$R$-matrices and the Yang-Baxter equation on GNS representations of C$^{*}$-bialgebras

Abstract

A new construction method of $R$-matrix is given. Let $A$ be a C$^{*}$-bialgebra with a comultiplication $Δ$. For two states $ω$ and $ψ$ of $A$ which satisfy certain conditions, we construct a unitary $R$-matrix $R(ω,ψ)$ of the C$^{*}$-bialgebra $(A,Δ)$ on the tensor product of GNS representation spaces associated with $ω$ and $ψ$. The set $\{R(ω,ψ):ω,ψ\}$ satisfies a kind of Yang-Baxter equation. Furthermore, we show a nontrivial example of such $R$-matrices for a non-quasi-cocommutative C$^{*}$-bialgebra.

15 pages