Mathematics – Quantum Algebra
Scientific paper
1997-07-22
Mathematics
Quantum Algebra
4 pages, amstex; in the revised version there are minor changes; in particular, the set X is assumed to be finite
Scientific paper
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.
Etingof Pavel
Schedler Travis
Soloviev Alexandre
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