Volume 5, issue 2 (2005)

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Yang–Baxter deformations of quandles and racks

Michael Eisermann

Algebraic & Geometric Topology 5 (2005) 537–562
 arXiv: math.QA/0409202
Abstract

Given a rack $Q$ and a ring $\mathbb{A}$, one can construct a Yang–Baxter operator ${c}_{Q}:\phantom{\rule{0.3em}{0ex}}V\otimes V\to V\otimes V$ on the free $\mathbb{A}$–module $V=\mathbb{A}Q$ by setting ${c}_{Q}\left(x\otimes y\right)=y\otimes {x}^{y}$ 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
Mathematical Subject Classification 2000
Primary: 17B37
Secondary: 18D10, 20F36, 20G42, 57M25