Triangular C∗-bialgebra defined as the direct sum of matrix algebras

Let M*(C) denote the C∗-algebra defined as the direct sum of all matrix algebras {Mn(C):n⩾1}. It is known that M*(C) has a non-cocommutative comultiplication Δφ. From a certain set of transformations of integers, we construct a universal R-matrix R of the C∗-bialgebra (M*(C),Δφ) such that the quasi-cocommutative C∗-bialgebra (M*(C),Δφ,R) is triangular. Furthermore, it is shown that certain linear Diophantine equations are corresponded to the Yang–Baxter equations of R.