Triangular C1-bialgebra defined as the direct sum of matrix algebras

Let $M_{*}(\mathbf{C})$ denote the ${\mathrm{C}}^1$-algebra defined as the direct sum of all matrix algebras $\{M_n(\mathbf{C}):n?1\}$. It is known that $M_{*}(\mathbf{C})$ has a non-cocommutative comultiplication ${\mathrm{\Delta }}_{\phi }$. From a certain set of transformations of integers, we construct a universal R-matrix R of the ${\mathrm{C}}^1$-bialgebra $(M_{*}(\mathbf{C})\text{,}{\mathrm{\Delta }}_{\phi })$ such that the quasi-cocommutative ${\mathrm{C}}^1$-bialgebra $(M_{*}(\mathbf{C})\text{,}{\mathrm{\Delta }}_{\phi }\text{,}R)$ is triangular. Furthermore, it is shown that certain linear Diophantine equations are corresponded to the Yang–Baxter equations of R.