Indecomposable Set-Theoretical Solutions to the Quantum Yang–Baxter Equation on a Set with a Prime Number of Elements

By Pavel Etingof, Robert M. Guralnick, A. Soloviev. 2000

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In this paper we show that all indecomposable nondegenerate set-theoretical solutions to the Quantum Yang-Baxter equation on a set of prime order are affine, which allows us to give a complete and very simple classification of such solutions. This result is a natural application of the general theory of set-theoretical solutions to the quantum Yang-Baxter equation. It is also a generalization of the corresponding statement for involutive set-theoretical solutions proved in an earlier paper of P.E. and A.S. with T.Schedler. In order to prove our main result, we use the classification theory developed by the third author (based on the ideas of Lu, Yan, and Zhu) to reduce the problem to a group-theoretical statement: a finite group with trivial center generated by a conjugacy class of prime order is a subgroup of the affine group. Unfortunalely, we did not find an elementary proof of this statement, and our proof relies on the classification of outer automorphisms of finite simple groups.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/78fd68dff359c69212dbf0b50e3d520d4440379e

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Citations in the corpus

Homology theory for the set-theoretic Yang–Baxter equation and knot invariants from generalizations of quandles (2002, background): “The set theoretic solutions to the Yang-Baxter equations are studied in detail in the papers [9, 10, 23, 30].”
Connected quandles and transitive groups (2014, methodology): “This has been known since [11] and, like in [11], our proof relies on a deep result of Kazarin about conjugacy classes of prime power order.”
Biquandles and virtual links (2004, background): “Work related to this can be found in [7,8,20,21,26].”
Problems on invariants of knots and 3-manifolds (2004, background): “Graña) It is shown, in [116] in terms of set theoretical solutions of the quantum Yang-Baxter equation, that a connected quandle of prime order p is isomorphic to the Alexander quandle Z[t±]/(p, t − a) for some a.”
From racks to pointed Hopf algebras (2002, background): “lian). It is immediate to see that (Z3,⊲2) is the only indecomposable crossed set with 3 elements; and that any crossed set with 3 elements is either trivial or isomorphic to (Z3,⊲2). It is proved in [EGS] that any indecomposable rack of prime order pis either isomorphic to (Zp,⊲q) for q∈ Z× p or it”
Problems on skew left braces (2018, background): “The classification of indecomposable solutions with a prime number of elements appears in [35, 37].”