On Hopf algebras with triangular decomposition

TL;DR

This survey reviews the representation theory of algebras with triangular decomposition, discusses a construction method for Hopf algebras with this structure, and extends key results on projective modules to these Hopf algebras.

Contribution

It provides a comprehensive overview of known results and introduces a construction recipe for Hopf algebras with triangular decomposition, extending existing module theory results.

Findings

Classification of simple modules for these algebras

A new construction method for Hopf algebras with triangular decomposition

Extension of projective module results to these Hopf algebras

Abstract

In this survey, we first review some known results on the representation theory of algebras with triangular decomposition, including the classification of the simple modules. We then discuss a recipe to construct Hopf algebras with triangular decomposition. Finally, we extend to these Hopf algebras the main results of arXiv:1612.09220 regarding projective modules over Drinfeld doubles of bosonizations of Nichols algebras and groups.