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.

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Michael Eisermann
Institut Fourier, Universite Grenoble I, 38402 St Martin d'Heres, France

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