arXiv subjects

El-kaïoum M. Moutuou

Publications and source records attributed to El-kaïoum M. Moutuou.

9 recordsLinked to original sources

arXiv2025

Kubo-Martin-Schwinger states of Path-structured Flow in Directed Brain Synaptic Networks

The brain's synaptic network, characterized by parallel connections and feedback loops, drives interaction pathways between neurons through a large system with infinitely many degrees of freedom. This system is best modeled by the graph C*-algebra of the underlying directed graph, the Toeplitz-Cuntz-Krieger (TCK) algebra, which captures the diversity of path-structured flow connectivity. Equipped with the gauge action, the TCK algebra defines an {\em algebraic quantum system}, and here we demonstrate that its thermodynamic properties provide a natural framework for describing the dynamic mappings of potential flow pathways within the network. Specifically, the KMS states of this system represent the stationary distributions of a non-Markovian stochastic process with memory decay, capturing how influence propagates along exponentially weighted paths, and yield global statistical measures of neuronal interactions. Applied to the {\em C. elegans} synaptic network, our framework reveals that neurolocomotor neurons emerge as the primary hubs of incoming path-structured flow at inverse temperatures where the entropy of KMS states peaks. This finding aligns with experimental evidence of the foundational role of locomotion in {\em C. elegans} behavior, suggesting that functional centrality may arise from the topological embedding of neurons rather than solely from local physiological properties. Our results highlight the potential of algebraic quantum methods and graph algebras to uncover patterns of functional organization in complex systems and neuroscience.

q-bio.NC↗

arXiv2018

Crossed extensions and equivalences of topological 2-groupoids

We provide concrete models for generalized morphisms and Morita equivalences of topological 2-groupoids by introducing the notions of crossings and crossed extensions of groupoid crossed modules. A systematic study of these objects is elaborated and an explicit description of how they do yield a groupoid and geometric picture of weak 2-groupoid morphisms is presented. Specifically, we construct a weak 3-category whose objects are crossed modules of topological groupoids and in which weak 1-isomorphisms correspond to Morita equivalences in the "category" of topological 2-groupoids.

math.AT↗

arXiv2017

Finitely stable racks and rack representations

We define a new class of racks, called finitely stable racks, which, to some extent, share various flavors with Abelian groups. Characterization of finitely stable Alexander quandles is established. Further, we study twisted rack dynamical systems, construct their cross-products, and introduce representation theory of racks and quandles. We prove several results on the {\em strong} representations of finite connected involutive racks analogous to the properties of finite Abelian groups. Finally, we define the {\em Pontryagin} dual of a rack as an Abelian group which, in the finite involutive connected case, coincides with the set of its strong irreducible representations.

math.RT↗

arXiv2015

Foundations of topological racks and quandles

We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then introduced providing a first step towards a general continuous cohomology theory for topological racks and quandles.

math.GT↗

arXiv2014

Graded Brauer groups of a groupoid with involution

We define a group $RBr(\mathcal{G})$ containing, in a sense, the graded complex and orthogonal Brauer groups of a locally compact groupoid $\mathcal{G}$ equipped with an involution. When the involution is trivial, we show that the new group naturally provides a generalization of Donovan-Karoubi's graded orthogonal Brauer group $GBrO$. More generally, it is shown to be a direct summand of the well-known graded complex Brauer goup. In addition, we prove that $RBr(\mathcal{G})$ identifies with a direct sum of a Real cohomology group and the abelian group $RExt(\mathcal{G},U(1))$ of Real graded $U(1)$-central extensions. A cohomological picture is then given.

math.FA↗

arXiv2013

Equivariant $\KK$-theory for generalised actions and Thom isomorphism in groupoid twisted $\K$-theory

We develop equivariant KK-theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K-theory.

math.KT↗

arXiv2012

On groupoids with involutions and their cohomology

We extend the definitions and main properties of graded extensions to the category of locally compact groupoids endowed with involutions. We introduce Real Čech cohomology, which is an equivariant-like cohomology theory suitable for the context of groupoids with involutions. The Picard group of such a groupoid is discussed and is given a cohomological picture. Eventually, we generalize Crainic's result, about the differential cohomology of a proper Lie groupoid with coefficients in a given representation, to the topological case.

math.OA↗

arXiv2011

Twistings of KR for Real groupoids

B-fields over a groupoid with involution are defined as Real graded Dixmier-Douady bundles. We use these to introduce the Real graded Brauer group which constitutes the set of twistings for Atiyah's KR-functor in the category of locally compact groupoids with involutions. We interpret this group in terms of groupoid extensions and elements of some equivariant Cech cohomology theory. The construction of the twisted KR-functor is outlined.

math.KT↗

arXiv2011

Equivalence of Fell Systems and their Reduced C*-Algebras

This paper is aimed at investigating links between Fell bundles over Morita equivalent groupoids and their corresponding reduced C*-algebras. Mainly, we review the notion of Fell pairs over a Morita equivalence of groupoids, and give the analogue of the Renault's Equivalence Theorem for the reduced C*-algebras of equivalent Fell systems. Eventually, we will use this theorem to connect the reduced C*-algebra of an S^1-central groupoid extension to that of its associated Dixmier-Douady bundle.

math.OA↗