Hom-quantum groups: I. Quasi-triangular Hom-bialgebras

We introduce a twisted 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. The results here are related to q-deformations of Lie algebras and have potential applications to invariants of knots and quandle cohomology. (paper)