Quasitriangular Hopf algebra

In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of ${\displaystyle H\otimes H}$ such that

where ${\displaystyle R_{12}=\phi _{12}(R)}$, ${\displaystyle R_{13}=\phi _{13}(R)}$, and ${\displaystyle R_{23}=\phi _{23}(R)}$, where ${\displaystyle \phi _{12}:H\otimes H\to H\otimes H\otimes H}$, ${\displaystyle \phi _{13}:H\otimes H\to H\otimes H\otimes H}$, and ${\displaystyle \phi _{23}:H\otimes H\to H\otimes H\otimes H}$, are algebra morphisms determined by

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