Talk:Quantum group

From Wikipedia, the free encyclopedia

[edit] 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 λ 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)

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)

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)
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)