  We say that a Hopf algebra H is semicocommutative if the right adjoint
coaction factorizes through the tensor product of H with the center of H. For
instance the commutative and the cocommutative Hopf algebras are
semicocommutative. The quasitriangular Hopf algebras generalize the
cocommutative Hopf algebras. In this paper we introduce and begin the study of
a similar generalization for the semicocommutative ones. These algebras, which
we call semiquasitriangular Hopf algebras have many of the basic properties of
the quasitriangular ones. In particular, they have associated braided
categories of representations in a natural way.
