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An Introduction to Hopf Algebras

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An Introduction to Hopf Algebras, 9780387727653 (0387727655), Springer, 2011
The purpose of this book is to provide an introduction to Hopf algebras. This book differs from other texts in that Hopf algebras are developed from notions of topological spaces, sheaves, and representable functors. This approach has certain pedagogical advantages, the foremost being that algebraic geometry and category theory provide a smooth transition from modern algebra to Hopf algebras. For example, the motivation for the definition of an exact sequence of Hopf algebras is best established by first defining exactness for a sequence of representable group functors.

Hopf algebras are attributed to the German mathematician Heinz Hopf (1894–1971). The study of Hopf algebras spans many fields in mathematics, including topology, algebraic geometry, algebraic number theory, Galois module theory, cohomology of groups, and formal groups. In this work, we focus on applications of Hopf algebras to algebraic number theory and Galois module theory. By the end of the book, readers will be familiar with established results in the field and should be poised to phrase research questions of their own.

An effort has been made to make this book as self-contained as possible. That said, readers should have an understanding of the material on groups, rings, and fields normally covered in a basic course in modern algebra. Also, most of the groups given here are Abelian. All of the rings are non-zero commutative rings with unity, and we only consider commutative algebras.
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