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

• 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.

related items

• Yang-Baxter equation (YBE)
• R-matrix in quantum affine algebra

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxVlBzUTYxOXdicXc/edit

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Braided_monoidal_category

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