An identification of Lusztig's modified forms of quantum algebras and their analogues

We introduce a multi-parameter twisted Hopf algebra associated with a root datum and show that its modified form is isomorphic to Lusztig's modified quantum algebra under certain restrictions on the parameters. By taking various specializations of the parameters, we obtain similar results for two/multi-parameter quantum algebras and Kang-Kashiwara-Oh's multi-parameter quantum superalgebras. As a consequence, the categories of weight modules of these algebras are identical.