Yang–Baxter deformations of quandles and racks

By Michael Eisermann. 2004

46 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/85d7549c54d74cede3ae29e4214dc2c3873b6408

How researchers see this paper

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In Algebra and Number Theory

1. Victoria Lebed
2. Petr Vojtvechovsk'y
3. Alexander Hulpke
4. David Stanovsk'y
5. Leandro Vendramin
6. Agata Smoktunowicz

In Geometry and Topology

1. Victoria Lebed
2. Leandro Vendramin
3. Petr Vojtvechovsk'y
4. Alexander Hulpke
5. David Stanovsk'y
6. Agata Smoktunowicz

In Mathematical Physics

1. Victoria Lebed
2. Leandro Vendramin
3. Petr Vojtvechovsk'y
4. Alexander Hulpke
5. David Stanovsk'y

Citations in the corpus

• Homologies of Algebraic Structures via Braidings and Quantum Shuffles (2012, methodology): “The last approach to “braided” cohomologies to be mentioned here is M.Eisermann’s Yang-Baxter cochain complex , cf. [12].”
• Connected quandles and transitive groups (2014, methodology): “e all colors used in a knot coloring fall into the same orbit of transitivity. From a broader perspective, quandles are a special type of set-theoretical solutions to the quantum Yang-Baxter equation [10, 12] and can be used to construct Hopf algebras [1]. There are indications, such as [13], that u”
• Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation (2015, background): “The interaction of these two enhancements reserves many open questions: (1) How can one relate the deformation theories of the braidings corresponding to a cycle set and its extension? (2) Do cycle set extensions form a class of deformations of the corresponding braiding? (3) Can the cycle set cohom”