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.
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