Triangular C$^{*}$-bialgebra defined as the direct sum of matrix algebras

TL;DR

This paper constructs a triangular C*-bialgebra structure on the direct sum of all matrix algebras, demonstrating the existence of a universal R-matrix that makes it quasi-cocommutative.

Contribution

It introduces a triangular structure with a universal R-matrix on the C*-bialgebra formed by the direct sum of matrix algebras, expanding the understanding of quantum group structures.

Findings

Existence of a universal R-matrix for the bialgebra

The bialgebra is shown to be triangular

Construction of a quasi-cocommutative structure

Abstract

Let $M_{*}({\bf C})$ denote the C$^{*}$-algebra defined as the direct sum of all matrix algebras $\{M_{n}({\bf C}):n\geq 1\}$. It is known that $M_{*}({\bf C})$ has a non-cocommutative comultiplication $\Delta_{\varphi}$. We show that the C$^{*}$-bialgebra $(M_{*}({\bf C}),\Delta_{\varphi})$ has a universal $R$-matrix $R$ such that the quasi-cocommutative C$^{*}$-bialgebra $(M_{*}({\bf C}),\Delta_{\varphi},R)$ is triangular.