title: Quasi-triangular quasi-Hopf algebra
text: # Quasi-triangular quasi-Hopf algebra
  
  A quasi-triangular quasi-Hopf algebra is a specialized form of a quasi-Hopf algebra defined by the Ukrainian mathematician Vladimir Drinfeld in 1989. It is also a generalized form of a quasi-triangular Hopf algebra.
  
  A quasi-triangular quasi-Hopf algebra is a set ${\mathcal {H_{A}}}=({\mathcal {A}},R,\Delta ,\varepsilon ,\Phi )$ where ${\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi )$ is a quasi-Hopf algebra and $R\in {\mathcal {A\otimes A}}$ known as the R-matrix, is an invertible element such that
  
      Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle R \Delta(a) = \sigma \circ \Delta(a) R}
  
  for all $a\in {\mathcal {A}}$, where $\sigma \colon {\mathcal {A\otimes A}}\rightarrow {\mathcal {A\otimes A}}$ is the switch map given by $x\otimes y\rightarrow y\otimes x$, and
  
      $(\Delta \otimes \operatorname {id} )R=\Phi _{312}R_{13}\Phi _{132}^{-1}R_{23}\Phi _{123}$
      $(\operatorname {id} \otimes \Delta )R=\Phi _{231}^{-1}R_{13}\Phi _{213}R_{12}\Phi _{123}^{-1}$
  
  where $\Phi _{abc}=x_{a}\otimes x_{b}\otimes x_{c}$ and $\Phi _{123}=\Phi =x_{1}\otimes x_{2}\otimes x_{3}\in {\mathcal {A\otimes A\otimes A}}$.
  
  The quasi-Hopf algebra becomes triangular if in addition, $R_{21}R_{12}=1$.
  
  The twisting of ${\mathcal {H_{A}}}$ by $F\in {\mathcal {A\otimes A}}$ is the same as for a quasi-Hopf algebra, with the additional definition of the twisted R-matrix
  
  A quasi-triangular (resp. triangular) quasi-Hopf algebra with $\Phi =1$ is a quasi-triangular (resp. triangular) Hopf algebra as the latter two conditions in the definition reduce the conditions of quasi-triangularity of a Hopf algebra.
  
  Similarly to the twisting properties of the quasi-Hopf algebra, the property of being quasi-triangular or triangular quasi-Hopf algebra is preserved by twisting.
  
  