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.
%0 Journal Article
%1 Eisermann2004a
%A Eisermann, Michael
%D 2005
%J Algebr. Geom. Topol.
%K algebra homology quandles yang-baxter
%P 537-562
%T Yang-Baxter deformations of quandles and racks
%U http://dx.doi.org/10.2140/agt.2005.5.537
%V 5
%X 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.
@article{Eisermann2004a,
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.
},
added-at = {2009-05-22T17:01:48.000+0200},
author = {Eisermann, Michael},
biburl = {https://www.bibsonomy.org/bibtex/21de2e9691f15f75fc4b808cb87dbfccf/njj},
description = {Yang-Baxter deformations of quandles and racks},
interhash = {5e7fc265298e81189d120cc9e53964e7},
intrahash = {1de2e9691f15f75fc4b808cb87dbfccf},
journal = {Algebr. Geom. Topol.},
keywords = {algebra homology quandles yang-baxter},
pages = {537-562},
timestamp = {2009-05-22T17:01:48.000+0200},
title = {Yang-Baxter deformations of quandles and racks},
url = {http://dx.doi.org/10.2140/agt.2005.5.537},
volume = 5,
year = 2005
}