On set-theoretical solutions of the quantum Yang-Baxter equation

Recently V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions of the quantum Yang-Baxter equation, i.e. solutions given by a permutation \(R\) of the set \(X\times X\), where \(X\) is a fixed finite set. I...

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Bibliographic Details
Published in:arXiv.org 1997-07
Main Authors: Etingof, Pavel, Schedler, Travis, Soloviev, Alexandre
Format: Article
Language:English
Subjects:
Automorphisms
Group theory
Permutations
Online Access:Get full text
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Summary:Recently V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions of the quantum Yang-Baxter equation, i.e. solutions given by a permutation \(R\) of the set \(X\times X\), where \(X\) is a fixed finite set. In this note we study such solutions, which satisfy the unitarity and the crossing symmetry conditions -- natural conditions arising in physical applications. More specifically, we consider ``linear'' solutions: the set \(X\) is an abelian group, and the map \(R\) is an automorphism of \(X\times X\). We show that in this case, solutions are in 1-1 correspondence with pairs \(a,b\in \End X\), such that \(b\) is invertible and \(bab^{-1}=\frac{a}{a+1}\). Later we consider ``affine'' solutions (\(R\) is an automorphism of \(X\times X\) as a principal homogeneous space), and show that they have a similar classification. The fact that these classifications are so nice leads us to think that there should be some interesting structure hidden behind this problem.
ISSN:2331-8422