Set-theoretical solutions to the quantum Yang-Baxter equation

Abstract

In 1992 V$.$Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions to the quantum Yang-Baxter equation, i.e. solutions given by a permutation R of the set $X\times X$, where X is a fixed set. In this paper we study such solutions, which in addition satisfy the unitarity and nondegeneracy conditions. We discuss the geometric and algebraic interpretations of such solutions, introduce several constructions of them, and make first steps towards their classification.

36 pages, amstex, figures, revised and expanded, contains q-alg/9707027 as a small subset

References in corpus (1)

• Permutation-type solutions to the Yang-Baxter and other n-simplex equations

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• Cohomology and extensions of braces
• Finite quotients of groups of I-type
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• Localization of unitary braid group representations
• A noncommutative discrete potential KdV lift
• A characterization of finite multipermutation solutions of the Yang-Baxter equation
• Tetrahedron maps and symmetries of three dimensional integrable discrete equations
• Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation
• The retraction relation for biracks
• Nilpotency of skew braces and multipermutation solutions of the Yang-Baxter equation
• On skew braces and their ideals (with an Appendix by Agata Smoktunowicz)
• Poisson Yang-Baxter maps with binomial Lax matrices
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• Computing skew left braces of small orders
• Skew braces of size $pq$
• Set theoretic Yang-Baxter equation, braces and Drinfeld twists
• Enumeration of set-theoretic solutions to the Yang-Baxter equation
• Heaps of modules and affine spaces
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• The Floquet Baxterisation
• A combinatorial approach to noninvolutive set-theoretic solutions of the Yang-Baxter equation
• Decomposition theorems for involutive solutions to the Yang-Baxter equation
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• Set-theoretic solutions of the Yang-Baxter equation associated to weak braces
• Cocyclic braces and indecomposable cocyclic solutions of the Yang-Baxter equation
• Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses
• Yang-Baxter operators in symmetric categories
• Tropical R and Tau Functions
• Combinatorial solutions to the reflection equation
• Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew braces
• Abelian quandles and quandles with abelian structure group
• Inverse semi-braces and the Yang-Baxter equation
• Distributive biracks and solutions of the Yang-Baxter equation
• Nilpotency in left semi-braces
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• Etale representations for reductive algebraic groups with one-dimensional center
• Skew braces of size $p^{2}q$ I: abelian type
• Isotropy in Group Cohomology
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• A Universal algebraic approach to rack coverings
• Indecomposable involutive solutions of the Yang-Baxter equation of multipermutational level 2 with abelian permutation group
• "Set-theoretical" solutions of the quantum Yang-Baxter equation and a class of Garside groups
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• Crystal limits of compact semisimple quantum groups as higher-rank graph algebras
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