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I am a beginner trying to figure out what is a R-matrix, in relation to the Yang-Baxter Equation.

The entry at wikipedia is a little too short.

I have read that a solution to the Yang-Baxter equation satisfies:

$R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$

Does the above relation apply to R-matrices too?

Thanks a lot.

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1 Answer 1

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If I recall - and I seem to have misplaced my copy of Kassel - what you have written is the definition of the (quantum) Yang-Baxter equation.

What I know of this stuff is from the quantum group point of view. There's a bunch of physics that this comes from but unfortunately I know little of this.

An $R$-matrix in this setting is the following. Let $(H,\mu,\eta,\Delta,\epsilon,S)$ be a Hopf algebra. A universal R-Matrix is an invertible element $R=R^{(1)}\otimes R^{(2)}\in H\otimes H$, which satisfies the equations,

$$(\Delta \otimes 1)(R) = R_{13} \ R_{23},$$ $$(1 \otimes \Delta)(R) = R_{13} \ R_{12},$$

where $R_{13}$, for example, is given by $R^{(1)}\otimes 1\otimes R^{(2)}$. (The subscripts tell you where to "put" the components of $R$.)

One can prove, from these equations, that $R$ actually satisfies the Yang-Baxter equation. I recommend Christian Kassel's "Quantum Groups" for the proof - or if you're generally interested in this stuff.

What's cool (in my opinion) about $R$-matrices and this quantum group business (a Hopf algebra with an element $R$ as above is actually one definition of "quantum group") is that a certain subcategory of its finite dimensional representations has very interesting properties. I recommend Bakalov and Kirillov Jr.'s "Lectures on tensor categories and modular functors" if you want to know more about this, though Kassel gives a pretty good exposition of the tangle category stuff.

So... in a very roundabout way, I suppose my answer is yes, if you're talking about this stuff in the context of quantum groups. I think there are a number of ways of talking about $R$-matrices, and I can only speak for this one.

I think Hong and Kang's "Introduction to Quantum Groups and Crystal Bases" has a chapter that has a pretty good introduction to the physical stuff. XXZ or models or whatnot. It's a cool book in general, too.

Anyway, hope that helps.

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Thanks a lot. (The subscripts tell you where to "put" the components of R .) was very helpful. I was under the mistaken notion that $R_{13}$ means the (1,3) entry of the matrix. –  yoyostein May 13 '12 at 7:09

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