Lusztig isomorphisms for Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type

TL;DR

This paper extends Lusztig isomorphisms to Drinfel'd doubles of Nichols algebras of diagonal type, providing new tools for understanding their structure and classification, with implications for quantum groups and representation theory.

Contribution

It introduces Lusztig maps for these Drinfel'd doubles, establishing isomorphisms and Coxeter relations, and characterizes Nichols algebras with finite arithmetic root systems.

Findings

Lusztig maps induce isomorphisms between doubles of Nichols algebras

The isomorphisms satisfy generalized Coxeter relations

Characterization of Nichols algebras with finite arithmetic root systems

Abstract

In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel'd double for a class of graded Hopf algebras is investigated. Various quantum algebras, including multiparameter quantizations of semisimple Lie algebras and of Lie superalgebras, are covered by the given definition. For these Drinfel'd doubles Lusztig maps are defined. It is shown that these maps induce isomorphisms between doubles of Nichols algebras of diagonal type. Further, the obtained isomorphisms satisfy Coxeter type relations in a generalized sense. As an application, the Lusztig isomorphisms are used to give a characterization of Nichols algebras of diagonal type with finite arithmetic root system. Key words: Hopf algebra, quantum group, Weyl groupoid