Regularity for braided multiplicative unitaries
David Buecher, Sutanu Roy

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.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra
