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Appears in collection : Representation theory of generalized small quantum groups

We will begin the course by exploring the very rich Representation Theory of the graded finite-dimensional algebras with triangular decomposition. We will follow mainly [1]. These algebras have plenty of powerful and beautiful methods and results inspired in Lie Theory (characters, Verma module, BGG Reciprocity, etc).

Then we will introduce some examples of Hopf algebras with triangular decomposition based on [2]. Among these, we will focus on the Drinfeld doubles of the bosonization of a finite- dimensional Nichols algebra over a semisimple Hopf algebra. We will investigate peculiar features which appear in this setting (duality and tensor product).

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Bibliography

  • [1] G. Bellamy and U. Thiel. Highest weight theory for finite-dimensional graded algebras with triangular de- composition. Adv. Math. 330 (2018), 361 – 419.
  • [2] C. Vay, On Hopf algebras with triangular decomposition, Contemp. Math. 728 (2019), 181-199.

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