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

Abstract

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.