"R-matrix"의 두 판 사이의 차이

2013년 3월 12일 (화) 09:36 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)

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

$R(u,\eta)$
- $u$ 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

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

these are the Braid group relations.

derivation of \ref{braid} from the 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

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

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

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

@@ 24번째 줄: / 24번째 줄: @@
 * 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>
-* whenever <math>|i-j| \geq 2 </math>, we have
+* whenever <math>|i-j| \geq 2 </math>, we have <math>\bar R_i\bar R_j =\bar R_j\bar R_i</math>
-:<math>\bar R_i\bar R_j =\bar R_j\bar R_i</math>
 * the YBE reduces to
-:<math>\bar R_i\bar R_{i+1}\bar R_i= \bar R_{i+1}\bar R_i \bar R_{i+1}</math>
+:<math>\bar R_i\bar R_{i+1}\bar R_i= \bar R_{i+1}\bar R_i \bar R_{i+1}  \label{braid}</math>
 * these are the [[Braid group]] relations.
-===proof of the new YBE===
+===derivation of \ref{braid} from the YBE===
 * $\bar R_{2}\bar R_1 \bar R_{2}$ corresponding to $R_{12}R_{13}R_{23}$ can be written as
 $$
@@ 40번째 줄: / 39번째 줄: @@
 (1,2,3) \xrightarrow{R_{12}} (1,2,3) \xrightarrow{P_{12}} (2,1,3) \xrightarrow{R_{23}} (2,1,3) \xrightarrow{P_{23}} (2,3,1)\xrightarrow{R_{12}} (2,3,1)\xrightarrow{P_{12}} (3,2,1)
 $$
 ==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]]<br>
+* 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
+$$
+\left(
+\begin{array}{cccc}
+ u+1 & 0 & 0 & 0 \\
+& u & 1 & 0 \\
+& 1 & u & 0 \\
+& 0 & 0 & u+1
+\end{array}
+\right)
+$$
+* trigonometric R-matrix
+$$
+\left(
+\begin{array}{cccc}
+ \sin (u+\eta ) & 0 & 0 & 0 \\
+& \sin (u) & \sin (\eta ) & 0 \\
+& \sin (\eta ) & \sin (u) & 0 \\
+& 0 & 0 & \sin (u+\eta )
+\end{array}
+\right)
+$$