# A Hopf Algebra from Preprojective Modules

## TL;DR

This paper constructs a Hopf algebra from indecomposable modules over preprojective algebras of finite type quivers, revealing a connection to nilpotent Lie algebras and extending Lusztig's work.

## Contribution

It introduces a new algebra generated by indecomposable modules, identifying it as the universal enveloping algebra of a nilpotent Lie algebra related to Lusztig's schemes.

## Key findings

- The algebra forms the universal enveloping algebra of a nilpotent Lie algebra.
- Explicit relations for types A1, A2, A3, A4 are provided.
- The construction extends Lusztig's framework to preprojective modules.

## Abstract

Let $Q$ be a finite type quiver i.e. ADE Dynkin quiver. Denote by $\Lambda$ its preprojective algebra. It is known that there are finitely many indecomposable $\Lambda$-modules if and only if $Q$ is of type $A_1,A_2,A_3,A_4$. In this paper, extending Lusztig's construction of $U\frak{n}_+$, we study an algebra generated by these indecomposable submodules. It turns out that it forms the universal enveloping algebra of some nilpotent Lie algebra inside the function algebra on Lusztig's nilpotent scheme. The defining relations of the corresponding nilpotent Lie algebra for type $A_1, A_2,A_3,A_4$ are given here.