Patrice P. Ntumba

Pairings of Sheaves of A-Modules

It is proved that for any free A-modules F and ε of finite rank on some C-algebraized space (X,A) a degenerate A-bilinear morphism Φ : F × ε → A induces a non-degenerate A-bilinear morphism Φ : F/ε ⊥ × ε/F ⊥ → A, where ε ⊥ and F ⊥ are the orthogonal sub-A-modules associated with ε and F, respectively. This result generalizes the finite case of the classical result, which states that given two vector spaces W and V, paired into a field k, the induced vector spaces W/V ⊥ and V/W ⊥ have the same dimension. Some related results are discussed as well.

On the way to Frölicher Lie groups

In this paper, we start by showing that one can obtain by way of diffeomorphisms some classical smooth curves and surfaces. The notion of bundles is also introduced in the category of Frölicher spaces, and one may carry on investigating the analogue notion of ordinary G-bundles. We then review tangent and cotangent bundles of an arbitrary Frölicher space X and amend Cherenacks proof [4] that these notions coincide with the usual ones when X is a smooth manifold. Finally, we prove that the analogues of some results in [10] and [11] are true in the category of Frölicher Lie groups. But as for differential groups, the unicity of an integral curve of a vector field does pose problems. Nevertheless, one can show that the map c′ : R → TX, sending t ↦ c *t (1 t ), where c : R → X is a curve on a Frölicher space X, is smooth. Most of the results in our paper hold because the category of Frölicher spaces is Cartesian closed [5].

Cartan's proof for the Darboux theorem in A-modules

Abstract We refer to [4] for a proof of the (affine) Darboux theorem in the category A-Mod X of A-modules, defined on a fixed topological space X. Hereby, we present another proof of the same theorem, based on E. Cartans approach, keeping, as is done in [4], the condition affixed to the coefficient algebra sheaf A, that is, A satisfies the inverse-closed section condition.

Characterization of Vector Fields on the Frölicher Standard n-Simplex, Imbedded in R n , and Hamiltonian Formalism on Differential Spaces

This paper is an attempt to the symplectization of smooth unusual, but standard imbedded subspaces of R n like a n-simplex, for the purpose of modelling Hamiltonian and Lagrangian systems thereon. We show that these subspaces are obtained by the process of smoothly gluing portions of R n , all considered as Frölicher spaces. On working with some of them, we characterize smooth vector fields that have a flow. Then, we show that both the modern and classical mechanical systems can be written in a more larger category than that of Frölicher spaces, which is the category of differential spaces.

On filtration of Clifford 𝒜-algebras and localization of 𝒜-modules

Abstract Let (X, 𝒜) be an algebraized space. We consider sheaves of Clifford Aalgebras (Clifford 𝒜X -algebras, for short) on X associated with arbitrary quadratic 𝒜-modules and study the the natural filtration of Clifford 𝒜-algebras. We show that for every 𝒜-algebra sheaf Ɛ, endowed with a regular filtration, one obtains a new graded 𝒜-algebra sheaf, denoted Gr(Ɛ), which turns out to be 𝒜-isomorphic to Ɛ. Finally, we also consider localization of 𝒜-modules at prime ideal subsheaves and at subsheaves induced by maximal ideals.

Reflexivity of orthogonality in A-modules

Abstract In this paper, as part of a project initiated by A. Mallios consisting of exploring new horizons for Abstract Differential Geometry (à la Mallios), [5, 6, 7, 8], such as those related to the classical symplectic geometry, we show that essential results pertaining to biorthogonality in pairings of vector spaces do hold for biorthogonality in pairings of A-modules. We single out that orthogonality is reflexive for orthogonally convenient pairings of free A-modules of finite rank, governed by non-degenerate A-morphisms, and where A is a PID (Corollary 3.8). For the rank formula (Corollary 3.3), the algebra sheaf A is assumed to be a PID. The rank formula relates the rank of an A-morphism and the rank of the kernel (sheaf) of the same A-morphism with the rank of the source free A-module of the A-morphism concerned.