"R-matrix"의 두 판 사이의 차이
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imported>Pythagoras0 (→YBE) |
imported>Pythagoras0 |
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* from this intertwining property we need to consider <math>\bar R=p\circ R</math> instead of the <math>R</math> matrix where <math>p</math> is the permutation map | * from this intertwining property we need to consider <math>\bar R=p\circ R</math> instead of the <math>R</math> matrix where <math>p</math> is the permutation map | ||
* this is what makes the module category into braided monoidal category | * this is what makes the module category into braided monoidal category | ||
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==YBE== | ==YBE== | ||
| − | + | * R-matrix is a solution of the [[Yang-Baxter equation (YBE)]] | |
| − | * [[Yang-Baxter equation (YBE) | + | :<math>R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}</math><br> |
* $R(u,\eta)$ | * $R(u,\eta)$ | ||
** $u$ is called the spectral parameter | ** $u$ is called the spectral parameter | ||
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** found by Drinfeld and Jimbo | ** found by Drinfeld and Jimbo | ||
** see [[Drinfeld-Jimbo quantum groups (quantized UEA)]] | ** see [[Drinfeld-Jimbo quantum groups (quantized UEA)]] | ||
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| + | ==permuted R-matrix== | ||
| + | * For <math>R</math> matrix on <math>V \otimes V</math>, define the permuted R-matrix <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\bar R_j =\bar R_j\bar R_i</math> whenever <math>|i-j| \geq 2 </math><br><math>\bar R_i\bar R_{i+1}\bar R_i= \bar R_{i+1}\bar R_i \bar R_{i+1}</math><br> which are the [[Braid group]] relations.<br> | ||
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==R-matrix and Braid groups== | ==R-matrix and Braid groups== | ||
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* 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]]<br> | ||
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==related items== | ==related items== | ||
| − | + | * [[Yang-Baxter equation (YBE)]] | |
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* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
* http://en.wikipedia.org/wiki/Braided_monoidal_category | * http://en.wikipedia.org/wiki/Braided_monoidal_category | ||
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| 77번째 줄: | 51번째 줄: | ||
* R-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators<br> | * R-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators<br> | ||
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[[분류:quantum groups]] | [[분류:quantum groups]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
2013년 3월 11일 (월) 06:11 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 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.
\(\bar R_i=1\otimes \cdots \otimes\bar R \cdots \otimes 1\), \(\bar R_i\) 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.
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