Why are noncommutative nonassociative Hopf algebras called quantum groups? This seems to be a purely mathematical notion and there is no quantum anywhere in it prima facie.
Why are Hopf algebras called quantum groups?
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0Have you seen the Wikipedia article, http://en.wikipedia.org/wiki/Quantum_group? There is some explanation there. – 2010-07-28
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0@Akhil: Thanks. I went and looked and didn't see a sufficient explanation. – 2010-07-28
2 Answers
One way that Hopf algebras come up is as the algebra of (real or complex) functions on a topological group. The multiplication is commutative since it is just pointwise multiplication of functions. However, in non-commutative geometry you want to replace the algebra of functions on a space with a non-commutative algebra, giving a non-commutative Hopf algebra.
This relates to quantum mechanics because there the analog of the classical coordinate functions of position and momentum do not commute. Therefore we think of the algebra of functions on a quantum "space" as being non-commutative.
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0I would like you to add details about how you associate this non-commutative Hopf algebra to a space. In particular, how the spaces that you get are the same that appear in physics. – 2010-07-28
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0@BBischof: I"m no expert here, but I don't think it's the "noncommutative space" of a quantum group that we really care about. One of the physics-y things you can do with quantum groups is investigate quantum integrable systems. Another is to use them to define 3d topological quantum field theories; see Turaev's "Quantum Invariants of Knots and 3-Manifolds" for a thorough discussion of this. – 2010-07-28
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0@Qiaochu He says "we think of the algebra of functions on a quantum space as being non-commutative". I was requesting that he describes how one could start with a noncommuative algebra and get a quantum space. I know how in some cases that I am interested in, but I was hoping he could give more. I also claim that I am genuinely interested in the noncomm spaces sitting behind. A deeper explanation of why I care is not appropriate for a comment. – 2010-07-28
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0@BBischof I am also interested in this same issue and I do not know much more than what I said in my answer. When I said quantum space I really meant (rather circularly) a "space" whose algebra of functions is non-commutative. – 2010-07-28
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0@Eric Depending how serious you are about understanding this, you might take a look at the work of my advisor. http://ncatlab.org/nlab/show/Alexander+Rosenberg – 2010-07-28
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1I feel a little guilty about the plug... – 2010-07-28
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0@BBischof: thanks for the link. I'm really interested in this stuff and have seen Rosenberg's name pop-up quite a few times. – 2010-07-29
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2There is no such thing as a *non-commutative* space, just as there is no such thing as an actual *quantum group*... What there is is the ring of functions on such non-existent spaces. – 2010-07-29
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0@Mariano, I agree, and think of this as the philosophy of Serre and Grothendeick that we don't need a space, just the quasi-coherent sheaves on that space. – 2010-07-30
I cannot comment, and this should be a comment...
Observe that the question in your title and the question in the body of your question are quite different!
A non-commutative non-cocommutative Hopf algebra is not the same thing as a non-commutative group, and quantum groups are usually associative.
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0Many instances of "noncommutative" objects are given the adjective "quantum". That is what I had in mind. – 2010-07-28
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2@Lne Bundle, but the noncommutativity in these two cases is somewhat different. "Quantum noncommutativity" is mostly restricted to things that are regarded as non-commutative analogues of function spaces. – 2010-07-28
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0Well, I do not know if it is so important to merit an editing. If you have some better title in mind, when you reach enough reputation(which I'm sure you will shortly), feel free to edit the question. – 2010-07-28
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1"Why are Hopf algebras calld quantum groups?" is the usual way people ask that question :) – 2010-07-28
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0Is the title better now? – 2010-07-29
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1A noncommutative algebraic group is still not the same thing as a noncommutative Hopf algebra. – 2010-07-29
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0@Qiaochu: I beg your pardon, what is a noncommutative algebraic group? I don't know of a definition except that of a quantum group. – 2010-07-29
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1An algebraic group which is not commutative. Any such group is the dual of a _commutative_ Hopf algebra which is not _cocommutative_. – 2010-07-29
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1@Line Bundle: Hopf algebras can satisfy two different kinds of commutativity, one related to their multiplication and one related to their comultiplication. For ordinary algebraic groups the former is always commutative and the latter is cocommutative or noncocommutative depending on whether the group is. For quantum groups the former is not necessarily commutative; that is what the "quantum" means, to a first approximation. – 2010-07-29