Q-systems and compact W*-algebra objects

TL;DR

This paper establishes a direct categorical equivalence between normalized irreducible Q-systems and compact connected W*-algebra objects within rigid C*-tensor categories, bypassing subfactor theory.

Contribution

It provides a new, direct categorical proof of the equivalence between Q-systems and W*-algebra objects, expanding the understanding of algebraic structures in tensor categories.

Findings

Categorical equivalence between Q-systems and W*-algebra objects.

Direct proof avoiding subfactor theory.

Clarification of algebraic structures in tensor categories.

Abstract

We show that given a rigid C*-tensor category, there is an equivalence of categories between normalized irreducible Q-systems, also known as connected unitary Frobenius algebra objects, and compact connected W*-algebra objects. Although this result could be proved as a corollary of our previous article on realizations of algebra objects and discrete subfactors, we prove it here directly via categorical methods without passing through subfactor theory.