A monoidal structure on the category of relative Hom-Hopf modules
Volume 165 / 2021
Abstract
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.
