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

• $\bar R_{2}\bar R_1 \bar R_{2}$ corresponding to $R_{12}R_{13}R_{23}$ can be written as

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

• $\bar R_{1}\bar R_2 \bar R_{1}$ corresponding to $R_{23}R_{13}R_{12}$ 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)

computational resource

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

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Braided_monoidal_category

articles

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