  We provide an explicit quantization of dynamical r-matrices for semisimple
Lie algebras, classified earlier by the third author, which includes the
Belavin-Drinfeld r-matrices. We do so by constructing an appropriate
(dynamical) twist in the tensor square of the Drinfeld-Jimbo quantum group
U_q(g), which twists the R-matrix of U_q(g) into the desired quatization. The
construction of this twist is based on the method stemming from the work of
Jimbo-Konno-Odake-Shiraishi and Arnaudon-Buffenoir-Ragoucy-Roche, i.e. on
defining the twist as a unique solution of a suitable difference equation. This
yields a simple closed formula for the twist. This construction allows one to
confirm the alternate version of the Gerstenhaber-Giaquinto-Schack conjecture
(about quantization of Belavin-Drinfeld r-matrices for sl(n) in the vector
representation), which was stated earlier by the second author on the basis of
computer evidence. It also allows one to define new quantum groups associated
to semisimple Lie algebras. We expect them to have a rich structure and
interesting representation theory.
