Braided Objects: Unifying Algebraic Structures and Categorifying Virtual Braids
type: Dataset
description: This thesis is devoted to an abstract theory of braided objects and its applications to a study of algebraic and topological structures. Part I presents our general homology theory for braided vector spaces and braided modules, based on the quantum co-shuffle coproduct. The construction of structural braidings characterizing different algebraic structures - self-distributive (SD) structures, associative / Leibniz algebras, their representations - allows then to generalize and unify familiar homologies. Loday's hyper-boundaries and certain homology operations are efficiently treated via our braided tools. We further introduce a concept of braided system and multi-braided module over it. This enables a thorough study of bialgebras, crossed products, bimodules, Yetter-Drinfel'd and Hopf (bi)modules: their braided interpretation, homologies and adjoint actions. A theory of multi-braided tensor products of algebras gives a unifying context for Heisenberg and Drinfel'd doubles, the algebras X of Cibils-Rosso and Y and Z of Panaite. Part III is topology-oriented. We start with a hom-set type categorification of virtual braid groups in terms of braided objects in a symmetric category (SC). This double braiding approach provides a source of representations of V Bn and a new categorical treatment for Manturov's virtual racks and the twisted Burau representation. We then define SD structures in an arbitrary SC and endow them with a braiding. The associativity and Jacobi identities in an SC are interpreted as SD conditions. Hopf algebras enter in the SD framework as well. Braided techniques from part I give a homology theory of categorical SD structures.
landing page: https://theses.hal.science/tel-00775857
keyword: algebra-x
keyword: algebraic-homology
keyword: algebre-de-battage-quantique
keyword: algebre-de-leibniz
keyword: algebre-x
keyword: auto-distributivite-categorique
keyword: bimodule-de-hopf
keyword: braided-module
keyword: braided-object
keyword: braided-system
keyword: caractere
keyword: categorical-self-distributivity
keyword: character
keyword: complexe-de-koszul
keyword: crossed-product
keyword: double-de-drinfel-
keyword: double-de-heisenberg
keyword: drinfeld-double
keyword: free-virtual-quandle
keyword: free-virtual-shelf
keyword: groupes-de-tresses-virtuelles
keyword: heisenberg-double
keyword: hochschild-homology
keyword: homologie-algebrique
keyword: homologie-de-hochschild
keyword: homologie-de-quandlerack
keyword: hopf-bimodule
keyword: hyper-bord-de-loday
keyword: infoeu-reposemanticsdoctoralthesis
keyword: koszul-complex
keyword: leibniz-algebra
keyword: lodays-hyper-boundaries
keyword: mathmath-atmathematics-mathalgebraic-topology-mathat
keyword: mathmath-ctmathematics-mathcategory-theory-mathct
keyword: mathmath-ktmathematics-mathk-theory-and-homology-mathkt
keyword: mathmath-qamathematics-mathquantum-algebra-mathqa
keyword: module-de-yetter-drinfel-d
keyword: module-tresse
keyword: multi-braided-tensor-product
keyword: objet-tresse
keyword: produit-croise
keyword: produit-tensoriel-multi-tresse
keyword: quantum-shuffle-algebra
keyword: r-matrice
keyword: r-matrix
keyword: rack-virtuel
keyword: rackquandle-homology
keyword: representation-de-burau-tordue
keyword: structures-auto-distributives-libres
keyword: systeme-tresse
keyword: theses
keyword: twisted-burau-representation
keyword: virtual-braid-groups
keyword: virtual-rack
keyword: yetter-drinfel-d-module
issued: 2026-05-15T07:39:11.600881
modified: 2026-05-15T07:39:11.600900
language: en
contact point: Organization
  fn: CCSD

test_moissonnage_selune
type: Agent

Braided Objects: Unifying Algebraic Structures and Categorifying Virtual Braids
type: Distribution
format: HTML
access URL: https://theses.hal.science/tel-00775857
issued: 2026-05-15T07:39:11.768044
modified: 2026-05-15T07:39:11.417802