Pentagon equation arising from state equations of a C$^{*}$-bialgebra

TL;DR

This paper constructs an operator satisfying the pentagon equation from state equations of a non-cocommutative C$^{*}$-bialgebra, revealing new algebraic structures related to Cuntz algebras.

Contribution

It introduces a novel operator derived from state equations of the C$^{*}$-bialgebra ${ m O}_*$ that satisfies the pentagon equation, expanding understanding of algebraic structures in quantum algebra.

Findings

Constructed an operator W satisfying the pentagon equation.

Showed W* is an isometry and relates to the comultiplication.

Connected state equations of ${ m O}_*$ with algebraic equations like the pentagon.

Abstract

The direct sum ${\cal O}_{*}$ of all Cuntz algebras has a non-cocommutative comultiplication $\Delta_{\varphi}$ such that there exists no antipode of any dense subbialgebra of the C$^{*}$-bialgebra $({\cal O}_{*},\Delta_{\varphi})$. From states equations of ${\cal O}_{*}$ with respect to the tensor product, we construct an operator $W$ for $({\cal O}_{*},\Delta_{\varphi})$ such that $W^{*}$ is an isometry, $W(x\otimes I)W^{*}=\Delta_{\varphi}(x)$ for each $x\in {\cal O}_{*}$ and $W$ satisfies the pentagon equation.