Regularity for braided multiplicative unitaries

TL;DR

This paper extends the concepts of regularity for braided multiplicative unitaries within W*-categories, showing that under bi-regularity, leg-algebras form braided C*-bialgebras and the braiding properties are preserved in Yetter-Drinfeld categories.

Contribution

It generalizes regularity notions to braided unitaries in W*-categories and demonstrates the preservation of braiding properties in related categories.

Findings

Leg-algebras form braided C*-bialgebras under bi-regularity.

Braiding on Yetter-Drinfeld categories remains (semi-)regular.

Properties are established for solutions of the braided Pentagon equation.

Abstract

We generalise the notions of semi-regularity, regularity, and bi-regularity to unitary solutions of the braided Pentagon equation in concrete W*-categories with semi-regular/regular/bi-regular braiding, and study their properties. We show, for example, that under bi-regularity-assumptions, the leg-algebras form braided C*-bialgebras. Moreover, we "close the circle" for the representation categories: The braiding on the Yetter-Drinfeld category of a (semi-)regular multiplicative unitary in a category with (semi-)regular braiding is again (semi-)regular.