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Braided Objects: Unifying Algebraic Structures and Categorifying Virtual Braids

Abstract : 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.
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Contributor : Victoria Lebed <>
Submitted on : Monday, January 14, 2013 - 4:00:49 PM
Last modification on : Wednesday, December 9, 2020 - 3:10:37 PM
Long-term archiving on: : Monday, April 15, 2013 - 4:04:45 AM


  • HAL Id : tel-00775857, version 1


Victoria Lebed. Braided Objects: Unifying Algebraic Structures and Categorifying Virtual Braids. Category Theory [math.CT]. Université Paris-Diderot - Paris VII, 2012. English. ⟨tel-00775857⟩



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