  We introduce a Hom-type generalization of quantum groups, called
quasi-triangular Hom-bialgebras. They are non-associative and non-coassociative
analogues of Drinfel'd's quasi-triangular bialgebras, in which the
non-(co)associativity is controlled by a twisting map. A family of
quasi-triangular Hom-bialgebras can be constructed from any quasi-triangular
bialgebra, such as Drinfel'd's quantum enveloping algebras. Each
quasi-triangular Hom-bialgebra comes with a solution of the quantum
Hom-Yang-Baxter equation, which is a non-associative version of the quantum
Yang-Baxter equation. Solutions of the Hom-Yang-Baxter equation can be obtained
from modules of suitable quasi-triangular Hom-bialgebras.
