"R-matrix"의 두 판 사이의 차이

2010년 8월 10일 (화) 11:41 판

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

For \(R\) matrix on \(V \otimes V\), define \(\bar R=p\circ R\) where \(p\) is the permutation map.
\(\bar R_i=1\otimes \cdots \otimes\bar R \cdots \otimes 1\), \(\bar R_i\) sitting in i and i+1 th slot.
Then YB reduces to

\(\bar R_i\bar R_j =\bar R_j\bar R_i\) whenever \(|i-j| \geq 2 \)

\(\bar R_i\bar R_{i+1}\bar R_i= \bar R_{i+1}\bar R_i \bar R_{i+1}\)

which are the Braid group relations.

@@ 24번째 줄: / 24번째 줄: @@
 <h5 style="margin: 0px; line-height: 2em;">R-matrix and Braid groups</h5>
-For <math>R</math> matrix on <math>V \otimes V</math>, define <math>\bar R=p\circ R</math> where <math>p</math> is the permutation map.
+*  For <math>R</math> matrix on <math>V \otimes V</math>, define <math>\bar R=p\circ R</math> where <math>p</math> is the permutation map.<br><math>\bar R_i=1\otimes \cdots \otimes\bar R \cdots \otimes 1</math>, <math>\bar R_i</math> sitting in i and i+1 th slot.<br>
+*  Then YB reduces to<br>
-<math>\bar R_i=1\otimes \cdots \otimes\bar R \cdots \otimes 1</math>, <math>\bar R_i</math> sitting in i and i+1 th slot.
-Then YB reduces to
 <math>\bar R_i\bar R_j =\bar R_j\bar R_i</math> whenever <math>|i-j| \geq 2 </math>