Classifying (weak) coideal subalgebras of weak Hopf C⁎-algebras

Abstract

We develop a general approach to the problem of classification of weak coideal C˚-subalgebras of weak Hopf C˚-algebras. As an example, we consider weak Hopf C˚-algebras and their weak coideal C˚-subalgebras associated with Tambara Yamagami categories.

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

“…This paper continues the study of coactions of weak Hopf C*-algebras on C*-algebras and their applications which was initiated in [25] and [26]. Let us first recall our motivation.…”

“…In [26] we introduced the notion of a weak coideal I of B, the difference of which from a coideal is that 1 I is not necessarily equal to 1 B . One can show that if I is invariant and…”

“…The description of the Hayashi's functor for Tambara-Yamagami categories and the corresponding WHA's was originally obtained in [8]. Below we follow [26], 2.3 and 4.1, where one can find more details. Given a finite abelian group G, a non degenerate symmetric bicharacter χ on it and a number τ = ±|G| −1/2 , one can define a fusion category denoted by T Y(G, χ, τ ) [24].…”

“…Using now the Hayashi's functor H : T Y(G, χ, τ ) → Corr f (R), where R ∼ = C |G|+1 (see [8], [26]), one can apply Theorem 2.9 in order to construct a biconnected regular WHA G…”

“…In the Tambara-Yamagami case any invariant coideal is not only isomorphic but is itself of quotient type: Proof. Due to [26], Lemma 3.3, b) there is a partition (Γ i ) i∈I 0 of Ω such that X 0 = V ec < v 0 Γ i /i ∈ I 0 >. Moreover, putting K := {g ∈ G/Dim(X g ) = 0}, one has by Lemmas 5.8 and 5.9: X m = {0}, and X g = Cv g m for all g ∈ K, g = 0.…”