Quasitriangular Hopf algebra

<h2 id="twisting">Twisting</h2>
<p>The property of being a <a href="/facts/Quasi-triangular_Hopf_algebra/Y6acUkBu">quasi-triangular Hopf algebra</a> is preserved by <a href="/facts/Quasi-bialgebra/5n7nU0EN">twisting</a> via an invertible element 
  
    
      
        F
        =
        
          ∑
          
            i
          
        
        
          f
          
            i
          
        
        ⊗
        
          f
          
            i
          
        
        ∈
        
          
            A
            ⊗
            A
          
        
      
    
    {\displaystyle F=\sum _{i}f^{i}\otimes f_{i}\in {\mathcal {A\otimes A}}}
  
 such that 
  
    
      
        (
        ε
        ⊗
        i
        d
        )
        F
        =
        (
        i
        d
        ⊗
        ε
        )
        F
        =
        1
      
    
    {\displaystyle (\varepsilon \otimes id)F=(id\otimes \varepsilon )F=1}
  
 and satisfying the cocycle condition
</p>

(
        F
        ⊗
        1
        )
        ⋅
        (
        Δ
        ⊗
        i
        d
        )
        (
        F
        )
        =
        (
        1
        ⊗
        F
        )
        ⋅
        (
        i
        d
        ⊗
        Δ
        )
        (
        F
        )
      
    
    {\displaystyle (F\otimes 1)\cdot (\Delta \otimes id)(F)=(1\otimes F)\cdot (id\otimes \Delta )(F)}

<p>Furthermore, 
  
    
      
        u
        =
        
          ∑
          
            i
          
        
        
          f
          
            i
          
        
        S
        (
        
          f
          
            i
          
        
        )
      
    
    {\displaystyle u=\sum _{i}f^{i}S(f_{i})}
  
 is invertible and the twisted antipode is given by 
  
    
      
        
          S
          ′
        
        (
        a
        )
        =
        u
        S
        (
        a
        )
        
          u
          
            −
            1
          
        
      
    
    {\displaystyle S'(a)=uS(a)u^{-1}}
  
, with the twisted comultiplication, R-matrix and co-unit change according to those defined for the <a href="/facts/Quasi-triangular_quasi-Hopf_algebra/GPpQQnV6">quasi-triangular quasi-Hopf algebra</a>. Such a twist is known as an admissible (or Drinfeld) twist.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Quasi-triangular_quasi-Hopf_algebra/GPpQQnV6">Quasi-triangular quasi-Hopf algebra</a></li>
<li><a href="/facts/Ribbon_Hopf_algebra/DctC2Y8w">Ribbon Hopf algebra</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Susan_Montgomery/aQgMzxcp">Montgomery, Susan</a> (1993). <i>Hopf algebras and their actions on rings</i>. Regional Conference Series in Mathematics. Vol. 82. Providence, RI: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0738-2. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0793.16029">0793.16029</a>.</li>
<li><a href="/facts/Susan_Montgomery/aQgMzxcp">Montgomery, Susan</a>; Schneider, Hans-Jürgen (2002). <i>New directions in Hopf algebras</i>. Mathematical Sciences Research Institute Publications. Vol. 43. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-81512-3. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0990.00022">0990.00022</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Montgomery & Schneider (2002), p. 72. <a href="https://books.google.com/books?id=I3IK9U5Co_0C&pg=PA72&dq=%22Quasitriangular%22" target="_blank">https://books.google.com/books?id=I3IK9U5Co_0C&pg=PA72&dq=%22Quasitriangular%22</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Quasitriangular Hopf algebra open-in-new

Quasitriangular Hopf algebra