"R-matrix"의 두 판 사이의 차이

2012년 10월 29일 (월) 09:53 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

Yang-Baxter equation
$R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$
$R(u,\eta)$
- $u$ is called the spectral parameter
- $\eta$ quantum paramter
ignoring $\eta$, we get 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

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

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