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10 pages, 2003

Abstract:

An introductory lecture about the author's book on Hopf Galois structures; this is a rigorously written lecture. The title comes from Ch8 = Childs' paper in Trans AMS in 1987, which suggests that "Hopf algebras could fruitfully broaden the domain of Galois module theory"; the Galois module theory is then described as "a branch of algebraic number theory which studies rings of integers of Galois extensions of number fields as modules over the integral group ring of the Galois group." It presents Hopf algebras and Galois extensions--an important topic for both algebraists and quantum theoreticians; also discusses Noether's theorem. Subsequent chapters present Hopf algebras of various ranks, as well as formal groups in relation to Hopf algebras, thus laying the ground work for quantum groups related to Hopf algebras. It concludes with a discussion on the question: Which is the correct Galois structure ? Open rights http://nyjm.albany.edu:8000/~lc802/mono

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