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

TL;DR

This paper establishes isomorphisms between Lusztig's modified quantum algebras and their multi-parameter and superalgebra analogues, revealing that their weight module categories are equivalent.

Contribution

It introduces a multi-parameter twisted Hopf algebra framework and demonstrates its isomorphism to Lusztig's modified quantum algebra, extending to superalgebras and various specializations.

Findings

Isomorphism between multi-parameter twisted Hopf algebra and Lusztig's modified quantum algebra

Equivalence of weight module categories across different algebra types

Generalization to multi-parameter quantum superalgebras

Abstract

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.