CATEGORICAL ASPECTS OF VIRTUALITY AND SELF-DISTRIBUTIVITY
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.
References
- J. Knot Theory Ramifications 17(2), 231 (2008), DOI: 10.1142/S021821650800604X. Link, ISI, Google Scholar
-
J. S. Birman , Braids, Links, and Mapping Class Groups ,Annals of Mathematics Studies 82 ( Princeton University Press , Princeton, N.J. , 1974 ) . Google Scholar - Abh. Math. Sem. Univ. Hamburg 11, 171 (1936), DOI: 10.1007/BF02940721. Crossref, Google Scholar
- J. Homotopy Relat. Struct. 3(1), 13 (2008). ISI, Google Scholar
- A. S. Crans, Lie 2-algebras, ProQuest LLC, Ann Arbor, MI, Ph.D. Thesis, University of California, Riverside (2004) . Google Scholar
- C. R. Acad. Sci. Paris Sér. I Math. 313(9), 569 (1991). Google Scholar
S. Eilenberg and G. M. Kelly , Closed categories, Proc. Conf. Categorical Algebra (Springer, New York, 1966) pp. 421–562. Google Scholar- Topology 36(1), 123 (1997), DOI: 10.1016/0040-9383(95)00072-0. Crossref, ISI, Google Scholar
- J. Knot Theory Ramifications 1(4), 343 (1992), DOI: 10.1142/S0218216592000203. Link, Google Scholar
- J. Pure Appl. Algebra 23(1), 37 (1982), DOI: 10.1016/0022-4049(82)90077-9. Crossref, ISI, Google Scholar
- Adv. Math. 102(1), 20 (1993), DOI: 10.1006/aima.1993.1055. Crossref, ISI, Google Scholar
S. Kamada , Invariants of Knots and 3-Manifolds,Geometry and Topology Monographs 4 (2002) pp. 103–117. Crossref, Google Scholar- Osaka J. Math. 44(2), 441 (2007). ISI, Google Scholar
-
C. Kassel and V. Turaev , Braid Groups ,Graduate Texts in Mathematics 247 ( Springer , New York , 2008 ) . Crossref, Google Scholar - European J. Combin. 20(7), 663 (1999), DOI: 10.1006/eujc.1999.0314. Crossref, ISI, Google Scholar
- Comm. Algebra 39(12), 4679 (2011), DOI: 10.1080/00927872.2011.617280. Crossref, ISI, Google Scholar
L. H. Kauffman and D. Radford , Diagrammatic Morphisms and Applications Contemporary Mathematics 318 (American Mathematical Society, Providence, RI, 2003) pp. 113–140. Crossref, Google Scholar- J. Algebra 391, 152 (2013), DOI: 10.1016/j.jalgebra.2013.06.009. Crossref, ISI, Google Scholar
- V. Lebed, Braided Objects: Unifying Algebraic Structures and Categorifying Virtual Braids, Ph.D. Thesis, Université Paris 7 (2012) . Google Scholar
-
J.-L. Loday , Cyclic Homology ,Grundlehren der Mathematischen Wissenschaften 301 ( Springer-Verlag , Berlin , 1992 ) . Crossref, Google Scholar - Acta Appl. Math. 72(3), 295 (2002), DOI: 10.1023/A:1016258728022. Crossref, ISI, Google Scholar
- Mat. Sb. (N.S.) 119(161), 78 (1982). Google Scholar
- S. Nelson, Virtual knots, http://www.esotericka.org/cmc/vknots.html . Google Scholar
- Demonstratio Math. 44(4), 823 (2011). Google Scholar
- Commun. Algbra (2011). Google Scholar
- Pacific J. Math. 197(1), 241 (2001), DOI: 10.2140/pjm.2001.197.241. Crossref, ISI, Google Scholar
-
V. G. Turaev , Quantum Invariants of Knots and 3-Manifolds ,de Gruyter Studies in Mathematics 18 ( Walter de Gruyter & Co. , Berlin , 1994 ) . Google Scholar - J. Knot Theory Ramifications 10(5), 795 (2001), DOI: 10.1142/S0218216501001165. Link, ISI, Google Scholar
- Lett. Math. Phys. 23(2), 143 (1991), DOI: 10.1007/BF00703727. Crossref, ISI, Google Scholar
