arXiv subjects

Laiachi El Kaoutit

Publications and source records attributed to Laiachi El Kaoutit.

17 recordsLinked to original sources

arXiv2025

Asymmetries of thermal processes in open quantum systems

An intriguing phenomenon in non-equilibrium quantum thermodynamics is the asymmetry of thermal processes. Relaxation to thermal equilibrium is the most important dissipative process, being a key concept for the design of heat engines and refrigerators, contributing to the study of foundational questions of thermodynamics, and being relevant for quantum computing through the process of algorithmic cooling. Despite their importance, the dynamics of these processes are far from being understood. We show that the free relaxation to thermal equilibrium follows intrinsically different paths depending on whether it involves the temperature of the system to increase or to decrease. Our theory is exemplified using the recently developed thermal kinematics based on information geometry theory, utilizing three prototypical examples: a two-level system, the quantum harmonic oscillator, and a trapped quantum Brownian particle, in all cases showing faster heating than cooling under the appropriate conditions. A general understanding is obtained based on the spectral decomposition of the Liouvillian and the spectral gap of reciprocal processes.

quant-ph↗

arXiv2022

Correspondence theorems for Hopf algebroids with applications to affine groupoids

We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid schemes) in connection with the study of normal affine subgroupoids.

math.RA↗

arXiv2021

On Burnside Theory for groupoids

We explore the concept of conjugation between subgroupoids, providing several characterizations of the conjugacy relation (Theorem A in §1.2). We show that two finite groupoid-sets, over a locally strongly finite groupoid, are isomorphic, if and only if, they have the same number of fixed points with respect to any subgroupoid with a single object (Theorem B in §1.2). Lastly, we examine the ghost map of a finite groupoid and the idempotents elements of its Burnside algebra. The exposition includes an Appendix where we gather the main general technical notions that are needed along the paper.

math.GR↗

arXiv2020

Towards differentiation and integration between Hopf algebroids and Lie algebroids

In this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie-Rinehart algebras over the same algebra, the differentiation functor, which can be seen as an algebraic counterpart to the differentiation process from Lie groupoids to Lie algebroids. The other way around, we provide two interrelated contravariant functors form the category of Lie-Rinehart algebras to that of commutative Hopf algebroids, the integration functors. One of them yields a contravariant adjunction together with the differentiation functor. Under mild conditions, essentially on the base algebra, the other integration functor only induces an adjunction at the level of Galois Hopf algebroids. By employing the differentiation functor, we also analyse the geometric separability of a given morphism of Hopf algebroids. Several examples and applications are presented along the exposition.

math.RA↗

arXiv2020

The Hopf algebroid structure of differentially recursive sequences

A differentially recursive sequence over a differential field is a sequence of elements satisfying a homogeneous differential equation with non-constant coefficients (namely, Taylor expansions of elements of the field) in the differential algebra of Hurwitz series. The main aim of this paper is to explore the space of all differentially recursive sequences over a given field with a non-zero differential. We show that these sequences form a two-sided vector space that admits, in a canonical way, a structure of Hopf algebroid over the subfield of constant elements. We prove that it is the direct limit, as a left comodule, of all spaces of formal solutions of linear differential equations and that it satisfies, as Hopf algebroid, an additional universal property. When the differential on the base field is zero, we recover the Hopf algebra structure of linearly recursive sequences.

math.AG↗

arXiv2019

Categorified groupoid-sets and their Burnside ring

We explore the category of internal categories in the usual category of (right) group-sets, whose objects are referred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relation between two categorified group-sets is given by a particular equivalence between the underlying categories. We also exhibit some of the differences between the old Burnside theory and the new one. Lastly, we briefly explain how to extend these new techniques and concepts to the context of groupoids, employing the categories of (right) groupoid-sets, aiming by this to give an alternative approach to the classical Burnside ring of groupoids.

math.GR↗

arXiv2018

Mackey formula for bisets over groupoids

In this paper we establish the Mackey formula for groupoids, extending the well known formula in abstract groups context. This formula involves the notion of groupoid-biset, its orbit set and the tensor product over groupoids, as well as cosets by subgroupoids.

math.GR↗

arXiv2018

Finite dual of a cocommutative Hopf algebroid. Application to linear differential matrix equations and Picard-Vessiot theory

A fundamental tool of Differential Galois Theory is the assignment of an algebraic group to each finite-dimensional differential module over differential field in such a way that the category of differential modules it generates is equivalent, as a symmetric monoidal category, to the category of representations of the group. Its underlying set is then recognized as the group of differential automorphisms of the Picard-Vessiot field extension of the base field for this differential module. These results can be obtained by means of a Tannaka reconstruction process, applied to the abelian category of finite-dimensional differential modules. In this paper, we explore the possibility of extending this theory when the differential field is replaced by a more general differential ring $A$. In this case, it is reasonable to deal with differential modules which are finitely generated and projective over $A$. A major obstacle is that this category is not abelian, in contrast with the classical case when $A$ is a field. To overcome this difficulty, we develop some fundamental results concerning the \emph{finite dual}, hereby introduced, of a cocommutative Hopf algebroid, and a canonical monoidal functor sending (differential) modules to comodules. This functor is proved to be an equivalence of categories whenever a canonical ring homomorphism, which is introduced in this work, with domain in the aforementioned finite dual and with values in the convolutional ring, is injective. Module-theoretical sufficient conditions to get its injectivity are investigated. This machinery is applied to differential modules and their Picard-Vessiot theory.

math.RA↗

arXiv2018

Topological tensor product of bimodules, complete Hopf Algebroids and convolution algebras

Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. These results, in particular, apply to the smooth global sections of any Lie algebroid over a smooth (connected) manifold and they lead a new formal groupoid scheme to enter into the picture. In the Appendix we develop the necessary machinery behind complete Hopf algebroid constructions, which involves also the topological tensor product of filtered bimodules over filtered rings.

math.AC↗

arXiv2017

On geometrically transitive Hopf algebroids

This paper contributes to the characterization of a certain class of commutative Hopf algebroids. It is shown that a commutative flat Hopf algebroid with a non zero base ring and a nonempty character groupoid is geometrically transitive if and only if any base change morphism is a weak equivalence (in particular, if any extension of the base ring is Landweber exact), if and only if any trivial bundle is a principal bi-bundle, and if and only if any two objects are fpqc locally isomorphic. As a consequence, any two isotropy Hopf algebras of a geometrically transitive Hopf algebroid (as above) are weakly equivalent. Furthermore, the character groupoid is transitive and any two isotropy Hopf algebras are conjugated. Several other characterizations of these Hopf algebroids in relation to transitive groupoids are also given.

math.AC↗

arXiv2017

Comparing topologies on linearly recursive sequences

The space of linearly recursive sequences of complex numbers admits two distinguished topologies. Namely, the adic topology induced by the ideal of those sequences whose first term is $0$ and the topology induced from the Krull topology on the space of complex power series via a suitable embedding. We show that these topologies are not equivalent.

math.AC↗

arXiv2017

Morita theory for Hopf algebroids, principal bibundles, and weak equivalences

We show that two flat commutative Hopf algebroids are Morita equivalent if and only if they are weakly equivalent and if and only if there exists a principal bibundle connecting them. This gives a positive answer to a conjecture due to Hovey and Strickland. We also prove that principal (left) bundles lead to a bicategory together with a 2-functor from flat Hopf algebroids to trivial principal bundles. This turns out to be the universal solution for 2-functors which send weak equivalences to invertible 1-cells. Our approach can be seen as an algebraic counterpart to Lie groupoid Morita theory.

math.AT↗

arXiv2015

Functorial Constructions for Non-associative Algebras with Applications to Quasi-bialgebras

The aim of this paper is to establish a contravariant adjunction between the category of quasi-bialgebras and a suitable full subcategory of dual quasi-bialgebras, adapting the notion of finite dual to this framework. Various functorial constructions involving non-associative algebras and non-coassociative coalgebras are then carried out. Several examples illustrating our methods are expounded as well.

math.QA↗

arXiv2015

Hereditary triangular matrix comonads

We recognise Harada's generalized categories of diagrams as a particular case of modules over a monad defined on a finite direct product of additive categories. We work in the dual (albeit formally equivalent) situation, that is, with comodules over comonads. With this conceptual tool at hand, we obtain several of the Harada results with simpler proofs, some of them under more general hypothesis, besides with a characterization of the normal triangular matrix comonads that are hereditary, that is, of homological dimension less or equal than $1$. Our methods rest on a matrix representation of additive functors and natural transformations, which allows us to adapt typical algebraic manipulations from Linear Algebra to the additive categorical setting.

math.RA↗

arXiv2014

Invertible Bimodules, Miyashita Action in Monoidal Categories and Azumaya Monoids

In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automorphisms. In the symmetric case, we show that for certain Azumaya monoids, which are abundant in practice, the corresponding Miyashita action is always an isomorphism of groups. This generalizes Miyashita's classical result and sheds light on other applications of geometric nature which can not be treated using the classical theory. In order to illustrate our methods, we give a concrete application to the category of comodules over commutative (flat) Hopf algebroids. This obviously includes the special cases of split Hopf algebroids (action groupoids), which for instance cover the situation of the action of an affine algebraic group on an affine algebraic variety.

math.RA↗

arXiv2012

Morita base change in Hopf-cyclic (co)homology

In this paper, we establish the invariance of cyclic (co)homology of left Hopf algebroids under the change of Morita equivalent base algebras. The classical result on Morita invariance for cyclic homology of associative algebras appears as a special example of this theory. In our main application we consider the Morita equivalence between the algebra of complex-valued smooth functions on the classical 2-torus and the coordinate algebra of the noncommutative 2-torus with rational parameter. We then construct a Morita base change left Hopf algebroid over this noncommutative 2-torus and show that its cyclic (co)homology can be computed by means of the homology of the Lie algebroid of vector fields on the classical 2-torus.

math.QA↗

arXiv2009

Non-commutative integral forms and twisted multi-derivations

Non-commutative connections of the second type or hom-connections and associated integral forms are studied as generalisations of right connections of Manin. First, it is proven that the existence of hom-connections with respect to the universal differential graded algebra is tantamount to the injectivity, and that every finitely cogenerated injective module admits a hom-connection with respect to any differential graded algebra. The bulk of the paper is devoted to describing a method of constructing hom-connections from twisted multi-derivations. The notion of a free twisted multi-derivation is introduced and the induced first order differential calculus is described. It is shown that any free twisted multi-derivation on an algebra A induces a unique hom-connection on A that vanishes on the dual basis for the module of one-forms. To any flat hom-connection \nabla on A one associates a chain complex, termed a complex of integral forms on A. The canonical cokernel morphism to the zeroth homology space is called a \nabla-integral. Examples of free twisted multi-derivations, hom-connections and corresponding integral forms are provided by covariant calculi on Hopf algebras or quantum groups and on quantum spaces. The example of a flat hom-connection within the 3D left-covariant differential calculus on the quantum group SL(2) is described in full detail. A descent of hom-connections to the base algebra of a faithfully flat Hopf-Galois extension or a principal comodule algebra is studied. As an example, a hom-connection on the standard quantum Podles sphere is presented. In both cases the complex of integral forms is shown to be isomorphic to the de Rham complex, and the \nabla-integrals coincide with Hopf-theoretic integrals or invariant (Haar) measures.

math.QA↗