# nLab coquasitriangular bialgebra

Coquasitriangularity is dual property to quasitriangularity.

A $k$-bialgebra (or, in particular, Hopf algebra) $\left(H,m,\eta ,\Delta ,ϵ\right)$ is coquasitriangular (or dual quasitriangular) if it is equipped with a $k$-linear map $R:H\otimes H\to k$ which is invertible in convolution algebra ${\mathrm{Hom}}_{k}\left(H\otimes H,k\right)$ (with respect to the convolution-unit $ϵ\otimes ϵ$) with a convolution inverse denoted $\overline{R}$ such that the opposite multiplication ${m}_{{H}_{\mathrm{op}}}:=m\circ \tau$ is given by

${m}_{{H}_{\mathrm{op}}}=R\star m\star \overline{R}$m_{H_{op}} = R\star m\star \bar{R}

and the following two identities hold when applied on $H\otimes H\otimes H$:

$R\left(m\otimes \mathrm{id}\right)={R}_{13}{R}_{23}$R(m\otimes id) = R_{13} R_{23}
$R\left(\mathrm{id}\otimes m\right)={R}_{12}{R}_{23}$R(id\otimes m) = R_{12} R_{23}

with the subscript notation as explained in the $n$lab entry quasitriangular Hopf algebra. The main examples come from quantized function algebras (that is, roughly, dual of quantized enveloping algebras).

Revised on August 16, 2009 19:10:38 by Toby Bartels (71.104.230.172)