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Where can I find references that discuss important classes of Infinite Hopf Algebras. By important classes, I mean heavily used in research and of relevance to Hopf Algebraist(s),Physicists, Analysts(Real/Complex),..etc.

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Is "infinite" a special term here or do you mean "infinite-dimensional"? –  Julian Kuelshammer Aug 29 '11 at 7:51
I mean that the number of elements is not finite. –  Ahmed Roman Aug 29 '11 at 9:05
I would have thought that Hopf algebras with only finitely many elements are rare (you'd need the ground field to be finite, for a start) –  Yemon Choi Aug 29 '11 at 10:11
(Clarification: rare compared to all possible Hopf algebras) –  Yemon Choi Aug 29 '11 at 10:12
I don't have enough rep to edit, but "Algebras" in the title is misspelled. I also retagged because it seems any textbook on Hopf algebras would do for this, and that's what I suspect all the answers will be. –  David White Aug 29 '11 at 13:40

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Shahn Majid's book Foundations of quantum group theory (Cambridge Univ. Press 1995, 2000) has lots of examples and of classes of examples. These are not only examples of quantum groups in the narrow sense (cf. the $n$Lab page for other references on quantum groups in various senses).

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