$\mathrm{ı}$Hopf algebras associated with self-dual Hopf algebras

Motivated by the construction of $\mathrm{ı}$Hall algebras and $\mathrm{\Delta }$-Hall algebras, we introduce $\mathrm{ı}$Hopf algebras associated with symmetrically self-dual Hopf algebras. We prove that the $\mathrm{ı}$Hopf algebra is an associative algebra with a unit, where the associativity relies on an analogue of Green’s formula in the framework of Hopf algebras. As an application, we construct the $\mathrm{ı}$Taft algebra of dimension 4, which is proved to be isomorphic to the group algebra of $\mathbb{Z}\mathrm{/}4\mathbb{Z}$.