iQuantum groups and iHopf algebras I: foundation

We introduce the notion of iHopf algebra, a new associative algebra structure defined on a Hopf algebra equipped with a Hopf pairing. The iHopf algebra on a Borel quantum group endowed with a $\tau$-twisted Hopf pairing is shown to be a quasi-split universal iquantum group. In particular, the Drinfeld double quantum group is realized as the iHopf algebra on the double Borel. This iHopf approach allows us to develop connections between Lusztig's braid group action and ibraid group action. It will further lead to the construction of dual canonical basis in a sequel.