%0 Journal Article
%T $R$-matrices and the Yang-Baxter equation on GNS representations of C$^{*}$-bialgebras
%A Katsunori Kawamura
%J Mathematics
%D 2009
%I arXiv
%X A new construction method of $R$-matrix is given. Let $A$ be a C$^{*}$-bialgebra with a comultiplication $\Delta$. For two states $\omega$ and $\psi$ of $A$ which satisfy certain conditions, we construct a unitary $R$-matrix $R(\omega,\psi)$ of the C$^{*}$-bialgebra $(A,\Delta)$ on the tensor product of GNS representation spaces associated with $\omega$ and $\psi$. The set $\{R(\omega,\psi):\omega,\psi\}$ satisfies a kind of Yang-Baxter equation. Furthermore, we show a nontrivial example of such $R$-matrices for a non-quasi-cocommutative C$^{*}$-bialgebra.
%U http://arxiv.org/abs/0907.2280v3