Yang–Baxter deformations of quandles and racks

Eisermann, Michael¹

¹Institut Fourier, Université Grenoble I, 38402 St Martin d’Hères, France

Algebraic and Geometric Topology, Tome 5 (2005) no. 2, pp. 537-562

Résumé

Given a rack Q and a ring A, one can construct a Yang–Baxter operator cQ: V ⊗ V → V ⊗ V on the free A–module V = AQ by setting cQ(x ⊗ y) = y ⊗ xy for all x,y ∈ Q. In answer to a question initiated by D N Yetter and P J Freyd, this article classifies formal deformations of cQ in the space of Yang–Baxter operators. For the trivial rack, where xy = 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 cQ. In many cases this allows us to conclude that cQ 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.

Détail

DOI : 10.2140/agt.2005.5.537

Keywords: Yang–Baxter operator, $r$–matrix, braid group representation, deformation theory, infinitesimal deformation, Yang–Baxter cohomology

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