"R-matrix"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 24번째 줄: | 24번째 줄: | ||
* define <math>\bar R_i</math> sitting in i and i+1 th slot by | * 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> | :<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> |
| − | |||
* the YBE reduces to | * 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. | * 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 | * $\bar R_{2}\bar R_1 \bar R_{2}$ 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) | (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== | ==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]] | + | * 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 | ||
| + | $$ | ||
| + | \left( | ||
| + | \begin{array}{cccc} | ||
| + | \sin (u+\eta ) & 0 & 0 & 0 \\ | ||
| + | 0 & \sin (u) & \sin (\eta ) & 0 \\ | ||
| + | 0 & \sin (\eta ) & \sin (u) & 0 \\ | ||
| + | 0 & 0 & 0 & \sin (u+\eta ) | ||
| + | \end{array} | ||
| + | \right) | ||
| + | $$ | ||
2013년 3월 12일 (화) 10:36 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
$$ \left( \begin{array}{cccc} \sin (u+\eta ) & 0 & 0 & 0 \\ 0 & \sin (u) & \sin (\eta ) & 0 \\ 0 & \sin (\eta ) & \sin (u) & 0 \\ 0 & 0 & 0 & \sin (u+\eta ) \end{array} \right) $$
computational resource
encyclopedia
articles
- R-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators