Let X and Y be Hilbert C *-modules over a C *-algebra, and ϕ : X × Y → [0, ∞) be a function. A (not necessarily linear) mapping f : X → Y is called a ϕ-perturbed adjointable mapping if there exists a (not necessarily linear) mapping g : Y → X such that In this paper, we investigate the stability of adjointable mappings on Hilbert C *-modules and discuss the case where the modules are self-dual....

We extend the Larson–Sweedler theorem for weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We establish the autonomous monoidal category of the modules of a weak Hopf algebra A and show the semisimplicity of the unit and the invertible modules of A. We also reveal the connection of these modules to lef...

Let Λ and Γ be artin algebras and ΛUΓ a faithfully balanced selforthogonal bimodule. In this paper, we first introduce the notion of k-Gorenstein modules with respect to ΛUΓ and then characterize it in terms of the U -resolution dimension of some special injective modules and the property of the functors Ext(Ext(−, U), U) preserving monomorphisms, which develops a classical result of Auslander....

Let Λ and Γ be artin algebras and ΛUΓ a faithfully balanced selforthogonal bimodule. In this paper, we first introduce the notion of k-Gorenstein modules with respect to ΛUΓ and then characterize it in terms of the U -resolution dimension of some special injective modules and the property of the functors Exti(Exti(−, U), U) preserving monomorphisms, which develops a classical result of Auslande...

It is well-known that a ring R is semiperfect if and only if RR (or RR) is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if RR (or RR) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie d...

Let H be a finite-dimensional quasibialgebra. We show that H is a quasi-Hopf algebra if and only if the monoidal category of its finite-dimensional left modules is rigid, if and only if a structure theorem for Hopf modules over H holds. We also show that a dual structure theorem for Hopf modules over a coquasibialgebra H holds if and only if the category of finite-dimensional right H-comodules ...

It is known that to every (1|1) dimensional supercurve X there is associated a dual supercurve X̂ , and a superdiagonal ∆ ⊂ X × X̂. We establish that the categories of D-modules on X , X̂ and ∆ are equivalent. This follows from a more general result about D-modules and purely odd submersions. The equivalences preserve tensor products, and take vector bundles to vector bundles. Line bundles with co...

We study the basic monoidal properties of the category of Hopf modules for a coquasi Hopf algebra. In particular we discuss the so called fundamental theorem that establishes a monoidal equivalence between the category of comodules and the category of Hopf modules. We present a categorical proof of Radford’s S formula for the case of a finite dimensional coquasi Hopf algebra, by establishing a ...

The aim of this work is to describe some operations in the category of regular holonomic D-modules with support a normal crossing and variation zero introduced in [2]. These operations will allow us to compute the characteristic cycle of the local cohomology supported on homogeneous prime ideals of these modules. In particular, we will be able to describe their Bass and dual Bass numbers.

Software Platform is a middleware component of Smart Space to coordinate and manage all modules. Location-awareness is a common feature of many modules. Current several typical methods for distributed systems can hardly be competent for both the role of Software Platform and accommodating locationawareness simultaneously. Aiming at this, we present our method: SLAP (Smart Location-awareness-Acc...