R-matrix

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imported>Pythagoras0님의 2012년 10월 26일 (금) 11:23 판 (→‎YBE)
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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
    \(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\)
  • $R(u,\eta)$
    • $u$ 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

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



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