Talk:Yang–Baxter equation

Figure
Does someone have an animation showing the train argument in a six-vertex model to show why this makes the transfer matrices commute? I saw that as a gif before but I can't find it now. AHusain314 22:47, 8 September 2013 (UTC)

Discussion of braids
The discussion of braids contains the sentence "Since I can swap three strand two different ways, the Yang-Baxter equation enforces that both paths are the same." which to me makes no sense. 138.38.141.220 (talk) 15:25, 13 June 2013 (UTC)

Algebra link
shouldn't the algebra link go to the Lie algebra article rather than the algebra disambiguation page? Eraserhead1 10:21, 30 July 2007 (UTC)

Simplest analogy and simplest true application.
someone (probably me) should write about "star-triangle equation" in electronic circuits, since it's definitly the simplest example of YBE. and about "true" YBE in some simple model, like 1/2-Heisenberg, underlining the significance of YBE in finding commuting integrals. —Preceding unsigned comment added by Virage (talk • contribs) 02:33, 19 October 2008 (UTC)
 * Do go ahead. A simple example would elucidate the topic for the rest of us. Xxanthippe (talk) 00:43, 23 November 2008 (UTC).

Different Formulation
Could somebody add how the standard YB eqn. is related to the one that can be found for example in hep-th/9306089 (i.e. R_23 R_12 R_23 = R_12 R_23 R_12)? IIRC these are related by R -> PR or something similar.


 * Indeed, I'm confused about that, too... In particular, the article itself seems to use two different conventions with $$R_{12} R_{23} R_{12} = R_{23} R_{12} R_{23}$$, while the main text uses $$R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$$. - Saibod (talk) 01:09, 23 July 2014 (UTC)


 * I tried to edit this but my edit got reverted. The braided form is $$\check{R}_{12}(u) \check{R}_{23}(u+v) \check{R}_{12}(v) = \check{R}_{23}(v) \check{R}_{12}(u+v) \check{R}_{23}(u)$$, where $$ \check{R} = T \circ R $$ using the same T as in the current version of the article.


 * In particular, for any parametrized matrix R(u), you can prove that $$R_{12}(u) R_{13}(v) R_{23}(w) = \check{R}_{12}(u) \check{R}_{23}(v) \check{R}_{12}(w) T_{12} T_{23} T_{12} $$, and analogously for the RHS, by repeatedly using the fact that you can change the subscripts by conjugating with T matrices and simplifying.


 * In particular, the very first statement of the equation is wrong & horribly confusing, unless you put a check on the R to indicate that they are different forms of the equation that apply to different R's. - 24.47.139.204 (talk) 06:07, 25 February 2019 (UTC)

Editors' Continuing Campaign of Intrusive Stupidity.
This article is marred by an intrusion by the soi disant editors, which reads "This article provides insufficient context for those unfamiliar with the subject. Please help improve the article with a good introductory style. (October 2009)" I take it they want some childish journalistic burble before one actually gets to the meat of the article. (Implicitly I am calling the "editor" a liar. I think any reasonable person would hold that no context beyond the title "Yang-Baxter equation" is necessary.)

Most people will get here by having already read all the context they need. Any more bumf is simply a waste of their time. A minority who get here in other ways -- and who could they possibly be? -- can simply click through to Yang, or Baxter, or maybe fershlugginer equation. I would suggest a.) that the idiot suggestion box be removed; and b.) that the person responsible for putting it there in the first place be removed from any role with Wikipedia. David Lloyd-Jones (talk) 14:24, 12 January 2014 (UTC)

Feb 2019 rewrite
A previous edit suggested a major rewrite of the article. I'm submitting a draft for a major rewrite of the article body. I reworked the sections into a form that can be absorbed more directly while eliminating possible sources of confusion while also describing the more general form of the YBE and using a notation consistent with most modern research papers.

I aimed to erase no information that was covered in the old form of the article. In particular, I left some vague references to quantum determinants needing clarification, and may eventually ask a colleague that happens to have discovered them to add a few sentences about this section if warranted, otherwise they could be erased or moved to their own stub article.

The main question I'd raise for possible future edits is whether or not the RTT relations should be considered within the scope of this article and included. They are among the main motivations for the parameter-dependent form of the YBE in most contexts as the R matrix acts on the T matrices, and are important to the concept of a Yangian (the page about Yangians includes them). Saolof (talk) 23:01, 25 February 2019 (UTC)
 * You need to quote authoritative source for your edits. Xxanthippe (talk) 23:12, 25 February 2019 (UTC).
 * Everything I included in the article is mentioned in section 1 of "Introduction to the Yang Baxter Equation", published by Michio Jimbo from the department of mathematics at Kyoto University, in International Journal of Modern Physics A, Volume 4, Issue 15, pp. 3759-3777 (1989) , and republished in Vol 9 of "Advanced Series in Mathematical Physics" by world scientific. With that said, everything in this article is either just definitions that mostly have to be consistent with mainstream literature to ensure that all terms have their proper meanings, or things that you can derive from those in two or three lines of relatively trivial algebra. Saolof (talk) 23:47, 25 February 2019 (UTC)
 * OK then put the sources in the article. Xxanthippe (talk) 00:01, 26 February 2019 (UTC).