Hopf Algebras I

Hopf algebras arise naturally in many different areas of mathematics, such as combinatorics, topology, and mathematical physics. Many commonly studied algebraic structures are either Hopf algebras themselves or are directly related to them: for example, groups, Lie algebras, and quantum groups. Hopf algebras act on rings, generalizing the notion of a group of automorphisms. Modules for a Hopf algebra can be added (direct sum) and multiplied (tensor product), giving their categories of modules the structure of tensor categories. In this first talk, we will define Hopf algebras and their actions on rings, give examples, and explain how their modules fit into this larger picture of tensor categories.

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