"R-matrix"의 두 판 사이의 차이

2014년 3월 19일 (수) 03:40 판

introduction

R-matrix has entries from Boltzman weights.
From quantum group point of view, R-matrix naturally appears as intertwiners of tensor product of two evaluation modules
from this intertwining property we need to consider $\bar R=p\circ R$ instead of the $R$ matrix where $p$ is the permutation map
this is what makes the module category into braided monoidal category

YBE

R-matrix is a solution of the Yang-Baxter equation (YBE)

$$R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)$$

$R(u,\eta)$
- $u$ is called the spectral parameter
- $\eta$ quantum paramter (or semi-classical parameter)
ignoring $\eta$, we get the classical R-matrix $R(u)$ in $U(\mathfrak{g})$
ignoring $u$, we get $R(\eta)$ in $U_{q}(\mathfrak{g})$ where $q=e^{\eta}$
- found by Drinfeld and Jimbo
- see Drinfeld-Jimbo quantum groups (quantized UEA)

permuted R-matrix

For $R$ matrix on $V \otimes V$, define the permuted R-matrix $\bar R=p\circ R$ where $p$ is the permutation map.
define $\bar R_i$ sitting in i and i+1 th slot by

\[\bar R_i=1\otimes \cdots \otimes\bar R\otimes \cdots \otimes 1\]

whenever $|i-j| \geq 2 $, we have $\bar R_i\bar R_j =\bar R_j\bar R_i$
the YBE reduces to

\[\bar R_i\bar R_{i+1}\bar R_i= \bar R_{i+1}\bar R_i \bar R_{i+1} \label{braid}\]

these are the Braid group relations.

derivation of \ref{braid} from the YBE

$\bar R_{2}(u)\bar R_1(u+v) \bar R_{2}(v)$ corresponding to $R_{12}(u)R_{13}(u+v)R_{23}(v)$ can be written as

$$ (1,2,3) \xrightarrow{R_{23}} (1,2',3') \xrightarrow{P_{23}} (1,3',2') \xrightarrow{R_{12}} (1',3'',2') \xrightarrow{P_{12}} (3'',1',2')\xrightarrow{R_{23}} (3'',1'',2'')\xrightarrow{P_{23}} (3'',2'',1'') $$

$\bar R_{1}(v)\bar R_2(u+v) \bar R_{1}(u)$ corresponding to $R_{23}(v)R_{13}(u+v)R_{12}(u)$ can be written as

$$ (1,2,3) \xrightarrow{R_{12}} (1',2',3) \xrightarrow{P_{12}} (2',1',3) \xrightarrow{R_{23}} (2,1'',3') \xrightarrow{P_{23}} (2,3',1'')\xrightarrow{R_{12}} (2'',3'',1'')\xrightarrow{P_{12}} (3'',2'',1'') $$

R-matrix and Braid groups

with an R-matrix satisfying the YBE, we obtain a representation of the Braid group, which then gives a link invariant in Knot theory

examples of R-matrix

rational R-matrix

$$ \left( \begin{array}{cccc} u+1 & 0 & 0 & 0 \\ 0 & u & 1 & 0 \\ 0 & 1 & u & 0 \\ 0 & 0 & 0 & u+1 \end{array} \right) $$

trigonometric R-matrix

$$ \left( \begin{array}{cccc} \sin (u+\eta ) & 0 & 0 & 0 \\ 0 & \sin (u) & \sin (\eta ) & 0 \\ 0 & \sin (\eta ) & \sin (u) & 0 \\ 0 & 0 & 0 & \sin (u+\eta ) \end{array} \right) $$

explicit R-matrices

taken from http://mathoverflow.net/questions/5103/solutions-of-the-quantum-yang-baxter-equation
The general theory (due to Jimbo) is that each irreducible finite dimensional representation of the quantised enveloping algebra of a Kac-Moody algebra (not of finite type) gives a trigonometric R-matrix.
There is substantial information on these representations but the R-matrices are not explicit.

tensor product graph method

There is a special case which is explicit and is given by the "tensor product graph" method (this was worked out by Niall MacKay and Gustav Delius).
I used this in my paper: R-matrices and the magic square. J. Phys. A, 36(7):1947–1959, 2003. and you can find the references there.
If you want to go beyond this special case and be explicit then you can use "cabling" a.k.a "fusion".

beyond the tensor product graph method

The only papers which deal with R-matrices not covered by the tensor product graph method that I know of are
Vyjayanthi Chari and Andrew Pressley. Fundamental representations of Yangians and singularities of R-matrices. J. Reine Angew. Math., 417:87–128, 1991.
G'abor Tak'acs. The R-matrix of the Uq(d(3)4 ) algebra and g(1)2 affine Toda field theory. Nuclear Phys. B, 501(3):711–727, 1997.
Bruce W. Westbury. An R-matrix for D(3) 4 . J. Phys. A, 38(2):L31–L34, 2005
Deepak Parashar, Bruce W. Westbury R-matrices for the adjoint representations of Uq(so(n)) arXiv:0906.3419
The Chari & Pressley paper deals with rational R-matrices.
The last preprint was an incomplete attempt to try and find the trigonometric analogues of these R-matrices.

related items

Yang-Baxter equation (YBE)

computational resource

https://docs.google.com/file/d/0B8XXo8Tve1cxVlBzUTYxOXdicXc/edit

encyclopedia

articles

R-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators

@@ 23번째 줄: / 23번째 줄: @@
 * For <math>R</math> matrix on <math>V \otimes V</math>, define the permuted R-matrix <math>\bar R=p\circ R</math> where <math>p</math> is the permutation map.
 * define <math>\bar R_i</math> sitting in i and i+1 th slot by
-:<math>\bar R_i=1\otimes \cdots \otimes\bar R \cdots \otimes 1</math>
+:<math>\bar R_i=1\otimes \cdots \otimes\bar R\otimes \cdots \otimes 1</math>
 * whenever <math>|i-j| \geq 2 </math>, we have <math>\bar R_i\bar R_j =\bar R_j\bar R_i</math>
 * the YBE reduces to