Yang-Baxter deformations of quandles and racks

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.

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-23.abs.html

References in corpus (2)

• Set-theoretical solutions to the quantum Yang-Baxter equation
• On set-theoretical solutions of the quantum Yang-Baxter equation

Cited by in corpus (12)

• Homology of left non-degenerate set-theoretic solutions to the Yang-Baxter equation
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• Quandle cohomology, extensions and automorphisms
• Torsion in one-term distributive homology
• Dehn quandles of groups and orientable surfaces
• Idempotents, free products and quandle coverings
• Presentations of Dehn quandles
• Quantum invariants of framed links from ternary self-distributive cohomology
• Zero-divisors and idempotents in quandle rings
• Quantum Cocycle Invariants of Knots from Yang-Baxter Cohomology
• Cocycle Deformations of Algebraic Identities and R-matrices
• Enumeration of involutory latin quandles, Bruck loops and commutative automorphic loops of odd prime power order