Algebraic and Geometric Topology 5 (2005), paper no. 23, pages 537-562.

Yang-Baxter deformations of quandles and racks

Michael Eisermann


Abstract. Given a rack Q and a ring A, one can construct a Yang-Baxter operator c_Q: V tensor V --> V tensor V on the free A-module V = AQ by setting c_Q(x tensor y) = y tensor x^y for all x,y in Q. In answer to a question initiated by D.N. Yetter and P.J. Freyd, this article classifies formal deformations of c_Q in the space of Yang-Baxter operators. For the trivial rack, where x^y = x for all x,y, one has, of course, the classical setting of r-matrices and quantum groups. In the general case we introduce and calculate the cohomology theory that classifies infinitesimal deformations of c_Q. In many cases this allows us to conclude that c_Q is rigid. In the remaining cases, where infinitesimal deformations are possible, we show that higher-order obstructions are the same as in the quantum case.

Keywords. Yang-Baxter operator, r-matrix, braid group representation, deformation theory, infinitesimal deformation, Yang-Baxter cohomology

AMS subject classification. Primary: 17B37. Secondary: 18D10,20F36,20G42,57M25.

E-print: arXiv:math.QA/0409202

DOI: 10.2140/agt.2005.5.537

Submitted: 16 September 2004. (Revised: 18 May 2005.) Accepted: 3 June 2005. Published: 19 June 2005.

Notes on file formats

Michael Eisermann
Institut Fourier, Universite Grenoble I, 38402 St Martin d'Heres, France
Email: Michael.Eisermann@ujf-grenoble.fr
URL: www-fourier.ujf-grenoble.fr/~eiserm

AGT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to http://msp.warwick.ac.uk/.