  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.
