Pentagon Equation and Compact Quantum Semigroups

TL;DR

This paper explores the extension of multiplicative unitaries from compact quantum groups to semigroups, introducing a new operator framework based on the pentagon equation that generalizes the concept.

Contribution

It proposes a novel operator approach using the pentagon equation to define comultiplication on C*-algebras, extending the multiplicative unitary concept to quantum semigroups.

Findings

Examples of C*-algebras with non-trivial comultiplication lacking multiplicative unitaries

Introduction of a new operator framework based on the pentagon equation

Proof of comultiplication representation for some compact quantum semigroups

Abstract

The generalization of multiplicative unitary notion from compact quantum groups to compact quantum semigroups is considered. We show why the same construction doesn't work in this case by giving examples of C*-algebras with non-trivial comultiplication which do not admit multiplicative unitaries. By the use of the pentagon equation we suggest a notion of an operator which gives comultiplication on any C*-algebra. The multiplicative unitary turns out to be its special case. We prove for some compact quantum semigroups that the comultiplication is given by such operator.