"R-matrix"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 8개는 보이지 않습니다) | |||
| 10번째 줄: | 10번째 줄: | ||
==YBE== | ==YBE== | ||
* R-matrix is a solution of the [[Yang-Baxter equation (YBE)]] | * R-matrix is a solution of the [[Yang-Baxter equation (YBE)]] | ||
| − | + | :<math>R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)</math> | |
| − | * | + | * <math>R(u,\eta)</math> |
| − | ** | + | ** <math>u</math> is called the spectral parameter |
| − | ** | + | ** <math>\eta</math> quantum paramter (or semi-classical parameter) |
| − | * ignoring | + | * ignoring <math>\eta</math>, we get the classical R-matrix <math>R(u)</math> in <math>U(\mathfrak{g})</math> |
| − | * ignoring | + | * ignoring <math>u</math>, we get <math>R(\eta)</math> in <math>U_{q}(\mathfrak{g})</math> where <math>q=e^{\eta}</math> |
** found by Drinfeld and Jimbo | ** found by Drinfeld and Jimbo | ||
** see [[Drinfeld-Jimbo quantum groups (quantized UEA)]] | ** see [[Drinfeld-Jimbo quantum groups (quantized UEA)]] | ||
| 23번째 줄: | 23번째 줄: | ||
* 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. | * 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. | ||
* 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\otimes \cdots \otimes 1</math> |
* whenever <math>|i-j| \geq 2 </math>, we have <math>\bar R_i\bar R_j =\bar R_j\bar R_i</math> | * 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 | ||
| 31번째 줄: | 31번째 줄: | ||
===derivation of \ref{braid} from the YBE=== | ===derivation of \ref{braid} from the YBE=== | ||
| − | * | + | * <math>\bar R_{2}(u)\bar R_1(u+v) \bar R_{2}(v)</math> corresponding to <math>R_{12}(u)R_{13}(u+v)R_{23}(v)</math> can be written as |
| − | + | :<math> | |
(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'') | (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'') | ||
| − | + | </math> | |
| − | * | + | * <math>\bar R_{1}(v)\bar R_2(u+v) \bar R_{1}(u)</math> corresponding to <math>R_{23}(v)R_{13}(u+v)R_{12}(u)</math> can be written as |
| − | + | :<math> | |
(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'') | ||
| − | + | </math> | |
==R-matrix and Braid groups== | ==R-matrix and Braid groups== | ||
| 46번째 줄: | 46번째 줄: | ||
==examples of R-matrix== | ==examples of R-matrix== | ||
* rational R-matrix | * rational R-matrix | ||
| − | + | :<math> | |
\left( | \left( | ||
\begin{array}{cccc} | \begin{array}{cccc} | ||
| 55번째 줄: | 55번째 줄: | ||
\end{array} | \end{array} | ||
\right) | \right) | ||
| − | + | </math> | |
* trigonometric R-matrix | * trigonometric R-matrix | ||
| − | + | :<math> | |
\left( | \left( | ||
\begin{array}{cccc} | \begin{array}{cccc} | ||
| 66번째 줄: | 66번째 줄: | ||
\end{array} | \end{array} | ||
\right) | \right) | ||
| − | + | </math> | |
| 90번째 줄: | 90번째 줄: | ||
==related items== | ==related items== | ||
* [[Yang-Baxter equation (YBE)]] | * [[Yang-Baxter equation (YBE)]] | ||
| − | + | * [[R-matrix in quantum affine algebra]] | |
==computational resource== | ==computational resource== | ||
| 104번째 줄: | 104번째 줄: | ||
==articles== | ==articles== | ||
| − | + | * Lentner, Simon, and Daniel Nett. “New <math>R</math>-Matrices for Small Quantum Groups.” arXiv:1409.5824 [math], September 19, 2014. http://arxiv.org/abs/1409.5824. | |
| − | * R-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators | + | * R-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators |
| 112번째 줄: | 112번째 줄: | ||
[[분류:quantum groups]] | [[분류:quantum groups]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
| + | [[분류:migrate]] | ||
| + | |||
| + | == 메타데이터 == | ||
| + | |||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q2993305 Q2993305] | ||
| + | |||
| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q2993305 Q2993305] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'baum'}, {'OP': '*'}, {'LOWER': 'connes'}, {'LEMMA': 'conjecture'}] | ||
2021년 2월 17일 (수) 01:17 기준 최신판
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)
\[R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)\]
- \(R(u,\eta)\)
- \(u\) is called the spectral parameter
- \(\eta\) quantum paramter (or semi-classical parameter)
- 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
\[\bar R_i=1\otimes \cdots \otimes\bar R\otimes \cdots \otimes 1\]
- 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
- \(\bar R_{2}(u)\bar R_1(u+v) \bar R_{2}(v)\) corresponding to \(R_{12}(u)R_{13}(u+v)R_{23}(v)\) can be written as
\[ (1,2,3) \xrightarrow{R_{23}} (1,2',3') \xrightarrow{P_{23}} (1,3',2') \xrightarrow{R_{12}} (1',3'',2') \xrightarrow{P_{12}} (3'',1',2')\xrightarrow{R_{23}} (3'',1'',2'')\xrightarrow{P_{23}} (3'',2'',1'') \]
- \(\bar R_{1}(v)\bar R_2(u+v) \bar R_{1}(u)\) corresponding to \(R_{23}(v)R_{13}(u+v)R_{12}(u)\) 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
\[ \left( \begin{array}{cccc} \sin (u+\eta ) & 0 & 0 & 0 \\ 0 & \sin (u) & \sin (\eta ) & 0 \\ 0 & \sin (\eta ) & \sin (u) & 0 \\ 0 & 0 & 0 & \sin (u+\eta ) \end{array} \right) \]
explicit R-matrices
- taken from http://mathoverflow.net/questions/5103/solutions-of-the-quantum-yang-baxter-equation
- The general theory (due to Jimbo) is that each irreducible finite dimensional representation of the quantised enveloping algebra of a Kac-Moody algebra (not of finite type) gives a trigonometric R-matrix.
- There is substantial information on these representations but the R-matrices are not explicit.
tensor product graph method
- There is a special case which is explicit and is given by the "tensor product graph" method (this was worked out by Niall MacKay and Gustav Delius).
- I used this in my paper: R-matrices and the magic square. J. Phys. A, 36(7):1947–1959, 2003. and you can find the references there.
- If you want to go beyond this special case and be explicit then you can use "cabling" a.k.a "fusion".
beyond the tensor product graph method
- The only papers which deal with R-matrices not covered by the tensor product graph method that I know of are
- Vyjayanthi Chari and Andrew Pressley. Fundamental representations of Yangians and singularities of R-matrices. J. Reine Angew. Math., 417:87–128, 1991.
- G'abor Tak'acs. The R-matrix of the Uq(d(3)4 ) algebra and g(1)2 affine Toda field theory. Nuclear Phys. B, 501(3):711–727, 1997.
- Bruce W. Westbury. An R-matrix for D(3) 4 . J. Phys. A, 38(2):L31–L34, 2005
- Deepak Parashar, Bruce W. Westbury R-matrices for the adjoint representations of Uq(so(n)) arXiv:0906.3419
- The Chari & Pressley paper deals with rational R-matrices.
- The last preprint was an incomplete attempt to try and find the trigonometric analogues of these R-matrices.
computational resource
encyclopedia
articles
- Lentner, Simon, and Daniel Nett. “New \(R\)-Matrices for Small Quantum Groups.” arXiv:1409.5824 [math], September 19, 2014. http://arxiv.org/abs/1409.5824.
- R-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators
메타데이터
위키데이터
- ID : Q2993305
메타데이터
위키데이터
- ID : Q2993305
Spacy 패턴 목록
- [{'LOWER': 'baum'}, {'OP': '*'}, {'LOWER': 'connes'}, {'LEMMA': 'conjecture'}]