On the Hopf algebra structure of the Lusztig quantum divided power algebras

TL;DR

This paper explores the Hopf algebra structure of Lusztig's quantum groups, revealing their zero part as a tensor product and constructing them via triangular decomposition methods.

Contribution

It provides a detailed structural analysis of Lusztig's quantum groups, especially their zero part and the construction from plus, minus, and zero components.

Findings

Zero part is a tensor product of a finite abelian group algebra and an abelian Lie algebra enveloping algebra.

Constructs Lusztig's quantum groups using actions and coactions within Sommerhauser's triangular decomposition framework.

Clarifies the algebraic structure and composition of Lusztig's quantum groups.

Abstract

We study the Hopf algebra structure of Lusztig's quantum groups. First we show that the zero part is the tensor product of the group algebra of a finite abelian group with the enveloping algebra of an abelian Lie algebra. Second we build them from the plus, minus and zero parts by means of suitable actions and coactions within the formalism presented by Sommerhauser to describe triangular decompositions.