CQG algebras: a direct algebraic approach to compact quantum groups

TL;DR

This paper introduces CQG algebras as a purely algebraic framework for compact quantum groups, establishing foundational theorems and connecting algebraic and operator algebraic perspectives.

Contribution

It provides a new algebraic approach to CQG algebras, proving the Peter-Weyl theorem and Haar functional existence within this framework.

Findings

Proved the Peter-Weyl theorem for CQG algebras

Established the existence of a unique positive Haar functional

Showed CQG algebras can be completed to C*-algebras

Abstract

The purely algebraic notion of CQG algebra (algebra of functions on a compact quantum group) is defined. In a straightforward algebraic manner, the Peter-Weyl theorem for CQG algebras and the existence of a unique positive definite Haar functional on any CQG algebra are established. It is shown that a CQG algebra can be naturally completed to a $C^\ast$-algebra. The relations between our approach and several other approaches to compact quantum groups are discussed.