"R-matrix"의 두 판 사이의 차이

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5번째 줄: 5번째 줄:
 
* 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==
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* R-matrix is a solution of the [[Yang-Baxter equation (YBE)]]
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* [[Yang-Baxter equation (YBE)|Yang-Baxter equation]]<br><math>R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}</math><br>
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:<math>R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)</math>
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* $R(u,\eta)$
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* <math>R(u,\eta)</math>
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** $u$ is called the spectral parameter
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** <math>u</math> is called the spectral parameter
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** $\eta$ quantum paramter
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** <math>\eta</math> quantum paramter (or semi-classical parameter)
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* ignoring $\eta$, we get classical R-matrix $R(u)$ in $U(\mathfrak{g})$
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* ignoring <math>\eta</math>, we get the classical R-matrix <math>R(u)</math> in <math>U(\mathfrak{g})</math>
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* ignoring $u$, we get $R(\eta)$ in $U_{q}(\mathfrak{g})$ where $q=e^{\eta}$
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* 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)]]
  
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==R-matrix and Braid groups==
 
  
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* For <math>R</math> matrix on <math>V \otimes V</math>, define <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>
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==permuted R-matrix==
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*  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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* 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.
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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>
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* 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\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>
 +
* the YBE reduces to
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:<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>
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* these are the [[Braid group]] relations.
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===derivation of \ref{braid} from the YBE===
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* <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
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:<math>
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(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'')
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</math>
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* <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
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:<math>
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(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'')
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</math>
  
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==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]]
  
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==related items==
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==examples of R-matrix==
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* rational R-matrix
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:<math>
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\left(
 +
\begin{array}{cccc}
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u+1 & 0 & 0 & 0 \\
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0 & u & 1 & 0 \\
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0 & 1 & u & 0 \\
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0 & 0 & 0 & u+1
 +
\end{array}
 +
\right)
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</math>
 +
* trigonometric R-matrix
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:<math>
 +
\left(
 +
\begin{array}{cccc}
 +
\sin (u+\eta ) & 0 & 0 & 0 \\
 +
0 & \sin (u) & \sin (\eta ) & 0 \\
 +
0 & \sin (\eta ) & \sin (u) & 0 \\
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0 & 0 & 0 & \sin (u+\eta )
 +
\end{array}
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\right)
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</math>
  
 
   
 
   
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==explicit R-matrices==
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* taken from http://mathoverflow.net/questions/5103/solutions-of-the-quantum-yang-baxter-equation
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* 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.
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* There is substantial information on these representations but the R-matrices are not explicit.
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===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.
  
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==books==
 
  
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* [[2009년 books and articles|찾아볼 수학책]]
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==related items==
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* http://gigapedia.info/1/
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* [[Yang-Baxter equation (YBE)]]
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* http://gigapedia.info/1/
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* [[R-matrix in quantum affine algebra]]
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* http://gigapedia.info/1/
 
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* http://gigapedia.info/1/
 
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
  
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==computational resource==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxVlBzUTYxOXdicXc/edit
  
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==encyclopedia==
 
==encyclopedia==
56번째 줄: 100번째 줄:
 
* 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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* http://en.wikipedia.org/wiki/
 
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* http://en.wikipedia.org/wiki/
 
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* http://en.wikipedia.org/wiki/
 
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==blogs==
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==articles==
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* 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
  
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*  구글 블로그 검색<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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** http://blogsearch.google.com/blogsearch?q=
 
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** http://blogsearch.google.com/blogsearch?q=
 
  
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==articles==
 
  
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*  R-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators<br>
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[[분류:quantum groups]]
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* [[2010년 books and articles|논문정리]]
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[[분류:math and physics]]
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* http://www.ams.org/mathscinet/search/publications.html?pg4=ALLF&s4=
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[[분류:migrate]]
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
  
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* http://www.ams.org/mathscinet
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== 메타데이터 ==
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* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
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* http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
 
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* http://dx.doi.org/
 
  
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===위키데이터===
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* ID : [https://www.wikidata.org/wiki/Q2993305 Q2993305]
  
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==TeX ==[[분류:개인노트]]
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==메타데이터==
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[[분류:quantum groups]]
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===위키데이터===
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[[분류:math and physics]]
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* ID :  [https://www.wikidata.org/wiki/Q2993305 Q2993305]
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[[분류:math and physics]]
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===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

\[R_{12}(u)R_{13}(u+v)R_{23}(v)=R_{23}(v)R_{13}(u+v)R_{12}(u)\]

  • \(R(u,\eta)\)
    • \(u\) 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 \(q=e^{\eta}\)


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

\[\bar R_i\bar R_{i+1}\bar R_i= \bar R_{i+1}\bar R_i \bar R_{i+1} \label{braid}\]


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

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.


related items

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

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'baum'}, {'OP': '*'}, {'LOWER': 'connes'}, {'LEMMA': 'conjecture'}]
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