Symmetric Categories as a Means of Virtualization for Braid Groups and Racks

Victoria Lebed (1)

1 IMJ - Institut de Mathématiques de Jussieu

In this paper we propose, firstly, a categorification of virtual braid groups and groupoids in terms of "locally" braided objects in a symmetric category (SC), and, secondly, a definition of self-distributive structures (SDS) in an arbitrary SC. SDS are shown to produce braided objects in a SC. As for examples, we interpret associativity and the Jacobi identity in a SC as generalized self-distributivity, thus endowing associative and Leibniz algebras with a "local" braiding. Our "double braiding" approach provides a natural interpretation for virtual racks and twisted Burau representation. Homology of categorical SDS is defined, generalizing rack, bar and Leibniz differentials. Free virtual SDS and the faithfulness of corresponding representations of virtual braid groups/groupoids are also discussed.

Mots clés: virtual braid groups; (free) virtual shelves and racks; braided categories; categorical self-distributivity; Leibniz algebras; rack homology; Chevalley-Eilenberg homology; (twisted) Burau representation; Dehornoy order

Domaines: Catégories et ensembles [math.CT]; Topologie algébrique [math.AT]; Théorie des représentations [math.RT]

Victoria Lebed. Symmetric Categories as a Means of Virtualization for Braid Groups and Racks. 2012. ⟨hal-00709240v2⟩