en dimension (2|1) pour des opérateurs de S...

Bulletin de la Société Royale des Sciences de Liège, 2020

We extend to the superalgebraic case the theory of Lie-Rinehart algebras and work out some exampl... more We extend to the superalgebraic case the theory of Lie-Rinehart algebras and work out some examples concerning the most popular samples of super manofilds

In d=2 with variables (x,t), the superalgebraic trick of adding a supplementary odd variable allo... more In d=2 with variables (x,t), the superalgebraic trick of adding a supplementary odd variable allows the construction of a "square root of time", an operator D satisfying D^2=∂/∂t in superspace of dimension (2|1). We used it to obtain a Miura transform in dimension (2|1) for non stationary Schrödinger type operators. We shall discuss here the construction of an algebra of pseudodifferential symbols in dimension (2|1); that algebra generalizes the one for d=1, used in construction of hierarchies from isospectral deformations of stationary Schrödinger type operators.

Publications du CRM, Université de Montreal, 2007

Letters in Mathematical Physics, 1989

Introducing the notion of an admissible graded Lie subalgebra M of the Nijenhuis-Richardson algeb... more Introducing the notion of an admissible graded Lie subalgebra M of the Nijenhuis-Richardson algebra A(V) of the vector space V, it is shown that each cohomology class of a subcomplex ~d of the Chevalley-Eilenbcrg complex A(M ~ V) extends in a cononical way as a graded cohomology class of weight-I of M. Applying this when V is the space N of smooth functions of a smooth manifold M, shows that the de Rham cohomology of M is induced by the graded cohomology of weight-I of the Schouten graded Lie algebra of M. This allows us to construct explicitly all l-differential, nc formal deformations of the Poisson bracket of a symplectic manifold. The construction also applies for an arbitrary Poisson manifold but leads to only part of these deformations when the structure degenerates, as shown by an example.

Функциональный анализ и его приложения, 1996

Theoretical and Mathematical Physics, 2011

Let S^lin:={a(t)(-2∂_t-∂_r^2+V(t,r) | a∈ C^∞(/2π), V∈ C^∞(/2π×)} be the space of Schrödinger oper... more Let S^lin:={a(t)(-2∂_t-∂_r^2+V(t,r) | a∈ C^∞(/2π), V∈ C^∞(/2π×)} be the space of Schrödinger operators in (1+1)-dimensions with periodic time-dependent potential. The action on S^lin of a large infinite-dimensional reparametrization group SV with Lie algebra RogUnt06,Unt08, called the Schrödinger-Virasoro group and containing the Virasoro group, is proved to be Hamiltonian for a certain Poisson structure on S^lin. More precisely, the infinitesimal action of appears to be part of a coadjoint action of a Lie algebra of pseudo-differential symbols, , of which is a quotient, while the Poisson structure is inherited from the corresponding Kirillov-Kostant-Souriau form.

We shall give a survey of classical examples, together with algebraic methods to deal with those ... more We shall give a survey of classical examples, together with algebraic methods to deal with those structures: graded algebra, cohomologies, cohomology operations. The corresponding geometric structures will be described(e.g., Lie algebroids), with particular emphasis on supergeometry, odd supersymplectic structures and their classification. Finally, we shall explain how BV-structures appear in Quantum Field Theory, as a version of functional integral quantization.

This article is concerned with an extensive study of a infinite-dimensional Lie algebra sv, intro... more This article is concerned with an extensive study of a infinite-dimensional Lie algebra sv, introduced in [14] in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra of invariance of the free Schrödinger equation and the central charge-free Virasoro algebra Vect(S1). We call sv the Schrödinger-Virasoro Lie algebra. We choose to present sv from a Newtonian geometry point of view first, and then in connection with conformal and Poisson geometry. We turn afterwards to its representation theory: realizations as Lie symmetries of field equations, coadjoint representation, coinduced representations in connection with Cartan’s prolongation method (yielding analogues of the tensor density modules for Vect(S1)), and finally Verma modules with a Kac determinant formula. We also present a detailed cohomogical study, providing in particular a classification of deformations and central extensions; there appears a non-local cocycle. in memory of Dani...

Geometric Methods in Physics XXXV, 2018

In d = 2 with variables (x, t), the superalgebraic trick of adding a supplementary odd variable a... more In d = 2 with variables (x, t), the superalgebraic trick of adding a supplementary odd variable allows the construction of a "square root of time", an operator D satisfying D 2 = ∂/∂t in superspace of dimension (2|1). We already used that trick to obtain a Miura transform in dimension(2|1) for non stationary Schrödinger type operators[5]. We shall discuss here the construction of an algebra of pseudodifferential symbols in dimension (2|1); that algebra generalizes the one for d = 1, used in construction of hierarchies from isospectral deformations of stationary Schrödinger type operators.

Advances in Pure and Applied Mathematics

We give an introduction to superalgebra, founded on the difference between even (commuting) and o... more We give an introduction to superalgebra, founded on the difference between even (commuting) and odd (anti-commuting) variables. We give a sketch of Graßmann’s work, and show how derivations of those structures induce various superalgebra structures, Lie superalgebras of Cartan type being obtained with even derivations, while odd derivations induce Jordan-type superalgebras.

Let S := {−2i∂t − ∂ 2 r + V (t, r) | V ∈ C (R/2πZ × R)} be the space of Schrödinger operators in ... more Let S := {−2i∂t − ∂ 2 r + V (t, r) | V ∈ C (R/2πZ × R)} be the space of Schrödinger operators in (1 + 1)-dimensions with periodic time-dependent potential. The action on S of a large infinitedimensional reparametrization group SV with Lie algebra sv [8, 10], called the Schrödinger-Virasoro group and containing the Virasoro group, is proved to be Hamiltonian for a certain symplectic structure on S . More precisely, the infinitesimal action of sv appears to be a projected coadjoint action of a Lie algebra of pseudo-differential symbols, g, of which sv is a quotient, while the symplectic structure is inherited from the corresponding Kirillov-Kostant-Souriau form.

Journal of Geometry and Symmetry in Physics

Given an associative supercommutative algebra equipped with an odd derivation, one considers the ... more Given an associative supercommutative algebra equipped with an odd derivation, one considers the space of vector fields it defines, and show, under suitable hypothesis, they form a Jordan superalgebra; in contrast with the Lie superalgebras of Virasoro type constructed from even derivations. Relations with AntiLie algebras studied by Ovsienko and collaborators are then shown.

Russian Mathematical Surveys - RUSS MATH SURVEY-ENGL TR, 1992

CONTENTSIntroduction § 1. Main theoremsChapter I. Algebra § 2. Moyal deformations of the Poisson ... more CONTENTSIntroduction § 1. Main theoremsChapter I. Algebra § 2. Moyal deformations of the Poisson bracket and *-product on \mathbb R^{2n} § 3. Algebraic construction § 4. Central extensions § 5. ExamplesChapter II. Deformations of the Poisson bracket and *-product on an arbitrary symplectic manifold § 6. Formal deformations: definitions § 7. Graded Lie algebras as a means of describing deformations § 8. Cohomology computations and their consequences § 9. Existence of a *-productChapter III. Extensions of the Lie algebra of contact vector fields on an arbitrary contact manifold §10. Lagrange bracket §11. Extensions and modules of tensor fieldsAppendix 1. Extensions of the Lie algebra of differential operatorsAppendix 2. Examples of equations of Korteweg-de Vries typeReferences Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Abstract Text Return: Query Results Return items starting with number Query Form Dat...

Theoretical and Mathematical Physics, 2011

Annales de l’institut Fourier, 1973

Étude des Γ-structures de codimension 1 sur la sphère S 2 Annales de l'institut Fourier, tome 23,... more Étude des Γ-structures de codimension 1 sur la sphère S 2 Annales de l'institut Fourier, tome 23, n o 4 (1973), p. 213-227 <http://www.numdam.org/item?id=AIF_1973__23_4_213_0> © Annales de l'institut Fourier, 1973, tous droits réservés. L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Annales de l’institut Fourier, 1973

Étude des Γ-structures de codimension 1 sur la sphère S 2 Annales de l'institut Fourier, tome 23,... more Étude des Γ-structures de codimension 1 sur la sphère S 2 Annales de l'institut Fourier, tome 23, n o 4 (1973), p. 213-227 <http://www.numdam.org/item?id=AIF_1973__23_4_213_0> © Annales de l'institut Fourier, 1973, tous droits réservés. L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

International audienceIn d = 2 with variables (x, t), the superalgebraic trick of adding a supple... more International audienceIn d = 2 with variables (x, t), the superalgebraic trick of adding a supplementary odd variable allows the construction of a " square root of time " , an operator D satisfying D 2 = ∂/∂t in superspace of dimension (2|1). We already used that trick to obtain a Miura transform in dimension(2|1) for non stationary Schrödinger type operators[5]. We shall discuss here the construction of an algebra of pseudodifferential symbols in dimension (2|1); that algebra generalizes the one for d = 1, used in construction of hierarchies from isospectral deformations of stationary Schrödinger type operators.En dimension d=2 en les variables (x,t), l'astuce superalgébrique d'ajouter une variable impaire supplémentaire permet la construction d'une racine carrée du temps, un opérateur D vérifiant D^2=∂/∂t dans le superespace de dimension (2|1). Nous avons utilisé cet artifice pour construire une transformée de Miura en dimension (2|1) pour des opérateurs de S...

Bulletin de la Société Royale des Sciences de Liège, 2020

We extend to the superalgebraic case the theory of Lie-Rinehart algebras and work out some exampl... more We extend to the superalgebraic case the theory of Lie-Rinehart algebras and work out some examples concerning the most popular samples of super manofilds

In d=2 with variables (x,t), the superalgebraic trick of adding a supplementary odd variable allo... more In d=2 with variables (x,t), the superalgebraic trick of adding a supplementary odd variable allows the construction of a "square root of time", an operator D satisfying D^2=∂/∂t in superspace of dimension (2|1). We used it to obtain a Miura transform in dimension (2|1) for non stationary Schrödinger type operators. We shall discuss here the construction of an algebra of pseudodifferential symbols in dimension (2|1); that algebra generalizes the one for d=1, used in construction of hierarchies from isospectral deformations of stationary Schrödinger type operators.

Publications du CRM, Université de Montreal, 2007

Letters in Mathematical Physics, 1989

Introducing the notion of an admissible graded Lie subalgebra M of the Nijenhuis-Richardson algeb... more Introducing the notion of an admissible graded Lie subalgebra M of the Nijenhuis-Richardson algebra A(V) of the vector space V, it is shown that each cohomology class of a subcomplex ~d of the Chevalley-Eilenbcrg complex A(M ~ V) extends in a cononical way as a graded cohomology class of weight-I of M. Applying this when V is the space N of smooth functions of a smooth manifold M, shows that the de Rham cohomology of M is induced by the graded cohomology of weight-I of the Schouten graded Lie algebra of M. This allows us to construct explicitly all l-differential, nc formal deformations of the Poisson bracket of a symplectic manifold. The construction also applies for an arbitrary Poisson manifold but leads to only part of these deformations when the structure degenerates, as shown by an example.

Функциональный анализ и его приложения, 1996

Theoretical and Mathematical Physics, 2011

Let S^lin:={a(t)(-2∂_t-∂_r^2+V(t,r) | a∈ C^∞(/2π), V∈ C^∞(/2π×)} be the space of Schrödinger oper... more Let S^lin:={a(t)(-2∂_t-∂_r^2+V(t,r) | a∈ C^∞(/2π), V∈ C^∞(/2π×)} be the space of Schrödinger operators in (1+1)-dimensions with periodic time-dependent potential. The action on S^lin of a large infinite-dimensional reparametrization group SV with Lie algebra RogUnt06,Unt08, called the Schrödinger-Virasoro group and containing the Virasoro group, is proved to be Hamiltonian for a certain Poisson structure on S^lin. More precisely, the infinitesimal action of appears to be part of a coadjoint action of a Lie algebra of pseudo-differential symbols, , of which is a quotient, while the Poisson structure is inherited from the corresponding Kirillov-Kostant-Souriau form.

We shall give a survey of classical examples, together with algebraic methods to deal with those ... more We shall give a survey of classical examples, together with algebraic methods to deal with those structures: graded algebra, cohomologies, cohomology operations. The corresponding geometric structures will be described(e.g., Lie algebroids), with particular emphasis on supergeometry, odd supersymplectic structures and their classification. Finally, we shall explain how BV-structures appear in Quantum Field Theory, as a version of functional integral quantization.

This article is concerned with an extensive study of a infinite-dimensional Lie algebra sv, intro... more This article is concerned with an extensive study of a infinite-dimensional Lie algebra sv, introduced in [14] in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra of invariance of the free Schrödinger equation and the central charge-free Virasoro algebra Vect(S1). We call sv the Schrödinger-Virasoro Lie algebra. We choose to present sv from a Newtonian geometry point of view first, and then in connection with conformal and Poisson geometry. We turn afterwards to its representation theory: realizations as Lie symmetries of field equations, coadjoint representation, coinduced representations in connection with Cartan’s prolongation method (yielding analogues of the tensor density modules for Vect(S1)), and finally Verma modules with a Kac determinant formula. We also present a detailed cohomogical study, providing in particular a classification of deformations and central extensions; there appears a non-local cocycle. in memory of Dani...

Geometric Methods in Physics XXXV, 2018

In d = 2 with variables (x, t), the superalgebraic trick of adding a supplementary odd variable a... more In d = 2 with variables (x, t), the superalgebraic trick of adding a supplementary odd variable allows the construction of a "square root of time", an operator D satisfying D 2 = ∂/∂t in superspace of dimension (2|1). We already used that trick to obtain a Miura transform in dimension(2|1) for non stationary Schrödinger type operators[5]. We shall discuss here the construction of an algebra of pseudodifferential symbols in dimension (2|1); that algebra generalizes the one for d = 1, used in construction of hierarchies from isospectral deformations of stationary Schrödinger type operators.

Advances in Pure and Applied Mathematics

We give an introduction to superalgebra, founded on the difference between even (commuting) and o... more We give an introduction to superalgebra, founded on the difference between even (commuting) and odd (anti-commuting) variables. We give a sketch of Graßmann’s work, and show how derivations of those structures induce various superalgebra structures, Lie superalgebras of Cartan type being obtained with even derivations, while odd derivations induce Jordan-type superalgebras.

Let S := {−2i∂t − ∂ 2 r + V (t, r) | V ∈ C (R/2πZ × R)} be the space of Schrödinger operators in ... more Let S := {−2i∂t − ∂ 2 r + V (t, r) | V ∈ C (R/2πZ × R)} be the space of Schrödinger operators in (1 + 1)-dimensions with periodic time-dependent potential. The action on S of a large infinitedimensional reparametrization group SV with Lie algebra sv [8, 10], called the Schrödinger-Virasoro group and containing the Virasoro group, is proved to be Hamiltonian for a certain symplectic structure on S . More precisely, the infinitesimal action of sv appears to be a projected coadjoint action of a Lie algebra of pseudo-differential symbols, g, of which sv is a quotient, while the symplectic structure is inherited from the corresponding Kirillov-Kostant-Souriau form.

Journal of Geometry and Symmetry in Physics

Given an associative supercommutative algebra equipped with an odd derivation, one considers the ... more Given an associative supercommutative algebra equipped with an odd derivation, one considers the space of vector fields it defines, and show, under suitable hypothesis, they form a Jordan superalgebra; in contrast with the Lie superalgebras of Virasoro type constructed from even derivations. Relations with AntiLie algebras studied by Ovsienko and collaborators are then shown.

Russian Mathematical Surveys - RUSS MATH SURVEY-ENGL TR, 1992

CONTENTSIntroduction § 1. Main theoremsChapter I. Algebra § 2. Moyal deformations of the Poisson ... more CONTENTSIntroduction § 1. Main theoremsChapter I. Algebra § 2. Moyal deformations of the Poisson bracket and *-product on \mathbb R^{2n} § 3. Algebraic construction § 4. Central extensions § 5. ExamplesChapter II. Deformations of the Poisson bracket and *-product on an arbitrary symplectic manifold § 6. Formal deformations: definitions § 7. Graded Lie algebras as a means of describing deformations § 8. Cohomology computations and their consequences § 9. Existence of a *-productChapter III. Extensions of the Lie algebra of contact vector fields on an arbitrary contact manifold §10. Lagrange bracket §11. Extensions and modules of tensor fieldsAppendix 1. Extensions of the Lie algebra of differential operatorsAppendix 2. Examples of equations of Korteweg-de Vries typeReferences Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Abstract Text Return: Query Results Return items starting with number Query Form Dat...

Theoretical and Mathematical Physics, 2011

Annales de l’institut Fourier, 1973

Étude des Γ-structures de codimension 1 sur la sphère S 2 Annales de l'institut Fourier, tome 23,... more Étude des Γ-structures de codimension 1 sur la sphère S 2 Annales de l'institut Fourier, tome 23, n o 4 (1973), p. 213-227 <http://www.numdam.org/item?id=AIF_1973__23_4_213_0> © Annales de l'institut Fourier, 1973, tous droits réservés. L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Annales de l’institut Fourier, 1973

Étude des Γ-structures de codimension 1 sur la sphère S 2 Annales de l'institut Fourier, tome 23,... more Étude des Γ-structures de codimension 1 sur la sphère S 2 Annales de l'institut Fourier, tome 23, n o 4 (1973), p. 213-227 <http://www.numdam.org/item?id=AIF_1973__23_4_213_0> © Annales de l'institut Fourier, 1973, tous droits réservés. L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/