Yang–Baxter deformations of quandles and racks

Eisermann, Michael

doi:10.2140/agt.2005.5.537

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Yang–Baxter deformations of quandles and racks

Michael Eisermann

Algebraic & Geometric Topology 5 (2005) 537–562

DOI: 10.2140/agt.2005.5.537

arXiv: math.QA/0409202

Abstract

Given a rack $Q$ and a ring $A$ , one can construct a Yang–Baxter operator $c_{Q} : V \otimes V \to V \otimes V$ on the free $A$ –module $V = A Q$ by setting $c_{Q} (x \otimes y) = y \otimes x^{y}$ 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

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Publication

Received: 16 September 2004

Revised: 18 May 2005

Accepted: 3 June 2005

Published: 19 June 2005

Authors

Michael Eisermann
Institut Fourier Université Grenoble I 38402 St Martin d’Hères France

Volume 5, issue 2 (2005)