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

TL;DR

This paper introduces a novel method for constructing R-matrices on GNS representations of C*-bialgebras, demonstrating their properties and providing a nontrivial example related to the Yang-Baxter equation.

Contribution

The paper presents a new construction of R-matrices for C*-bialgebras using states satisfying specific conditions, extending the understanding of solutions to the Yang-Baxter equation.

Findings

Constructed R-matrices on GNS spaces for C*-bialgebras.

Proved these R-matrices satisfy a Yang-Baxter type equation.

Provided an example for a non-quasi-cocommutative C*-bialgebra.

Abstract

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.