R-matrix
imported>Pythagoras0님의 2013년 3월 11일 (월) 10:24 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
\[\bar R_i\bar R_{i+1}\bar R_i= \bar R_{i+1}\bar R_i \bar R_{i+1}\]
- these are the Braid group relations.
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
encyclopedia
articles
- R-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators