R-matrix

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

• Yang-Baxter equation
  \(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\)
  

 

 

 

R-matrix and Braid groups

• For \(R\) matrix on \(V \otimes V\), define \(\bar R=p\circ R\) 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.
  
• with an R-matrix satisfying the YBE, we obtain a representation of the Braid group, which then gives a link invariant in Knot theory
  

 

 

 

 

books

• 찾아볼 수학책
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Braided_monoidal_category
• http://en.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=

 

articles

• R-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators
  
• 논문정리
• http://www.ams.org/mathscinet/search/publications.html?pg4=ALLF&s4=
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html

• http://www.ams.org/mathscinet
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• 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=
• http://dx.doi.org/