nLab
coquasitriangular bialgebra

Coquasitriangularity is dual property to quasitriangularity.

A kk-bialgebra (or, in particular, Hopf algebra) (H,m,η,Δ,ϵ)(H, m,\eta,\Delta,\epsilon) is coquasitriangular (or dual quasitriangular) if it is equipped with a kk-linear map R:HHkR:H\otimes H\to k which is invertible in convolution algebra Hom k(HH,k)Hom_k(H\otimes H,k) (with respect to the convolution-unit ϵϵ\epsilon\otimes\epsilon) with a convolution inverse denoted R¯\bar{R} such that the opposite multiplication m H op:=mτm_{H_{op}} := m\circ \tau is given by

m H op=RmR¯ m_{H_{op}} = R\star m\star \bar{R}

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

R(mid)=R 13R 23 R(m\otimes id) = R_{13} R_{23}
R(idm)=R 12R 23 R(id\otimes m) = R_{12} R_{23}

with the subscript notation as explained in the nnlab 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)