Ibrahim Nonkane | Université d'Abomey_calavi (UAC) (original) (raw)
Papers by Ibrahim Nonkane
In this paper, we develop a new approach to the discriminant of a complete intersection curve in ... more In this paper, we develop a new approach to the discriminant of a complete intersection curve in the 3-dimensional projective space. By relying on the resultant theory, we first prove a new formula that allows us to define this discriminant without ambiguity and over any commutative ring, in particular in any characteristic. This formula also provides a new method for evaluating and computing this discriminant efficiently, without the need to introduce new variables as with the well-known Cayley trick. Then, we obtain new properties and computational rules such as the covariance and the invariance formulas. Finally, we show that our definition of the discriminant satisfies to the expected geometric property and hence yields an effective smoothness criterion for complete intersection space curves. Actually, we show that in the generic setting, it is the defining equation of the discriminant scheme if the ground ring is assumed to be a unique factorization domain.
Journal of physics, Nov 1, 2021
In this note, we study the action of the rational quantum Calogero-Moser system on polynomials ri... more In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.
African Diaspora Journal of Mathematics. New Series, Dec 1, 2012
In this paper we generalize some affine completeness properties of abelian groups to modules over... more In this paper we generalize some affine completeness properties of abelian groups to modules over commutative domains.
Journal of Mathematical Sciences, Jul 28, 2023
In this paper, we study an irreducible decomposition structure of the D-module direct image + (O ... more In this paper, we study an irreducible decomposition structure of the D-module direct image + (O n) for the finite map ∶ ℂ n → ℂ n ∕(S n 1 × ⋯ × S n r). We explicitly construct the simple components of + (O ℂ n) by providing their generators and their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the polynomial ring localized at the discriminant of. Furthermore, we study the action of invariant differential operators on the higher Specht polynomials.
arXiv (Cornell University), Oct 9, 2021
In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system ... more In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type B n. We endow the polynomial ring C[x 1 ,. .. .. . , x n ] with a structure of module over the Weyl algebra associated with the ring C[x 1 ,. .. , x n ] W of invariant polynomials under a reflections group W of type B n. Then we study the polynomial representation of the ring of invariant differential operators under the reflections group W. We use the group representation theory namely the higher Specht polynomials associated with the reflection group W and establish a decomposition of that structure by providing explicitly the generators of the simple components.
arXiv (Cornell University), Feb 6, 2017
In this paper, we develop a new approach to the discriminant of a complete intersection curve in ... more In this paper, we develop a new approach to the discriminant of a complete intersection curve in the 3-dimensional projective space. By relying on the resultant theory, we first prove a new formula that allows us to define this discriminant without ambiguity and over any commutative ring, in particular in any characteristic. This formula also provides a new method for evaluating and computing this discriminant efficiently, without the need to introduce new variables as with the well-known Cayley trick. Then, we obtain new properties and computational rules such as the covariance and the invariance formulas. Finally, we show that our definition of the discriminant satisfies to the expected geometric property and hence yields an effective smoothness criterion for complete intersection space curves. Actually, we show that in the generic setting, it is the defining equation of the discriminant scheme if the ground ring is assumed to be a unique factorization domain.
Journal of Mathematics Research, 2021
Valtancoli in his paper entitled (P. Valtancoli, Canonical transformations and minimal length, J.... more Valtancoli in his paper entitled (P. Valtancoli, Canonical transformations and minimal length, J. Math. Phys. 56, 122107 2015) has shown how the deformation of the canonical transformations can be made compatible with the deformed Poisson brackets. Based on this work and through an appropriate canonical transformation, we solve the problem of one dimensional (1D) damped harmonic oscillator at the classical limit of the Snyder-de Sitter (SdS) space. We show that the equations of the motion can be described by trigonometric functions with frequency and period depending on the deformed and the damped parameters. We eventually discuss the influences of these parameters on the motion of the system.
International Journal of Mathematics and Mathematical Sciences, Feb 18, 2021
In this paper, we study a decomposition D-module structure of the polynomial ring. en, we illustr... more In this paper, we study a decomposition D-module structure of the polynomial ring. en, we illustrate a geometric interpretation of the Specht polynomials. Using Brauer's characterization, we give a partial generalization of the fact that factors of the discriminant of a finite map π: specB ⟶ specA generate the irreducible factors of the direct image of B under the map π. Peel gave the construction of irreducible submodules of C[x 1 ,. .. , x n ] in the following way [4]. By a partition of n, we mean a sequence λ � (λ 1 ,. .. , λ r) such that
Afrika Matematika, 2018
In this paper, we study an irreducible decomposition structure of the D-module direct image π + (... more In this paper, we study an irreducible decomposition structure of the D-module direct image π + (O C n) for the finite map π : C n → C n /(S n1 × • • • × S nr). We explicitly construct the simple component of π + (O C n) by providing their generators and their multiplicities. Using an equivalence categories and the higher Specht polynomials, we describe a D-module decomposition of the of the polynomial ring localized at the discriminant of π. Furthermore we study the action invariants differential operators on the higher Specht polynomials.
In this paper, we study an irreducible decomposition structure of the D-module direct image π + (... more In this paper, we study an irreducible decomposition structure of the D-module direct image π + (O C n) for the finite map π : C n → C n /(S n1 × • • • × S nr). We explicitly construct the simple component of π + (O C n) by providing their generators and their multiplicities. Using an equivalence categories and the higher Specht polynomials, we describe a D-module decomposition of the of the polynomial ring localized at the discriminant of π. Furthermore we study the action invariants differential operators on the higher Specht polynomials.
In this paper we study the decomposition of the direct image of π+(OX) the polynomial ring OX as ... more In this paper we study the decomposition of the direct image of π+(OX) the polynomial ring OX as a D-module, under the map π : specOX → specO G(r,n) X , where O G(r,n) X is the ring of invariant polynomial under the action of the wreath product G(r, p) := Z/rZ ≀ Sn. We first describe the generators of the simple components of π+(OX) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the of the polynomial ring localized at the discriminant of π. Furthermore, we study the action invariants differential operators on the higher Specht polynomials. keywords: Direct image, Differential structure, complex reflection groups, Representation theory, wreath product, Higher Specht polynomials, Primitive idempotents, Reflection groups, Young diagram. Mathematics Subject Classification: Primary 13N10, Secondary 20C30.
Journal of Physics: Conference Series
In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system ... more In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type Bn. We endowed the polynomial ring C[x 1,..., xn ] with a structure of module over the Weyl algebra associated with the ring C[x 1,..., xn]W of invariant polynomials under a reflections group W of type Bn . Then we study the polynomials representation of the ring of invariant differential operators under the reflections group W. We make use of the theory of representation of groups namely the higher Specht polynomials associated with the reflection group W to yield a decomposition of that structure by providing explicitly the generators of its simple components.
Journal of Physics: Conference Series
In this paper, we study the action of the rational quantum Calogero-Moser system on polynomials. ... more In this paper, we study the action of the rational quantum Calogero-Moser system on polynomials. In this vein, we study polynomials ring over the complex field ℂ as a module over a ring of differential operators by elaborating its irreducible submodules. we endowed the polynomial ring ℂ[x 1 , …, x n ] with a differential structure by using directly the action of the Weyl algebra associated with the ring of symmetric polynomial ℂ[x 1 , …, x n ] Sn after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the symmetric group. We use the representation theory of symmetric groups to exhibit the generators of its simple components.
In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system ... more In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type Bn. We endowed the polynomial ring C[x1,. .. , xn] with a structure of module over the Weyl algebra associated with the ring C[x1,. .. , xn] W of invariant polynomials under a reflections group W of type Bn. Then we study the polynomials representation of the ring of invariant differential operators under the reflections group W. We make use of the theory of representation of groups namely the higher Specht polynomials associated with the reflection group W to yield a decomposition of that structure by providing explicitly the generators of its simple components.
In this paper we study the decomposition of the direct image of π + (O X) the polynomial ring O X... more In this paper we study the decomposition of the direct image of π + (O X) the polynomial ring O X as a D-module, under the map π : spec O X → spec O G(r,n) X , where O G(r,n) X is the ring of invariant polynomial under the action of the wreath product G(r, p) := Z/rZ ≀ S n. We first describe the generators of the simple components of π + (O X) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the of the polynomial ring localized at the discriminant of π. Furthermore, we study the action invariants differential operators on the higher Specht polynomials.
In this paper we study the decomposition of the direct image of π + (O X) the polynomial ring O X... more In this paper we study the decomposition of the direct image of π + (O X) the polynomial ring O X as a D-module, under the map π : spec O X → spec O G(r,n) X , where O G(r,n) X is the ring of invariant polynomial under the action of the wreath product G(r, p) := Z/rZ ≀ S n. We first describe the generators of the simple components of π + (O X) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the of the polynomial ring localized at the discriminant of π. Furthermore, we study the action invariants differential operators on the higher Specht polynomials.
The thesis consists of two papers that treat decomposition of modules over the Weyl algebra. In t... more The thesis consists of two papers that treat decomposition of modules over the Weyl algebra. In the first paper decomposition of holonomic modules is used to prove that there is a finite set of Noe ...
In this paper, we develop a new approach to the discriminant of a complete intersection curve in ... more In this paper, we develop a new approach to the discriminant of a complete intersection curve in the 3-dimensional projective space. By relying on the resultant theory, we first prove a new formula that allows us to define this discriminant without ambiguity and over any commutative ring, in particular in any characteristic. This formula also provides a new method for evaluating and computing this discriminant efficiently, without the need to introduce new variables as with the well-known Cayley trick. Then, we obtain new properties and computational rules such as the covariance and the invariance formulas. Finally, we show that our definition of the discriminant satisfies to the expected geometric property and hence yields an effective smoothness criterion for complete intersection space curves. Actually, we show that in the generic setting, it is the defining equation of the discriminant scheme if the ground ring is assumed to be a unique factorization domain.
Journal of physics, Nov 1, 2021
In this note, we study the action of the rational quantum Calogero-Moser system on polynomials ri... more In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.
African Diaspora Journal of Mathematics. New Series, Dec 1, 2012
In this paper we generalize some affine completeness properties of abelian groups to modules over... more In this paper we generalize some affine completeness properties of abelian groups to modules over commutative domains.
Journal of Mathematical Sciences, Jul 28, 2023
In this paper, we study an irreducible decomposition structure of the D-module direct image + (O ... more In this paper, we study an irreducible decomposition structure of the D-module direct image + (O n) for the finite map ∶ ℂ n → ℂ n ∕(S n 1 × ⋯ × S n r). We explicitly construct the simple components of + (O ℂ n) by providing their generators and their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the polynomial ring localized at the discriminant of. Furthermore, we study the action of invariant differential operators on the higher Specht polynomials.
arXiv (Cornell University), Oct 9, 2021
In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system ... more In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type B n. We endow the polynomial ring C[x 1 ,. .. .. . , x n ] with a structure of module over the Weyl algebra associated with the ring C[x 1 ,. .. , x n ] W of invariant polynomials under a reflections group W of type B n. Then we study the polynomial representation of the ring of invariant differential operators under the reflections group W. We use the group representation theory namely the higher Specht polynomials associated with the reflection group W and establish a decomposition of that structure by providing explicitly the generators of the simple components.
arXiv (Cornell University), Feb 6, 2017
In this paper, we develop a new approach to the discriminant of a complete intersection curve in ... more In this paper, we develop a new approach to the discriminant of a complete intersection curve in the 3-dimensional projective space. By relying on the resultant theory, we first prove a new formula that allows us to define this discriminant without ambiguity and over any commutative ring, in particular in any characteristic. This formula also provides a new method for evaluating and computing this discriminant efficiently, without the need to introduce new variables as with the well-known Cayley trick. Then, we obtain new properties and computational rules such as the covariance and the invariance formulas. Finally, we show that our definition of the discriminant satisfies to the expected geometric property and hence yields an effective smoothness criterion for complete intersection space curves. Actually, we show that in the generic setting, it is the defining equation of the discriminant scheme if the ground ring is assumed to be a unique factorization domain.
Journal of Mathematics Research, 2021
Valtancoli in his paper entitled (P. Valtancoli, Canonical transformations and minimal length, J.... more Valtancoli in his paper entitled (P. Valtancoli, Canonical transformations and minimal length, J. Math. Phys. 56, 122107 2015) has shown how the deformation of the canonical transformations can be made compatible with the deformed Poisson brackets. Based on this work and through an appropriate canonical transformation, we solve the problem of one dimensional (1D) damped harmonic oscillator at the classical limit of the Snyder-de Sitter (SdS) space. We show that the equations of the motion can be described by trigonometric functions with frequency and period depending on the deformed and the damped parameters. We eventually discuss the influences of these parameters on the motion of the system.
International Journal of Mathematics and Mathematical Sciences, Feb 18, 2021
In this paper, we study a decomposition D-module structure of the polynomial ring. en, we illustr... more In this paper, we study a decomposition D-module structure of the polynomial ring. en, we illustrate a geometric interpretation of the Specht polynomials. Using Brauer's characterization, we give a partial generalization of the fact that factors of the discriminant of a finite map π: specB ⟶ specA generate the irreducible factors of the direct image of B under the map π. Peel gave the construction of irreducible submodules of C[x 1 ,. .. , x n ] in the following way [4]. By a partition of n, we mean a sequence λ � (λ 1 ,. .. , λ r) such that
Afrika Matematika, 2018
In this paper, we study an irreducible decomposition structure of the D-module direct image π + (... more In this paper, we study an irreducible decomposition structure of the D-module direct image π + (O C n) for the finite map π : C n → C n /(S n1 × • • • × S nr). We explicitly construct the simple component of π + (O C n) by providing their generators and their multiplicities. Using an equivalence categories and the higher Specht polynomials, we describe a D-module decomposition of the of the polynomial ring localized at the discriminant of π. Furthermore we study the action invariants differential operators on the higher Specht polynomials.
In this paper, we study an irreducible decomposition structure of the D-module direct image π + (... more In this paper, we study an irreducible decomposition structure of the D-module direct image π + (O C n) for the finite map π : C n → C n /(S n1 × • • • × S nr). We explicitly construct the simple component of π + (O C n) by providing their generators and their multiplicities. Using an equivalence categories and the higher Specht polynomials, we describe a D-module decomposition of the of the polynomial ring localized at the discriminant of π. Furthermore we study the action invariants differential operators on the higher Specht polynomials.
In this paper we study the decomposition of the direct image of π+(OX) the polynomial ring OX as ... more In this paper we study the decomposition of the direct image of π+(OX) the polynomial ring OX as a D-module, under the map π : specOX → specO G(r,n) X , where O G(r,n) X is the ring of invariant polynomial under the action of the wreath product G(r, p) := Z/rZ ≀ Sn. We first describe the generators of the simple components of π+(OX) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the of the polynomial ring localized at the discriminant of π. Furthermore, we study the action invariants differential operators on the higher Specht polynomials. keywords: Direct image, Differential structure, complex reflection groups, Representation theory, wreath product, Higher Specht polynomials, Primitive idempotents, Reflection groups, Young diagram. Mathematics Subject Classification: Primary 13N10, Secondary 20C30.
Journal of Physics: Conference Series
In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system ... more In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type Bn. We endowed the polynomial ring C[x 1,..., xn ] with a structure of module over the Weyl algebra associated with the ring C[x 1,..., xn]W of invariant polynomials under a reflections group W of type Bn . Then we study the polynomials representation of the ring of invariant differential operators under the reflections group W. We make use of the theory of representation of groups namely the higher Specht polynomials associated with the reflection group W to yield a decomposition of that structure by providing explicitly the generators of its simple components.
Journal of Physics: Conference Series
In this paper, we study the action of the rational quantum Calogero-Moser system on polynomials. ... more In this paper, we study the action of the rational quantum Calogero-Moser system on polynomials. In this vein, we study polynomials ring over the complex field ℂ as a module over a ring of differential operators by elaborating its irreducible submodules. we endowed the polynomial ring ℂ[x 1 , …, x n ] with a differential structure by using directly the action of the Weyl algebra associated with the ring of symmetric polynomial ℂ[x 1 , …, x n ] Sn after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the symmetric group. We use the representation theory of symmetric groups to exhibit the generators of its simple components.
In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system ... more In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type Bn. We endowed the polynomial ring C[x1,. .. , xn] with a structure of module over the Weyl algebra associated with the ring C[x1,. .. , xn] W of invariant polynomials under a reflections group W of type Bn. Then we study the polynomials representation of the ring of invariant differential operators under the reflections group W. We make use of the theory of representation of groups namely the higher Specht polynomials associated with the reflection group W to yield a decomposition of that structure by providing explicitly the generators of its simple components.
In this paper we study the decomposition of the direct image of π + (O X) the polynomial ring O X... more In this paper we study the decomposition of the direct image of π + (O X) the polynomial ring O X as a D-module, under the map π : spec O X → spec O G(r,n) X , where O G(r,n) X is the ring of invariant polynomial under the action of the wreath product G(r, p) := Z/rZ ≀ S n. We first describe the generators of the simple components of π + (O X) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the of the polynomial ring localized at the discriminant of π. Furthermore, we study the action invariants differential operators on the higher Specht polynomials.
In this paper we study the decomposition of the direct image of π + (O X) the polynomial ring O X... more In this paper we study the decomposition of the direct image of π + (O X) the polynomial ring O X as a D-module, under the map π : spec O X → spec O G(r,n) X , where O G(r,n) X is the ring of invariant polynomial under the action of the wreath product G(r, p) := Z/rZ ≀ S n. We first describe the generators of the simple components of π + (O X) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the of the polynomial ring localized at the discriminant of π. Furthermore, we study the action invariants differential operators on the higher Specht polynomials.
The thesis consists of two papers that treat decomposition of modules over the Weyl algebra. In t... more The thesis consists of two papers that treat decomposition of modules over the Weyl algebra. In the first paper decomposition of holonomic modules is used to prove that there is a finite set of Noe ...