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I want to learn more about hopf algebras but I am having trouble finding a down to earth introduction to the subject with lots of motivation and examples. My algebra knowledge ranges from Dummit and Foote, Atiyah-Macdonald commutative algebra, and Hartshorne level algebraic geometry.

What are good places to look at for a hopf algebra beginner? My main motivation is for studying groupoid schemes from an algebraic point of view.

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The following texts provide self-consistent introductions to the topic of Hopf algebras, almost from the scratch:

  1. Dascalescu, Nastasescu, Raianu, "Hopf algebras, an introduction",
  2. S. Montgomery, "Hopf algebras and their actions on rings",
  3. Moss E. Sweedler, "Hopf algebras",
  4. Kassel, "Quantum groups",
  5. S.Majid, "Foundations of Quantum Group theory",
  6. S.Majid, "A quantum groups primer",
  7. E.Abe, "Hopf algebras",
  8. D. Radford, "Hopf algebras"

1., 4., and 8. are the most elementary introductions in the sense that they start from the fundamentals and drive you up to some of the very high ends. 1. emphasizes structure theory and contains an extreme wealth of examples with detailed computations and solved exercises (it would be my personal recommendation for someone to begin with), while 4. emphasizes representation theory and the braided monoidal categories point of view. (4. contains an extensive general introduction as well).

3., 7., are the oldest books on the topic. Lots of today's hopf algebraists have started studying from these books. Still authoritative today. (7. contains an extensive general introduction on various topics from abstract algebra).

2., is a relatively small and dense book, addressed to students who feel experienced with abstract algebra texts. However it covers unbelievably many examples and details and sheds light into myriads of dark corners. Very strong in the category theoretical point of view, it covers -in sufficient detail- research results until the mid 90's.

5., 6. would be useful for someone interested in the mathematical physics motivations for studying the topic: the use of quantum groups and non-commutative geometry models in physics is hiding behind each example and each computation. Might be too heavy for a first introduction. They focus on various different aspects, but i would not recomend them as starting points but rather as complements to someone already comfortable with the basic notions. However, they are unique in the sense that they contain a braided graphical calculus -developed by the author- for performing computations in braided monoidal categories.

In conclusion, if i were to teach a general introductory course to first year postgraduate students, I would be based on 1. and 2. using 4. as a supplement (in the first place) and I would recomend 3.,7.,8. for further support. 5. and 6. would be good sources for further search in the bibliography and for mathematical physics or graphical calculus related projects.
If however, i would like to be oriented towards quasitriangularity and braided monoidal categories (i.e. the more representation theoretical aspects of the topic) then I would recommend 2., 4., 5., 6., 8. (in that order).

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  • 2
    $\begingroup$ These types of answers are my favorite because you just gave anyone a pathway towards understanding the theory. I wish answers like these existed for more topics! $\endgroup$ – 54321user Mar 19 '17 at 22:39

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