Non-existence of universal $R$-matrix for some C$^{*}$-bialgebras

Abstract

For a C$^{*}$-bialgebra $A$ with a comultiplication $Δ$, a universal $R$-matrix of $(A,Δ)$ is defined as a unitary element in the multiplier algebra $M(A\otimes A)$ of $A\otimes A$ which is an intertwiner between $Δ$ and its opposite comultiplication $Δ^{op}$. We show that there exists no universal $R$-matrix for some C$^{*}$-bialgebras.

8 pages