# Why are Hopf algebras called quantum groups?

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.

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Have you seen the Wikipedia article, en.wikipedia.org/wiki/Quantum_group? There is some explanation there. – Akhil Mathew Jul 28 '10 at 13:13
@Akhil: Thanks. I went and looked and didn't see a sufficient explanation. – user218 Jul 28 '10 at 13:15

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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I 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. – BBischof Jul 28 '10 at 15:53
@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. – Qiaochu Yuan Jul 28 '10 at 17:57
@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. – BBischof Jul 28 '10 at 18:37
I feel a little guilty about the plug... – BBischof Jul 28 '10 at 19:18
There 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. – Mariano Suárez-Alvarez Jul 29 '10 at 5:41

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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@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. – Mariano Suárez-Alvarez Jul 28 '10 at 15:29
"Why are Hopf algebras calld quantum groups?" is the usual way people ask that question :) – Mariano Suárez-Alvarez Jul 28 '10 at 16:30
A noncommutative algebraic group is still not the same thing as a noncommutative Hopf algebra. – Qiaochu Yuan Jul 29 '10 at 18:40
An algebraic group which is not commutative. Any such group is the dual of a commutative Hopf algebra which is not cocommutative. – Qiaochu Yuan Jul 29 '10 at 20:30
@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. – Qiaochu Yuan Jul 29 '10 at 21:50