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CATEGORICAL ASPECTS OF VIRTUALITY AND SELF-DISTRIBUTIVITY

VICTORIA LEBED

Institut de Mathématiques de Jussieu, Université Paris-Diderot (Paris 7), Case 7012, 2 place Jussieu, 75251 Paris Cedex 05, France

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https://doi.org/10.1142/S0218216513500454Cited by:5 (Source: Crossref)

Abstract

This paper revolves around two main results. First, we propose a "hom-set" type categorification of virtual braid groups and positive virtual braid monoids in terms of "locally" braided objects in a symmetric category (SC). This "double braiding" approach provides a rich source of representations, and offers a natural categorical interpretation for virtual racks and for the twisted Burau representation. Second, we define self-distributive (SD) structures in an arbitrary SC. SD structures are shown to produce braided objects in a SC. As for examples, we interpret the associativity and the Jacobi identity in a SC as generalized self-distributivity, thus endowing associative and Leibniz algebras with a (pre-)braiding. A homology theory of categorical SD structures is developed using the "braided" techniques from [Lebed, Homologies of algebraic structures via braidings and quantum shuffles, to appear in J. Algebra], generalizing rack, bar, Leibniz and other familiar complexes.

Keywords:

AMSC: 20F36, 18D35, 18D10, 17D99, 55N35, 57M27, 17A32

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