A MONOIDAL STRUCTURE ON THE CATEGORY OF RELATIVE HOM-HOPF MODULES

We first define a Hom-Yetter-Drinfeld category with a new compatibility relation and prove that it is a pre-braided monoidal category. Secondly, let (H,beta) be a Hom-bialgebra, and (A, alpha) a left (H,beta)-comodule algebra. Assume further that (A,alpha) is also a Hom-coalgebra, with a not necessarily Hom-associative or Hom-unital left (H,beta)-action which commutes with alpha,beta. Then we define a right (A,alpha)-action on the tensor product of two relative Hom-Hopf modules. Our main result is that this action gives a monoidal structure on the category of relative Hom-Hopf modules if and only if (A,alpha) is a braided Hom-bialgebra in the category of Hom-Yetter-Drinfeld modules over (H,beta). Finally, we give some examples and discuss the monoidal Hom-Doi-Hopf datum.