Algebraic and Geometric Topology 5 (2005),
paper no. 23, pages 537-562.
Yang-Baxter deformations of quandles and racks
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.
Yang-Baxter operator, r-matrix, braid group representation, deformation theory, infinitesimal deformation, Yang-Baxter cohomology
AMS subject classification.
Submitted: 16 September 2004.
(Revised: 18 May 2005.)
Accepted: 3 June 2005.
Published: 19 June 2005.
Notes on file formats
Institut Fourier, Universite Grenoble I, 38402 St Martin d'Heres, France
AGT home page
These pages are not updated anymore.
They reflect the state of
For the current production of this journal, please refer to