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Is that true that any pointed Hopf algebra is quantum group?


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I don't understand your question. The notion of "quantum group" has been given many definitions, and one of them is nothing more than "Hopf algebra". But others, like "deformation of a universal enveloping algebra", are very restrictive. (Note: neither of these is a particularly useful definition, but I've seen both of them.) So you need to expand your question: explain what you mean by "quantum group", and explain why you are asking your question, and so on. Please read carefully for more information on asking good MathOverflow questions. – Theo Johnson-Freyd Jul 10 '11 at 0:33
I agree with Theo Johnson-Freyd. Just one further remark: See e.g. work of Andruskiewitsch and Schneider on partial classification of pointed Hopf algebras. Depending on your definition, you can see them as quantum groups or not. (Some also define a quantum group to be exactly a "pointed Hopf algebra"). – Julian Kuelshammer Jul 10 '11 at 10:41
I think it's best to think of "quantum group" as a metaphor and not a definition. – Noah Snyder Jul 10 '11 at 15:27
ok, sorry! I have got the answer! ma question is really bad. Due to my definition pointed Hopf algebra is more general object, but the problem is that there is really no common definition. Sorry again for non-correct question. – Andriy Jul 11 '11 at 19:47

Acording to Drinfeld V. G., Quantum groups, Proc. of the ICM, Berkeley (1987), 798-820, you can think of a quantum group as an object in the dual category of Hopf algebras.

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