#### 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

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