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×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×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
ba
b
−1
=
a
a+1
. Later we consider ``affine'' solutions (
R
is an automorphism of
X×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.