Talk:Quantum group

Quantum groups at q a root of unity

In the information that I have supplied on quantum groups, there are some unanswered questions about the defining relations in certain cases. For example,in the relation $[e_{i},f_{i}]={\frac {k_{i}-k_{i}^{-1}}{q_{i}-q_{i}^{-1}}}$ , there is no discussion about what happens in the case that $q_{i}=\pm 1$ . I presume that if $q_{i}=\pm 1$ , then the appropriate relation becomes $k_{i}^{2}=1$ . Would it be right that there is another appropriate relation for $q_{i}=\pm 1$ , e.g. $[e_{i},f_{i}]=q_{i}k_{i}t_{\alpha _{i}}$ , where $t_{\alpha _{i}}$ is the element of the Cartan subalgebra such that $\lambda (t_{\alpha _{i}})=(\lambda ,\alpha _{i})$ for all $\lambda$ is the dual space to the Cartan subalgebra?

Also, I have not discussed representation theory or quasitriangularity in the case where q is a root of unity. Could somebody please supply the missing information. Thanks. Figaro 09:15, 2 January 2006 (UTC)[reply]

There really should be more background information on quantum groups - or at least links to background information. Algebras, coalgebras, bialgebras, hopf algebras, q-calculus, (infinite dimensional) Lie algebras, UEAs, the tensor algebra, etc. There is not really a good definition of a quantum group on this page; there are just defining relations. Anyone feel like doing this? Myrkkyhammas 17:32, 10 September 2006 (EST)

How are quantum groups connected to quantum mechanics? Are they? Scott Tillinghast, Houston TX 02:13, 9 May 2007 (UTC)[reply]

Good question. They are, but precise connections are rather complicated. Affine quantum groups first appeared in exactly solvable models of one-dimensional quantum field theory. Later it was realized that quantum groups corresponding to simple Lie algebras have similar interpretations, but again, the 'physical' context is rather quantum field theory, not quantum mechanics. More generally, quantum groups arise by the procedure of quantization from Poisson-Lie groups, so in this sense they are 'philosophically' related (meaning that some of the mathematical techniques used had been initially developed in quantum-mechanical contexts), but not necessarily 'physically' related. Arcfrk 04:36, 9 May 2007 (UTC)[reply]

This is true. It was highly disappointing when I found out that quantum groups are not really related to deformation quantization of Possion Lie groups in a physically motivated way - they just happen to be dual notions. I was expecting something deep and fascinating. Poo. Myrkkyhammas 09:39, 3 June 2007 (UTC)[reply]

Rubics cube

Having the "group" box with the Rubic's cube on this page seems strange to me:

- The group describing the cube has nothing to do with quantum groups

- The box also somehow suggests that quantum groups are groups, but they aren't.

129.199.98.79 (talk) 16:48, 6 February 2009 (UTC)[reply]

Well, it actually isn't completely wrong to think of quantum groups as "groups," although you are correct that they aren't (they're Hopf algebras). That is, it's "morally" okay to equate them with groups, because the structure of quantum groups is intimately connected with actual groups, namely algebraic groups. In fact one can use quantum groups to attack problems in algebraic group theory; there is a famous proof by Andersen, Jantzen, and Soergel of the so-called "Lusztig conjecture" for

p\gg 0

. This in particular is a conjecture that is very much about groups, but it was solved using quantum groups, by exploiting the fact that quantum groups are structurally similar to algebraic groups (in a way that I'll leave imprecise). FrobeniusTwist (talk) 17:26, 16 March 2009 (UTC)[reply]

Compact matrix quantum group

The part compact matrix quantum group is quite a copy-paste from the article compact quantum group we should link to there. For the moment added "see also". It is the occation to remove the flag orphan.

Imprecision of this article

The definition of quantum group uses the word 'kind' (kind of...) It seems to me like a less vague explanation is missing. — Preceding unsigned comment added by Yaron hadad (talk • contribs) 07:38, 2 October 2012 (UTC)[reply]

Huge Improvements Possible If ...

The readability of this article could enormously improve if we change from e_i & f_i to e_i and e_{-i} instead. Several pairs of formulas would collapse and the symmetry inherent to the structure would come out much better. I did not do this b/c I did not want to spend couple of hours just to see someone come on revert it back to where it is at now. — Preceding unsigned comment added by 99.241.86.114 (talk) 05:06, 9 October 2012 (UTC)[reply]

Yes agreed. The e/f notation is not the notation I see in the stuff I read, instead I see

e_{i}^{+}

and

e_{i}^{-}

which makes it clear that they are the raising and lower operators. Which, oh by the way, a quickie review of Lie algebraas using this notation would be excellent, because otherwise its opaque as to what it is that is being deformed. Hmm, actually, I see

E_{i}^{+}

and

E_{i}^{-}

because the lower-case e's are reserved as basis vectors for the vector space of the representation... so lower-case-e seems like a poor choice. 67.198.37.16 (talk) 05:31, 19 September 2016 (UTC)[reply]

How are groups related to QM?

I tried reading this article, but understanding it depends on exposure to many specific terms in advanced algebraic theory, terms I have never even heard of before, and I am a college-educated adult. Is it possible to have an initial paragraph that explains the rest of the lead? For example, is there a simple way that the elements of a quantum group relate to wave functions or bra-ket notation? Or, perhaps, is there a simple way that the noncummutative property of quantum groups can be related to some aspect of QM, such as the purity of states? David Spector (talk) 17:47, 23 March 2024 (UTC)[reply]