STNB2026 (39th edition)
Hopf algebras and their actions on modules
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Resumen
We start by an introduction to the theory of Hopf algebras over rings that will be eventually needed in Hopf-Galois theory. A Hopf algebra is a module that is a bialgebra (it admits compatible structures of algebra and coalgebra) and is equipped with a coinverse map or antipode. The prototypical example of Hopf algebra is the one of a group algebra, consisting in the linear combinations of elements of a group with scalars in a ring. We shall see the notion of duality in the theory of modules and we will study the dual of a Hopf algebra. In addition, we shall study the notion of module and comodule algebras over Hopf algebras, which is the setting where Hopf-Galois extensions will be defined.
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