R-matrix

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)

\[R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\]

$R(u,\eta)$
- $u$ is called the spectral parameter
- $\eta$ quantum paramter
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

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

whenever $|i-j| \geq 2 $, we have

\[\bar R_i\bar R_j =\bar R_j\bar R_i\]

the YBE reduces to

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

these are the Braid group relations.

proof of the new YBE

$\bar R_{2}\bar R_1 \bar R_{2}$ can be written as

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

$\bar R_{1}\bar R_2 \bar R_{1}$ 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

related items

Yang-Baxter equation (YBE)

encyclopedia

articles

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