Yetter-Drinfel'd algebras and coideals of Weak Hopf $C^*$-Algebras

TL;DR

This paper characterizes braided commutative Yetter-Drinfeld $C^*$-algebras over weak Hopf $C^*$-algebras, studies their quotient coideals, and provides explicit examples related to Tambara-Yamagami categories.

Contribution

It offers a categorical characterization of braided commutative Yetter-Drinfeld $C^*$-algebras over weak Hopf $C^*$-algebras and describes quotient coideal subalgebras explicitly.

Findings

Categorical characterization of braided commutative Yetter-Drinfeld $C^*$-algebras.

Description of quotient type coideal subalgebras.

Explicit examples related to Tambara-Yamagami categories.

Abstract

We characterize braided commutative Yetter-Drinfeld $C^*$-algebras over weak Hopf $C^*$-algebras in categorical terms. Using this, we then study quotient type coideal subalgebras of a given weak Hopf $C^*$-algebra $\mathcal G$ and coideal subalgebras invariant with respect to the adjoint action of $\mathcal G$. Finally, as an example, we explicitly describe quotient type coideal subalgebras of the weak Hopf $C^*$-algebras associated with Tambara-Yamagami categories.