  A class of non-associative and non-coassociative generalizations of cobraided
bialgebras, called cobraided Hom-bialgebras, is introduced. The
non-(co)associativity in a cobraided Hom-bialgebra is controlled by a twisting
map. Several methods for constructing cobraided Hom-bialgebras are given. In
particular, Hom-type generalizations of FRT quantum groups, including quantum
matrices and related quantum groups, are obtained. Each cobraided Hom-bialgebra
comes with solutions of the operator quantum Hom-Yang-Baxter equations, which
are twisted analogues of the operator form of the quantum Yang-Baxter equation.
Solutions of the Hom-Yang-Baxter equation can be obtained from comodules of
suitable cobraided Hom-bialgebras. Hom-type generalizations of the usual
quantum matrices coactions on the quantum planes give rise to non-associative
and non-coassociative analogues of quantum geometry.
