Classifying (Weak) Coideal Subalgebras of Weak Hopf C*-Algebras

TL;DR

This paper introduces a general method for classifying weak coideal subalgebras within weak Hopf C*-algebras, exemplified through structures linked to Tambara Yamagami categories.

Contribution

It provides a novel classification framework for weak coideal subalgebras in weak Hopf C*-algebras, with specific applications to Tambara Yamagami categories.

Findings

Developed a general classification approach.

Applied method to Tambara Yamagami categories.

Enhanced understanding of subalgebra structures in 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.

Equations

$$\Delta (bc)=\Delta (b)\Delta (c),\Delta (b^{\ast })=\Delta (b{)}^{\ast },$$

$$\epsilon (b{c}_1)\epsilon (c_{2}d)$$

$$(\Delta (1)\otimes 1)(1\otimes \Delta (1))$$

$$m(i{d}_B\otimes S)\Delta (b)$$

$$m(S\otimes i{d}_B)\Delta (b)$$

$$<\phi \psi ,b>$$

$$<\hat{\Delta }(\phi ),b\otimes c>$$

$$<\hat{S}(\phi ),b>$$

$$<{\varphi }^{\ast },b>$$

$$(i{d}_B\otimes h)\Delta =({\epsilon }_t\otimes h)\Delta ,h\circ S=h,h\circ {\epsilon }_t=\epsilon ,(i{d}_B\otimes h)\Delta (1_B)=1_B.$$

$$<b\cdot v,w>=<v,b^{\ast }\cdot w>,\text{for all}v,w\in H,b\in B.$$

$$R_H(1_B)={\Sigma }_i(G^{1/2}\cdot {\overline{e}}_i{\otimes }_{B_t}\cdot e_i),{\overline{R}}_H(1_B)={\Sigma }_i(e_i{\otimes }_{B_t}{G}^{-1/2}\cdot {\overline{e}}_i),$$

$$R_U(1_B)={\Sigma }_i({\hat{G}}^{1/2}\cdot {\overline{e}}_i{\otimes }_{B_s}{e}_i),{\overline{R}}_U(1_B)={\Sigma }_i(e_i{\otimes }_{B_s}{\hat{G}}^{-1/2}\cdot {\overline{e}}_i),$$

$$\mathcal{H}(x)=H^x=\underset{y,z\in \Omega }{⨁}Hom(z,y\otimes x),\text{for every}x\in \Omega ,$$

$$p_y\cdot H^x\cdot p_z=Hom(z,y\otimes x),\text{for all}x,y,z\in \Omega .$$

$$\mathcal{H}(f)(g)=(i{d}_z\otimes f)\cdot g,\text{for any}z,t\in \Omega andg\in p_z\cdot H^x\cdot p_t.$$

$${\mathcal{H}}_{x,y}(v\otimes w)=a_{z,x,y}\cdot (v\otimes i{d}_y)\cdot w\in p_z\cdot H^{(x\otimes y)}\cdot p_s,$$

$$B=\underset{x\in \Omega }{⨁}{H}_x\otimes {\overline{H}}_x,$$

$$(w\otimes \overline{v}{)}_x\cdot (g\otimes \overline{h}{)}_y=({\mathcal{H}}_{x,y}(w\otimes g)\otimes \overline{{\mathcal{H}}_{x,y}(v\otimes h)}{)}_{x\otimes y}\in H^{(x\otimes y)}\otimes \overline{H^{(x\otimes y)}}$$

$$\Delta (w\otimes \overline{v})=\underset{j}{⨁}(w\otimes \overline{e_j^x}{)}_x\otimes (e_j^x\otimes \overline{v}{)}_x,$$

$$\epsilon (w\otimes \overline{v})=<w,v{>}_x.$$

$${\Phi }_x(v)=(i{d}_y\otimes {\overline{R_x}}^{\ast })\cdot a_{y,x,x^{\ast }}\cdot (v\otimes i{d}_{x^{\ast }}),{\Psi }_x(\overline{v})=(\overline{v}\otimes i{d}_{x^{\ast }})\cdot a_{y,x,x^{\ast }}^{-1}\cdot (i{d}_y\otimes \overline{R_x}),$$

$$S(w\otimes \overline{v})={\Psi }_x(\overline{v})\otimes {\Phi }_x(w),$$

$$(w\otimes \overline{v}{)}^{\ast }=w^{♮}\otimes {\overline{v}}^{♭},\text{where}{w}^{♮}={\Psi }_x(\overline{w}),{\overline{v}}^{♭}={\Phi }_x(v).$$

$${\mathfrak{a}}_x(v)=\underset{j}{\Sigma }{e}_j^x\otimes \overline{e_j^x}\otimes v,\text{where}v\in H_x,$$

$$\hat{B}=\underset{x\in \Omega }{⨁}B(H_x),$$

$$<A,w\otimes \overline{v}>=<Aw,v{>}_x.$$

$$v\circ w={\mathcal{H}}_{x,y}(v{\otimes }_{R}w)$$

$$\mathfrak{a}(\alpha (x)a\beta (y))=(1_A\otimes x)\mathfrak{a}(a)(1_A\otimes y),$$

$$(\alpha (x)\otimes 1_B)\mathfrak{a}(a)(\beta (y)\otimes 1_B)=(1_A\otimes S(x))\mathfrak{a}(a)(1_A\otimes S(y)).$$

$$A_U:=\{a\in A|\mathfrak{a}(a)\in \mathfrak{a}(1_A)(A\otimes B_U)\}.$$

$$A=\underset{x\in \Omega }{⨁}{A}_{U^x}:=\underset{x\in \Omega }{⨁}(F(U^x)\otimes \overline{H^x})$$

$$\tilde{A}=\underset{U\in \Vert UCorep(\mathfrak{G})\Vert }{⨁}{A}_U:=\underset{U\in \Vert UCorep(\mathfrak{G})\Vert }{⨁}(\overline{F(U)}\otimes H_U),$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Weak Hopf $C^{*}$-algebras

4. 2.2 Unitary representations and corepresentations of a weak Hopf $C^{*}$-algebra

5. 2.3 The Hayashi’s fiber functor and reconstruction theorem.

6. 2.4 Coactions.

7. 2.5 Categorical duality.

8. 3 Classifying Indecomposable Weak Coideals

9. 4 Weak Hopf $C^{*}$-Algebras related to Tambara-Yamagami categories

10. 4.1 Tambara-Yamagami categories

11. 4.2 Classification of Indecomposable Finite Dimensional $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$-$C^{*}$-algebras

12. 5 Indecomposable Weak Coideals of $\mathfrak{G}_{TY}$

13. 5.1 The case $A^{m}=\{0\}$

14. 5.2 The case $A^{m}\neq\{0\}$ 111AMS Subject Classification [2010]: Primary 16T05, Secondary 18D10, Tertiary 46L05.222 Keywords : Coactions and corepresentations of quantum groupoids, $C^{*}$-categories, reconstruction theorem.

1 Introduction

It is known that any finite tensor category equipped with a fiber functor to the category of finite dimensional vector spaces is equivalent to the representation category of some Hopf algebra - see, for example, [5], Theorem 5.3.12. But many tensor categories do not admit a fiber functor, so they cannot be presented as representation categories of Hopf algebras. On the other hand, T. Hayashi [6] showed that any fusion category always admits a tensor functor to the category of bimodules over some semisimple (even commutative) algebra. Using this, it was proved in [6], [13], [15] that any fusion category is equivalent to the representation category of some algebraic structure generalizing Hopf algebras called a weak Hopf algebra [2] or a finite quantum groupoid [12]. The main difference between weak and usual Hopf algebra is that in the former the coproduct $\Delta$ is not necessarily unital.

Apart from tensor categories, weak Hopf algebras have interesting applications to the subfactor theory. In particular, for any finite index and finite depth $II_{1}$-subfactor $N\subset M$, there exists a weak Hopf $C^{*}$-algebra $\mathfrak{G}$ such that the corresponding Jones tower can be expressed in terms of crossed products of $N$ and $M$ with $\mathfrak{G}$ and its dual. Moreover, there is a Galois correspondence between intermediate subfactors in this Jones tower and coideal $C^{*}$-subalgebras of $\mathfrak{G}$ - see [11]. This motivates the study of coideal $C^{*}$-subalgebras of weak Hopf $C^{*}$-algebras which is the subject of the present paper. The abbreviation WHA will always mean a weak Hopf $C^{*}$-algebra.

A coideal $C^{*}$-subalgebra is a special case of the notion of a $\mathfrak{G}$-$C^{*}$-algebra, which is, by definition, a unital $C^{*}$ algebra $A$ equipped with a coaction $\mathfrak{a}$ of a WHA $\mathfrak{G}=(B,\Delta,S,\varepsilon)$. More exactly, we will use the following

Definition 1.1

A weak right coideal $C^{*}$-subalgebra of $B$ is a right $\mathfrak{G}$-$C^{*}$-algebra $(A,\mathfrak{a})$ with a $C^{*}$-algebra inclusion $i:A\mapsto B$ (not necessarily unital) satisfying $\Delta=(i\otimes id_{B})\mathfrak{a}$. One can think of $A$ as of a $C^{*}$-subalgebra of $B$ such that $\mathfrak{a}=\Delta$. If $i$ is unital, we call $A$ a coideal $C^{*}$-subalgebra of $B$.

For the sake of brevity, we will call a (weak) coideal $C^{*}$-subalgebra a (weak) coideal of $B$. Note that if $\mathfrak{G}$ is a usual Hopf $C^{*}$-algebra, then one can prove that necessarily $1_{A}=1_{B}$, so weak and usual coideals coincide.

It was shown in [18] that any $\mathfrak{G}$-$C^{*}$-algebra $(A,\mathfrak{a})$ corresponds to a pair $(\mathcal{M},M)$, where $\mathcal{M}$ is a module category with a generator $M$ over the category of unitary corepresentations of $\mathfrak{G}$.

In Preliminaries we recall definitions and facts needed for the exact formulation of this result expressed in Theorem 2.9. Note that similar categorical duality for compact quantum group coactions was obtained earlier in [4], [9].

Section 3 is devoted to necessary conditions which a pair $(\mathcal{M},M)$ satisfies if $(A,\mathfrak{a})$ is an indecomposable (weak) coideal.

In Sections 4 and 5 the above mentioned general approach is applied to the problem of classification of $\mathfrak{G}$-algebras and weak coideals of WHA’s associated with a concrete class of fusion categories - Tambara-Yamagami categories $\mathcal{T}\mathcal{Y}(G,\chi,\tau)$ [16].

Recall that simple objects of $\mathcal{T}\mathcal{Y}(G,\chi,\tau)$ are exactly the elements of a finite abelian group $G$ and one separate element $m$ satisfying the fusion rule $g\cdot h=gh,\ g\cdot m=m\cdot g=m,\ m^{2}=\underset{g\in G}{\Sigma}g,\ g^{*}=-g,\ m=m^{*}\ (g,h\in G)$. These categories are parameterized by non degenerate symmetric bicharacters $\chi:G\times G\to\mathbb{C}\backslash\{0\}$ and $\tau=\pm|G|^{-1/2}$. For any subset $K\subset G$, we shall denote $K^{\perp}:=\{g\in G|\chi(k,g)=1,\ \forall k\in K\}.$

The Hayashi’s reconstruction theorem allows to construct a WHA $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$ associated with $\mathcal{T}\mathcal{Y}(G,\chi,\tau)$ - see [8]. We recall this construction in slightly different form in Subsection 4.1. Then, using the methods elaborated in [7], we classify in Subsection 4.2 indecomposable module categories over representations of $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$, their autoequivalences and generators. Together with the above mentioned results this leads to the following classification theorem:

Theorem 1.2

There are two types of isomorphism classes of indecomposable finite dimensional $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$-$C^{*}$-algebras:

(i)* those parameterized by pairs $(K,\{m_{\lambda}\}^{orb})$, where $K<G$ and $\{m_{\lambda}\}^{orb}$ is the orbit of a nonzero collection $\{m_{\lambda}\in\mathbb{Z}_{+}|\lambda\in G/K\}$ under the action of the group of translations on $G/K$.*

(ii)* those parameterized by pairs $(K,(\{m_{\lambda}\},\{m_{\mu}\})^{orb})$, where $K<G$ and $(\{m_{\lambda}\},\{m_{\mu}\})^{orb}$ is the orbit of a nonzero double collection $(\{m_{\lambda}\in\mathbb{Z}_{+}|\lambda\in G/K\},\{m_{\mu}\in\mathbb{Z}_{+}|\mu\in G/K^{\perp}\})$ under the action of:*

a) the group of translations on $G/K\times G/K^{\perp}$ if $K\neq K^{\perp}$;

b) the semi-direct product $(G/K\times G/K)\underset{\sigma}{\ltimes}\mathbb{Z}_{2}$ generated by translations on $G/K\times G/K$ and the flip $\sigma:(\{m_{\lambda}\},\{m_{\mu}\})\leftrightarrow(\{m_{\mu}\},\{m_{\lambda}\})$ if $K=K^{\perp}$.

Finally, Section 5 is devoted to the classification of indecomposable (weak) coideals of $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$. Their classification is given by the following

Theorem 1.3

Isomorphism classes of indecomposable weak coideals of $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$ are parameterized by pairs $(K,(Z_{0},Z_{1})^{orb})$, where $K$ is a subgroup of $G$ and $(Z_{0},Z_{1})^{orb}$ is the orbit of a nonempty subset $(Z_{0},Z_{1})\subset G/K\times G/K^{\perp}$ such that either $|Z_{0}|\leq 1$ or $|Z_{1}|\leq 1$, under the action of:

a) the group of translations on $G/K\times G/K^{\perp}$ if $K\neq K^{\perp}$;

b) the semi-direct product $(G/K\times G/K)\underset{\sigma}{\ltimes}\mathbb{Z}_{2}$ generated by translations on $G/K\times G/K$ and the flip $\sigma:(Z_{0},Z_{1})\leftrightarrow(Z_{1},Z_{0})$ if $K=K^{\perp}$.

Given a subgroup $K<G$, the isomorphism classes containing coideals correspond exactly to the following orbits:

when $K\not=K^{\perp}$, to the four orbits $\{(\lambda,\emptyset)/\lambda\in G/K\}$, $\{(\emptyset,\mu),/\mu\in G/K^{\perp}\}$, $\{(G/K,\mu)/\mu\in G/K^{\perp}\}$, $\{(\lambda,G/K^{\perp})/\lambda\in G/K\}$,

when $K=K^{\perp}$, to the two orbits $\{(\lambda,\emptyset)\cup(\emptyset,\lambda),/\lambda\in G/K\}$ and $\{(G/K,\lambda)\cup(\lambda,G/K)/\lambda\in G/K\}$.

In fact, we give an explicit construction of representatives of all isomorphism classes of indecomposable finite dimensional $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$-$C^{*}$-algebras and indecomposable (weak) coideals of $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$.

Our references are: to [5] for tensor categories, to [10] for $C^{*}$-tensor categories and to [12] for weak Hopf algebras (finite quantum groupoids).

2 Preliminaries

2.1 Weak Hopf $C^{*}$-algebras

A weak Hopf $C^{*}$-algebra (WHA) $\mathfrak{G}=(B,\Delta,S,\varepsilon)$ is a finite dimensional $C^{*}$-algebra $B$ with the comultiplication $\Delta:B\to B\otimes B$, counit $\varepsilon:B\to\mathbb{C}$, and antipode $S:B\to B$ such that $(B,\Delta,\varepsilon)$ is a coalgebra and the following axioms hold for all $b,c,d\in B$ :

1. (1)

$\Delta$ is a (not necessarily unital) $*$-homomorphism :

[TABLE] 2. (2)

The unit and counit satisfy the identities (we use the Sweedler leg notation $\Delta(c)=c_{1}\otimes c_{2},\ (\Delta\otimes id_{B})\Delta(c)=c_{1}\otimes c_{2}\otimes c_{3}$ etc.):

[TABLE] 3. (3)

$S$ is an anti-algebra and anti-coalgebra map such that

[TABLE]

where $m$ denotes the multiplication.

The right hand sides of two last formulas are called target and source counital maps $\varepsilon_{t}$ and $\varepsilon_{s}$, respectively. Their images are unital $C^{*}$-subalgebras of $B$ called target and source counital subalgebras $B_{t}$ and $B_{s}$, respectively.

The dual vector space $\hat{B}$ has a natural structure of a weak Hopf $C^{*}$-algebra $\hat{\mathfrak{G}}=(\hat{B},\hat{\Delta},\hat{S},\hat{\varepsilon})$ given by dualizing the structure operations of $B$:

[TABLE]

for all $b,c\in B$ and $\varphi,\psi\in\hat{B}$. The unit of $\hat{B}$ is $\varepsilon$ and the counit is $1$.

The counital subalgebras commute elementwise, we have $S\circ\varepsilon_{s}=\varepsilon_{t}\circ S$ and $S(B_{t})=B_{s}$. We say that $B$ is connected if $B_{t}\cap Z(B)=\mathbb{C}$ (where $Z(B)$ is the center of $B$), coconnected if $B_{t}\cap B_{s}=\mathbb{C}$, and biconnected if both conditions are satisfied.

The antipode $S$ is unique, invertible, and satisfies $(S\circ*)^{2}=id_{B}$. We will only consider regular quantum groupoids, i.e., such that $S^{2}|_{B_{t}}=id$. In this case, there exists a canonical positive element $H$ in the center of $B_{t}$ such that $S^{2}$ is an inner automorphism implemented by $G=HS(H)^{-1}$, i.e., $S^{2}(b)=GbG^{-1}$ for all $b\in B$. The element $G$ is called the canonical group-like element of $B$, it satisfies the relation $\Delta(G)=(G\otimes G)\Delta(1)=\Delta(1)(G\otimes G)$.

There exists a unique positive functional $h$ on $B$, called a normalized Haar measure such that

[TABLE]

We will dehote by $H_{h}$ the GNS Hilbert space generated by $B$ and $h$ and by $\Lambda_{h}:B\to H_{h}$ the corresponding GNS map.

2.2 Unitary representations and corepresentations of a weak Hopf $C^{*}$-algebra

Let $\mathfrak{G}=(B,\Delta,S,\varepsilon)$ be a weak Hopf $C^{*}$-algebra. We denote by $\varepsilon_{t},\varepsilon_{s}$ its target and source counital maps, by $B_{t}$ and $B_{s}$ its target and source subalgebras, respectively, and by $G$ its canonical group-like element. We also denote by $h$ the normalized Haar measure of $\mathfrak{G}$.

Any object of the category $URep(\mathfrak{G})$ of unitary representations of $\mathfrak{G}$ is a left $B$-module of finite rank such that the underlying vector space is a Hilbert space $H$ with a scalar product $<\cdot,\cdot>$ satisfying

[TABLE]

$URep(\mathfrak{G})$ is a semisimple category whose morphisms are $B$-linear maps and simple objects are irreducible $B$-modules. One defines the tensor product of two objects $H_{1},H_{2}\in URep(\mathfrak{G})$ as the Hilbert subspace $\Delta(1_{B})\cdot(H_{1}\otimes H_{2})$ of the usual tensor product together with the action of $B$ given by $\Delta$. Here we use the fact that $\Delta(1_{B})$ is an orthogonal projection.

Tensor product of morphisms is the restriction of the usual tensor product of $B$-module morphisms. Let us note that any $H\in URep(\mathfrak{G})$ is automatically a $B_{t}$-bimodule via $z\cdot v\cdot t:=zS(t)\cdot v,\ \forall z,t\in B_{t},v\in E$, and that the above tensor product is in fact $\otimes_{B_{t}}$, moreover the $B_{t}$-bimodule structure for $H_{1}\otimes_{B_{t}}H_{2}$ is given by $z\cdot\xi\cdot t=(z\otimes S(t))\cdot\xi,\ \forall z,t\in B_{t},\xi\in H_{1}\otimes_{B_{t}}H_{2}$. The above tensor product is associative, so the associativity isomorphisms are trivial. The unit object of $URep(\mathfrak{G})$ is $B_{t}$ with the action of $B$ given by $b\cdot z:=\varepsilon_{t}(bz),\ \forall b\in B,z\in B_{t}$ and the scalar product $<z,t>=h(t^{*}z)$.

For any morphism $f:H_{1}\to H_{2}$, let $f^{*}:H_{2}\to H_{1}$ be the adjoint linear map: $<f(v),w>=<v,f^{*}(w)>,\ \forall v\in H_{1},w\in H_{2}$. Clearly, $f^{*}$ is $B$-linear, $f^{**}=f$, $(f\otimes_{B_{t}}g)^{*}=f^{*}\otimes_{B_{t}}g^{*}$, and $End(H)$ is a $C^{*}$-algebra, for any object $H$. So $URep(\mathfrak{G})$ is a finite $C^{*}$-multitensor category (${\bf 1}$ can be decomposable).

The conjugate object for any $H\in URep(\mathfrak{G})$ is the dual vector space $\hat{H}$ naturally identified ($v\mapsto\overline{v}$) with the conjugate Hilbert space $\overline{H}$ with the action of $B$ defined by $b\cdot\overline{v}=\overline{G^{1/2}S(b)^{*}G^{-1/2}\cdot v}$, where $G$ is the canonical group-like element of $\mathfrak{G}$. Then the rigidity morphisms defined by

[TABLE]

where $\{e_{i}\}_{i}$ is any orthogonal basis in $H$, satisfy all the needed properties - see [3], 3.6. Also, it is known that the $B$-module $B_{t}$ is irreducible if and only if $B_{s}\cap Z(B)=\mathbb{C}1_{B}$, i.e., if $\mathfrak{G}$ is connected. So that, we have

Proposition 2.1

$URep(\mathfrak{G})$* is a rigid finite $C^{*}$-multitensor category with trivial associativity constraints. It is $C^{*}$-tensor if and only if $\mathfrak{G}$ is connected.*

Definition 2.2

A right unitary corepresentation $U$ of $\mathfrak{G}$ on a Hilbert space $H_{U}$ is a partial isometry $U\in B(H_{U})\otimes B$ such that:

(i) $(id\otimes\Delta)(U)=U_{12}U_{13}$.

(ii) $(id\otimes\varepsilon)(U)=id$.

If $U$ and $V$ are two right corepresentations on Hilbert spaces $H_{U}$ and $H_{V}$, respectively, a morphism between them is a bounded linear map $T\in B(H_{U},H_{V})$ such that $(T\otimes 1_{B})U=V(T\otimes 1_{B})$. We denote by $UCorep(\mathfrak{G})$ the category whose objects are unitary corepresentations on finite dimensional vector spaces with morphisms as above.

If $\mathfrak{G}$ is coconnected (i.e., if $B_{t}\cap B_{s}=\mathbb{C}1_{B}$), $UCorep(\mathfrak{G})$ is a rigid $C^{*}$-tensor category with trivial associativities isomorphic to $URep(\hat{\mathfrak{G}})$. Namely, any $H_{U}$ is a right $B$-comodule via $v\mapsto U(v\otimes 1_{B})$, therefore, automatically a $(B_{s},B_{s})$-bimodule. Then tensor product $U\otop V:=U_{13}V_{23}$ acts on $H_{U}\otimes_{B_{s}}H_{V}$, the unit object $U_{\varepsilon}\in B(B_{s})\otimes B$ is defined by $z\otimes b\mapsto\Delta(1_{B})(1_{B}\otimes zb),\ \forall z\in B_{s},b\in B$, and the rigidity morphisms related to the conjugate $\overline{U}$ of an object $U$ which acts on the conjugate Hilbert space $\overline{H}_{U}$ of $H_{U}$, are

[TABLE]

where $\{e_{i}\}_{i}$ is any orthogonal basis in $H_{U}$. We denote by $\Omega$ an exhaustive set of representatives of the equivalence classes of irreducibles in $UCorep(\mathfrak{G})$.

Denote $H_{U^{x}}$ by $H^{x}$, then $U^{x}=\oplus_{i,j}m^{x}_{i,j}\otimes U^{x}_{i,j}$, where $m^{x}_{i,j}$ are the matrix units of $B(H^{x})$ with respect to some orthogonal basis $\{e_{i}\}\in H^{x}$ and $U^{x}_{i,j}$ are the corresponding matrix coefficients of $U^{x}$. Recall that $B=\oplus_{x\in\Omega}B_{U^{x}}$, where $B_{U^{x}}=Span(U^{x}_{i,j})$.

2.3 The Hayashi’s fiber functor and reconstruction theorem.

Let $\mathcal{C}$ be a rigid finite $C^{*}$-tensor category and $\Omega=Irr(\mathcal{C})$ be an exhaustive set of representatives of equivalence classes of its simple objects. Let $R$ be the $C^{*}$-algebra $R=\mathbb{C}^{\Omega}=\underset{x\in\Omega}{\bigoplus}\mathbb{C}p_{x}$, where $p_{x}=p_{x}^{*}$ are mutually orthogonal idempotents: $p_{x}p_{y}=\delta_{x,y}p_{x}$, for all $x,y\in\Omega$. Let us define a functor $\mathcal{H}$ from $\mathcal{C}$ to the category $Corr_{f}(R)$ of finite dimensional Hilbert $R$-bimodules (called $R$-correspondences) by:

[TABLE]

where $Hom(x,y)$ is the vector space of morphisms $x\to y$. The $R$-bimodule structure on $H^{x}$ is given by:

[TABLE]

If $f\in Hom(x,y)$, then $\mathcal{H}(f):H^{x}\to H^{y}$ is defined by:

[TABLE]

The tensor structure of $\mathcal{H}$ is a family of natural isomorphisms $\mathcal{H}_{x,y}:H^{x}\otimes H^{y}\to H^{x}\underset{R}{\otimes}H^{y}$ defined by:

[TABLE]

for all $v\in p_{z}\cdot H^{x}\cdot p_{t},w\in p_{t}\cdot H^{y}\cdot p_{s},z,s,t\in\Omega$. Here $a_{z,x,y}$ are the associativity isomorphisms of $\mathcal{C}$.

We define the scalar product on $H^{x}$ as follows. If $x,y,z\in\Omega$ and $f,g\in Hom(z,y\otimes x)$, then $g^{*}\in Hom(y\otimes x,z)$ and $g^{*}\cdot f\in End(z)=\mathbb{C}$, so one can put $<f,g>_{x}=g^{*}\cdot f$. The subspaces $Hom(z,y\otimes x)$ are declared to be orthogonal, so $H^{x}\in Corr_{f}(R)$. Dually, $\overline{H}^{x}\in Corr_{f}(R)$ via $z_{1}\cdot{\overline{v}}\cdot z_{2}=\overline{z_{2}^{*}\cdot v\cdot z_{1}^{*}}$, for all $z_{1},z_{2}\in R,v\in H^{x}$. Now one can check that $\mathcal{H}$ is a unitary tensor functor in the sense of [10] 2.1.3.

Theorem 2.3

(a $C^{*}$-version of the Hayashi’s theorem -see [6], [14])

Let $\mathcal{C}$ be a rigid finite $C^{*}$-tensor category, $\Omega=Irr(\mathcal{C})$ and $\mathcal{H}:\mathcal{C}\to Corr_{f}(R)$ be the Hayashi’s functor, where $R=\mathbb{C}^{|\Omega}|$. Then the vector space

[TABLE]

has a regular biconnected weak Hopf $C^{*}$-algebra structure $\mathfrak{G}$ such that $\mathcal{C}\cong UCorep(\mathfrak{G})$ as rigid $C^{*}$-tensor categories.

Explicitly, if $v,w\in H^{x},g,h\in H^{y}$ and $\{e^{x}_{j}\}$ is an orthonormal basis in $H^{x}$, for all $x,y\in\Omega$, then:

[TABLE]

[TABLE]

[TABLE]

Now define an antipode and an involution. Consider the natural isomorphisms $\Phi_{x}:H^{x}\to\overline{H}^{x^{*}}$ and $\Psi_{x}:\overline{H}^{x}\to H^{x^{*}}$, where $x^{*}$ is the dual of $x\in\Omega$:

[TABLE]

where $x,y,z\in\Omega$, we identify $y$ with $y\otimes 1$, $v\in{p_{y}}\cdot H^{x}\cdot{p_{z}}$, $\overline{R_{x}}$ and $a_{y,x,x^{*}}$ are, respectively, the rigidity morphisms and the associativities in $\mathcal{C}$. Then:

[TABLE]

[TABLE]

Any $H^{x}$ is a right $B$-comodule via

[TABLE]

one checks that it is unitary which gives the equivalence $\mathcal{C}\cong UCorep(\mathfrak{G})$.

The algebra of the dual quantum groupoid $\hat{\mathfrak{G}}$ is

[TABLE]

the duality is given, for all $x\in\Omega,A\in B(H_{x}),v,w\in H_{x}$ by:

[TABLE]

$\hat{B}$ is clearly a $C^{*}$-algebra with the obvious matrix product and involution,

Notations 2.4

For all $x,y\in\Omega$ and all $v\in H^{x}$, $w\in H^{y}$, we denote:

[TABLE]

Remark 2.5

Let [math] be the unit element of $\mathcal{C}$, and $H^{0}:=\underset{x\in\Omega}{\oplus}Hom(x,x)$, then using (3) and (8) it is easy to check that $(H^{0},\circ,\sharp)$ is a commutative $C^{*}$-algebra and if, for all $x\in\Omega$, $v^{0}_{x}$ is a normalized vector in $Hom(x,x)$, then $(v^{0}_{x})_{x\in\Omega}$ is a basis of mutually orthogonal projections in $H^{0}$.

Remark 2.6

Let $\mathcal{C}$ be a rigid finite $C^{*}$-tensor category and $\mathcal{F}:\mathcal{C}\to Corr_{f}(R)$ be a unitary tensor functor, where $R$ is a finite dimensional unital $C^{*}$-algebra. Then there exists [15] a regular biconnected finite quantum groupoid $\mathfrak{G}$ with $B_{t}\cong B_{s}\cong R$ such that $\mathcal{C}\cong UCorep(\mathfrak{G})$ as $C^{*}$-tensor categories. For any fixed $\mathcal{C}$, the set of $C^{*}$-algebras $R$ for which the above mentioned functor $\mathcal{F}$ exists, contains at least $R=\mathbb{C}^{|\Omega|}$ (where $\Omega=Irr(\mathcal{C})$), then $\mathcal{F}=\mathcal{H}$. In general, this set contains several elements, and the corresponding WHAs are called Morita equivalent.

In particular, if the above set of functors contains a fiber functor $\mathcal{F}:\mathcal{C}\to Hilb_{f}$, i.e., $R=\mathbb{C}$, the corresponding quantum groupoids are Morita equivalent to a usual $C^{*}$-Hopf algebra.

2.4 Coactions.

Definition 2.7

A right coaction of a WHA $\mathfrak{G}$ on a unital $*$-algebra $A$, is a $*$-homomorphism $\mathfrak{a}:A\to A\otimes B$ such that:

1) $(\mathfrak{a}\otimes i)\mathfrak{a}=(id_{A}\otimes\Delta)\mathfrak{a}$.

2) $(id_{A}\otimes\varepsilon)\mathfrak{a}=id_{A}.$

3) $\mathfrak{a}(1_{A})\in A\otimes B_{t}$.

One also says that $(A,\mathfrak{a})$ is a $\mathfrak{G}$-$*$-algebra.

If $A$ is a $C^{*}$-algebra, then $\mathfrak{a}$ is automatically continuous, even an isometry.

There are $*$-homomorphism $\alpha:B_{s}\to A$ and $*$-antihomomorphism $\beta:B_{s}\to A$ with commuting images defined by $\alpha(x)\beta(y):=(id_{A}\otimes\varepsilon)[(1_{A}\otimes x)\mathfrak{a}(1_{A})(1_{A}\otimes y)]$, for all $x,y\in B_{s}$. We also have $\mathfrak{a}(1_{A})=(\alpha\otimes id_{B})\Delta(1_{B})$,

[TABLE]

and

[TABLE]

The set $A^{\mathfrak{a}}=\{a\in A|\mathfrak{a}(a)=\mathfrak{a}(1_{A})(a\otimes 1_{B})\}$ is a unital $*$-subalgebra of $A$ (it is a unital $C^{*}$-subalgebra of $A$ when $A$ is a $C^{*}$-algebra) commuting pointwise with $\alpha(B_{s})$. A coaction $\mathfrak{a}$ is called ergodic if $A^{\mathfrak{a}}=\mathbb{C}1_{A}$.

Definition 2.8

A $\mathfrak{G}-C^{*}$-algebra $(A,\mathfrak{a})$ is said to be indecomposable if it cannot be presented as a direct sum of two $\mathfrak{G}-C^{*}$-algebras.

It is easy to see that $(A,\mathfrak{a})$ is indecomposable if and only if $Z(A)\cap A^{\mathfrak{a}}=\mathbb{C}1_{A}$. Clearly, any ergodic $\mathfrak{G}-C^{*}$-algebra is indecomposable.

For any $(U,H_{U})\in UCorep(\mathfrak{G})$, we define the spectral subspace of $A$ corresponding to $(U,H_{U})$ by

[TABLE]

Let us recall the properties of the spectral subspaces:

(i) All $A_{U}$ are closed.

(ii) $A=\oplus_{x\in\Omega}A_{U^{x}}$.

(iii) $A_{U^{x}}A_{U^{y}}\subset\oplus_{z}A_{U^{z}},$ where $z$ runs over the set of all irreducible direct summands of $U^{x}\otop U^{y}$.

(iv) $\mathfrak{a}(A_{U})\subset\mathfrak{a}(1_{A})(A_{U}\otimes B_{U})$ and $A_{\overline{U}}=(A_{U})^{*}$.

(v) $A_{\varepsilon}$ is a unital $C^{*}$-algebra.

2.5 Categorical duality.

Let us recall the main result of [18]:

Theorem 2.9

Given a regular coconnected WHA $\mathfrak{G}$, the following two categories are equivalent:

(i) The category of unital $\mathfrak{G}$-$C^{*}$-algebras with unital $\mathfrak{G}$-equivariant $*$-homomorphisms as morphisms.

(ii) The category of pairs $(\mathcal{M},M)$, where $\mathcal{M}$ is a left module $C^{*}$-category with trivial module associativities over the $C^{*}$-tensor category $UCorep(\mathfrak{G})$ and $M$ is a generator in $\mathcal{M}$, with equivalence classes of unitary module functors respecting the prescribed generators as morphisms.

In particular, given a unital $\mathfrak{G}$-$C^{*}$-algebra $A$, one constructs the $C^{*}$-category $\mathcal{M}=\mathcal{D}_{A}$ of finitely generated right Hilbert $A$-modules which are equivariant, that is, equipped with a compatible right coaction [1]. Any its object is automatically a $(B_{s},A)$-bimodule, and the bifunctor $U\boxtimes X:=H_{U}\otimes_{B_{s}}X\in\mathcal{D}_{A}$, for all $U\in UCorep(\mathfrak{G})$ and $X\in\mathcal{D}_{A}$, turns $\mathcal{D}_{A}$ into a left module $C^{*}$-category over $UCorep(\mathfrak{G})$ with generator $A$ and trivial associativities.

Vice versa, if a pair $(\mathcal{M},M)$ is given, the construction of a $\mathfrak{G}$-$C^{*}$-algebra $(A,\mathfrak{a})$ contains the following steps. First, denote by $R$ the unital $C^{*}$-algebra $End(M)$ and consider the functor $F:\mathcal{C}\to Corr(R)$ defined on the objects by $F(U)=Hom_{\mathcal{M}}(M,U\boxtimes M)\ \forall U\in\mathcal{C}$. Here $X=F(U)$ is a right $R$-module via the composition of morphisms, a left $R$-module via $rX=(id\otimes r)X$, the $R$-valued inner product is given by $<X,Y>=X^{*}Y$, the action of $F$ on morphisms is defined by $F(T)X=(T\otimes id)X$. The weak tensor structure of $F$ (in the sense of [9]) is given by $J_{X,Y}(X\otimes Y)=(id\otimes Y)X$, for all $X\in F(U),Y\in F(V),U,V\in UCorep(\mathfrak{G})$.

Then consider two vector spaces:

[TABLE]

and

[TABLE]

where $F(U)=\underset{i}{\bigoplus}F(U_{i})$ corresponds to the decomposition $U=\bigoplus U_{i}$ into irreducibles, and $\|UCorep(\mathfrak{G})\|$ is an exhaustive set of representatives of the equivalence classes of objects in $UCorep(\mathfrak{G})$ (these classes constitute a countable set). $\tilde{A}$ is a unital associative algebra with the product

[TABLE]

and the unit

[TABLE]

Note that $(id\otimes Y)X=J_{X,Y}(X\otimes Y)\in F(U\otop V)$. Then, for any $U\in UCorep(G)$, choose isometries $w_{i}:H_{i}\to H_{U}$ defining the decomposition of $U$ into irreducibles, and construct the projection $p:\tilde{A}\to A$ by

[TABLE]

which does not depend on the choice of $w_{i}$. Then $A$ is a unital $*$-algebra with the product $x\cdot y:=p(xy)$, for all $x,y\in A$ and the involution $x^{*}:=p(x^{\bullet})$, where $(X\otimes\overline{\xi})^{\bullet}:=(id\otimes X^{*})F(\overline{R}_{U})\otimes\overline{\hat{G}^{1/2}\xi}$, for all $\xi\in H_{U},X\in F(U),U\in UCorep(\mathfrak{G})$. Here $\overline{R}_{U}$ is the rigidity morphism from (2). Finally, the map

[TABLE]

where $\{\xi_{i}\}$ is an orthogonal basis in $H^{x}$ and $(U^{x}_{i,j}$ are the matrix elements of $U^{x}$ in this basis, is a right coaction of $\mathfrak{G}$ on $A$. Moreover, $A$ admits a unique $C^{*}$-completion $\overline{A}$ such that $\mathfrak{a}$ extends to a continuous coaction of $\mathfrak{G}$ on it.

Remark 2.10

1) We say that a $UCorep(\mathfrak{G})$-module category is indecomposable if it is not equivalent to a direct sum of two nontrivial $UCorep(\mathfrak{G})$-module subcategories. Theorem 2.9 implies that a $\mathfrak{G}-C^{*}$-algebra $(A,\mathfrak{a})$ is indecomposable if and only if the $UCorep(\mathfrak{G})$-module category $\mathcal{M}$ is indecomposable.

2) Let $I$ be a unital right coideal $*$-subalgebra of $B$. Then $I_{U^{x}}=I\cap B_{U^{x}}$ and $F(U^{x})$ can be identified with a Hilbert subspace of $H^{x}\ (\forall x\in\Omega)$ and the coaction is the restriction of $\Delta$.

Example 2.11

The $C^{*}$-algebra $B$ with coproduct $\Delta$ viewed as $\mathfrak{G}$-$C^{*}$-algebra, corresponds to the $UCorep(\mathfrak{G})$-module $C^{*}$-category $\mathcal{C}orr_{f}(B_{s})$ with generator $M=B_{s}$: for any element $U\in UCorep(\mathfrak{G})$ and $N\in Corr_{f}(B_{s})$, one defines $U\boxtimes N:=F(U)\otimes_{B_{s}}N$, where the functor $F:UCorep(\mathfrak{G})\to Corr_{f}(B_{s})\ (F(U)=H_{U})$ is the forgetful functor. Indeed, identifying $\mathcal{M}(B_{s},H_{U})$ with $H_{U}$, we get an isomorphism of the algebra $\tilde{A}$ constructed from the pair $(\mathcal{M},M)$ onto $\tilde{B}=\underset{U}{\bigoplus}(H_{U}\otimes\overline{H}_{U})$ and then an isomorphism $A\cong B=\underset{x\in\hat{G}}{\bigoplus}(H_{x}\otimes\overline{H}_{x})$ such that $p:\tilde{A}\to A$ turns into the map $\tilde{B}\to B$ sending $\xi\otimes\overline{\eta}\in H_{U}\otimes\overline{H}_{U}$ into the matrix coefficient $U_{\xi,\eta}$.

3 Classifying Indecomposable Weak Coideals

If $dim(A)<\infty$, we have the following remarks.

Remark 3.1

If $(A,\mathfrak{a})$ is a finite dimensional $\mathfrak{G}-C^{*}$-algebra, then $\mathcal{M}=\mathcal{D}_{A}$ is a semisimple $C^{*}$-category. Indeed, $dim(Hom_{\mathcal{M}}(\mathcal{E},\mathcal{E}))<\infty$, for any $\mathcal{E}\in D_{A}$ which is finitely generated. Then the proof of [4], Proposition 3.9 applies. As $A$ is a generator of $\mathcal{M}$, the set $\{M_{\lambda}|\lambda\in\Lambda\}$ of its (classes of) simple objects is finite and we have the corresponding fusion rule

[TABLE]

The associativity and the unit object conditions mean, respectively, that

[TABLE]

where $c^{z}_{x,y}$ are the fusion coefficients of $\mathcal{C}=UCorep(\mathfrak{G})$. Proposition 7.1.6 of [5] gives $n^{\mu}_{x,\lambda}=n^{\lambda}_{x^{*},\mu}$, for all $\lambda,\mu\in\Lambda,x\in\Omega$.

Remark 3.2

If $A$ is a coideal of $B$, then, due to [17], Theorem 1.1, there is an inclusion $j:\mathcal{M}\mapsto\mathcal{C}$ such that

[TABLE]

where $\mathcal{M}$ is the left $\mathcal{C}$-module category with generator $M$ coming from $(A,\Delta|_{A})$ and $\mathcal{C}=UCorep(\mathfrak{G})$ is viewed as a $\mathcal{C}$-module category with generator the $\underset{x\in\Omega}{\oplus}U^{x}$.

If $\Lambda$ is the set of irreducibles of $\mathcal{M}$ (we denote them by $M_{\lambda}$), we can write $j(M_{\lambda})=\underset{x\in\Omega}{\Sigma}a_{\lambda,x}U^{x}$, for all $\lambda\in\Lambda$, where $a_{\lambda,x}\in\mathbb{Z}_{+}$. Writing $M=\underset{\lambda\in\Lambda}{\Sigma}m_{\lambda}M_{\lambda}$, we must have:

[TABLE]

Recall that due to the reconstruction theorem for $\mathfrak{G}$, any $H^{x}(x\in\Omega)$ is the direct sum of 1-dimensional subspaces $Hom(z,y\otimes x)$, where $y,z\in\Omega$ are such that $z\subset(y\otimes x)$. In particular, $H^{0}=\underset{z\in\Omega}{\oplus}Hom(z,z)$ (where [math] denotes the trivial corepresentation of $\mathfrak{G}$); we will denote by $v^{0}_{z}$ a norm one vector generating $Hom(z,z)$ viewed as a subspace of $H^{0}$.

The following lemma allows to select weak coideals of $B$ from all $\mathfrak{G}-C^{*}$-algebras.

Lemma 3.3

Let us fix a $UCorep(\mathfrak{G})$-module category $\mathcal{M}$ and a generator $M$ in it, and let $(A,\mathfrak{a})$ be a $\mathfrak{G}$-algebra constructed from this data using the weak tensor functor $(F,J_{U,V})$. Then:

a) $(A,\mathfrak{a})$ is a weak coideal of $B$ if and only if each $F(U^{x})$ can be identified with a subspace $X^{x}\subset H^{x}$ such that the map $\zeta\mapsto\zeta^{\natural}=\Psi_{x}(\overline{\zeta})$ sends $X^{x}$ onto $X^{\overline{x}}\cong F(\overline{U^{x}})$ and $J_{U^{x},U^{y}}=\mathcal{H}_{x,y}$, for all $x,y\in\Omega$.

b) $X^{0}$ is a $C^{*}$-subalgebra of $H^{0}$. The unit of $X^{0}$ is $v^{0}_{\Gamma}:=\underset{x\in\Gamma}{\oplus}v^{0}_{x}$, where $\Gamma\subset\Omega$ is some nonempty subset. $A=\underset{x\in\Omega}{\oplus}(X^{x}\otimes\overline{H}^{x})$ is a coideal if and only if $\Gamma=\Omega$.

c) A weak coideal $A=\underset{x\in\Omega}{\oplus}(X^{x}\otimes\overline{H}^{x})$ is decomposable if and only if $Z(A)$ contains an element of the form $p=v^{0}_{\Gamma_{0}}\otimes v^{0}_{\Omega}$, where $\Gamma_{0}$ is a proper nonempty subset of $\Gamma$.

d) For any two identifications, $F(U^{x})\cong X^{x}$ and $F(U^{x})\cong\tilde{X}^{x},\forall x\in\Omega$, satisfying the above mentioned conditions, the corresponding weak coideals $\underset{x\in\Omega}{\oplus}(X^{x}\otimes\overline{H}^{x})$ and $\underset{x\in\Omega}{\oplus}(\tilde{X}^{x}\otimes\overline{H}^{x})$ are isomorphic as $\mathfrak{G}$-$C^{*}$-algebras.

Proof. a) If $(A,\mathfrak{a})$ is a weak coideal of $B$, then $A_{U}\subset B_{U}$, for any $U\in UCorep(\mathfrak{G})$. Indeed, by [18], Proposition 3.17 $A_{U}=\{a\in A|\Delta(a)\in\Delta(1_{A})(A\otimes B_{U})\}$, but $\Delta(1_{A})=\Delta(1_{B})(1_{A}\otimes 1_{B})$, hence $\Delta(a)\in\Delta(1_{B})(A\otimes B_{U})\subset\Delta(1_{B})(B\otimes B_{U})$, so that $A_{U}\subset B_{U}$. It follows from [18], Theorem 4.12 and Theorem 2.21, respectively, that $A_{U^{x}}\cong F(U^{x})\otimes\overline{H^{x}}$ and $B_{U^{x}}\cong H^{x}\otimes\overline{H^{x}}$, so the above inclusions mean that $F(U^{x})\subset H^{x}$, for all $x\in\Omega$.

The multiplication in $A$ is the restriction of that in $B$, therefore, comparing the formulas (15) and [18], (16) and using the relation $J_{U^{x},U^{y}}(X\otimes_{B_{s}}Y)=(id\otimes Y)X(\forall X\in F(U^{x}),Y\in F(U^{y}))$, we have $J_{U^{x},U^{y}}=\mathcal{H}_{x,y}$.

The involution in $A$ sends $X\otimes\overline{\eta}$ onto $X^{*}\otimes(\overline{\eta})^{\flat}$ (see Subsection 2.4) and is the restriction of that in $B$, the last one is defined by $\zeta\otimes\overline{\eta}\mapsto\zeta^{\natural}\otimes(\overline{\eta})^{\flat}\ \forall\zeta,\eta\in H^{x},x\in\Omega$. Then for $X\in F(U^{x})\subset H^{x}$ we have $X^{*}=X^{\natural}$.

Conversely, suppose that $F(U^{x})\subset H^{x}$ and $J_{U^{x},U^{y}}=\mathcal{H}_{x,y}$, for all $x,y\in\Omega$. It follows from the argument above that the multiplication in $A$ is the restriction of that in $B$. Next, compare the formulas [18], (29) for $\mathfrak{a}$ and [18], (14) for $\Delta$. Since $B_{U^{x}}=H^{x}\otimes\overline{H}^{x}$, for any $U^{x}$ - see [18], (12), the matrix coefficient $U^{x}_{\zeta,\eta}$ with respect to a basis $\{\zeta^{x}\}$ of $H^{x}$ can be identified with $\zeta^{x}\otimes\eta^{x}$, for all $x\in\Omega$. Now it is clear that $\mathfrak{a}$ is the restriction of $\Delta$. Finally, putting $X^{*}=X^{\natural}$ for any $X\in F(U^{x})$ and using the fact that $F(U^{x})^{\natural}=F(\overline{U^{x}})$, one checks that $(A,\mathfrak{a})$ is a coideal of $B$.

b) By Remark 2.5, $H^{0}=\underset{x\in\Omega}{\oplus}\mathbb{C}v^{0}_{x}$ is a commutative unital $C^{*}$-algebra, $v^{0}_{x}\ (x\in\Omega)$ are mutually orthogonal projections, and if $A=\underset{x\in\Omega}{\oplus}(X^{x}\otimes\overline{H}^{x})$ is a weak coideal of $B$, then $X^{0}$ is a $C^{*}$-subalgebra of $H^{0}$. Its spectral mutually orthogonal projectors are $v^{0}_{\Gamma_{i}}$, where $\Gamma_{i}\subset\Omega\ (i=1,...,k_{0}=dim(X^{0}))$ are disjoint subsets of $\Omega$, the unit of $X^{0}$, i.e., the image of $F(id_{M})$, is $v^{0}_{\Gamma}$, where $\Gamma=\sqcup^{k_{0}}_{i=1}\Gamma_{i}$. As $1_{A}=v^{0}_{\Gamma}\otimes\overline{v}^{0}_{\Omega}$ and $1_{B}=v^{0}_{\Omega}\otimes\overline{v}^{0}_{\Omega}$, $A$ is a coideal if and only if $\Gamma=\Omega$.

c) One checks that $B_{t}=H^{0}\otimes\overline{v}^{0}_{\Omega}$ and that any nontrivial orthoprojector $p\in[Z(A)\cap B_{t}]$ gives a decomposition $A=pA\oplus(1-p)A$ into the direct sum of two weak coideals of $B$. As $1_{A}=v^{0}_{\Gamma}\otimes\overline{v}^{0}_{\Omega}$, $p$ must be of the form $v^{0}_{\Gamma_{0}}\otimes\overline{v}^{0}_{\Omega}$, where $\Gamma_{0}$ is a proper nonempty subset of $\Gamma$.

d) The two $\mathfrak{G}$-$C^{*}$-algebras are isomorphic because they correspond to the same couple $(\mathcal{M},M)$. $\square$

Corollary 3.4

It follows from the definition of the functor $F$ that $X^{0}=F(U^{0})=End_{\mathcal{M}}(M)$. This finite dimensional $C^{*}$-algebra is commutative due to the statement b) which is only possible if $m_{\lambda}\in\{0,1\}$ for all $\lambda\in\Lambda$.

4 Weak Hopf $C^{*}$-Algebras related to Tambara-Yamagami categories

4.1 Tambara-Yamagami categories

These categories denoted by $\mathcal{T}\mathcal{Y}(G,\chi,\tau)$ ($G$ is a finite group; we consider them only over $\mathbb{C}$) are $\mathbb{Z}_{2}$-graded fusion categories whose [math]-component is $Vec_{G}$ - the category of finite dimensional $G$-graded vector spaces with trivial associativities (its simple objects are $g\in G$) and $1$-component is generated by single simple object $m$. The Grothendieck ring of $\mathcal{T}\mathcal{Y}(G,\chi,\tau)$ is isomorphic to the $\mathbb{Z}_{2}$-graded fusion ring $\mathcal{T}\mathcal{Y}_{G}=\mathbb{Z}G\oplus\mathbb{Z}\{m\}$ such that $g\cdot m=m\cdot g=m,\ m^{2}=\underset{g\in G}{\Sigma}g,\ m=m^{*}$. These categories exist if and only if $G$ is abelian, they are parameterized by non degenerate symmetric bicharacters $\chi:G\times G\to\mathbb{C}\backslash\{0\}$ and $\tau=\pm|G|^{-1/2}$ - see [16], [5], Example 4.10.5. The associativities $\phi(U,V,W):(U\otimes V)\otimes W\to U\otimes(V\otimes W)$ are

[TABLE]

[TABLE]

[TABLE]

where $g,h,k\in G$. The unit isomorphisms are trivial. $\mathcal{T}\mathcal{Y}(G,\chi,\tau)$ becomes a $C^{*}$-tensor category when $\chi:G\times G\to T=\{z\in\mathbb{C}||z|=1\}$, from now on we assume that this is the case. The dual objects are: $g^{*}=-g$, for all $g\in G$, and $m^{*}=m$. The rigidity morphisms are defined by $R_{g}:0\overset{id_{0}}{\to}g^{*}\otimes g$, $\overline{R_{g}}:0\overset{id_{0}}{\to}g\otimes g^{*}$, $R_{m}=\tau|G|^{1/2}\iota$, and $\overline{R_{m}}=|G|^{1/2}\iota$, where $\iota:0\to m\otimes m$ is the inclusion. Then $dim_{q}(g)=1$, for all $g\in G$, and $dim_{q}(m)=\sqrt{|G|}$.

Let us apply Theorem 2.3 to the category $\mathcal{T}\mathcal{Y}(G,\chi,\tau)$ in order to construct a biconnected regular WHA $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}=(B,\Delta,S,\varepsilon)$ with $UCorep(\mathfrak{G}_{\mathcal{T}\mathcal{Y}})\cong\mathcal{T}\mathcal{Y}(G,\chi,\tau)$. The Hayashi’s functor $\mathcal{H}:\mathcal{T}\mathcal{Y}(G,\chi,\tau)\to Corr_{f}(R)$, where $C^{*}$-algebra $R:=End(\underset{x\in\Omega}{\oplus}x)\cong\mathbb{C}^{|G|+1}$, was constructed in [8]. Denoting $\Omega_{g}=\Omega:=G\sqcup\{m\}$ and $\Omega_{m}:=G\sqcup\overline{G}$, where $g\in G$ and $\overline{G}$ is the second copy of $G$, one easily computes that $H^{g}\cong\mathbb{C}^{|G|+1}$, for all $g\in G$ and $H^{m}:\cong\mathbb{C}^{2|G|}$.

Let us fix a basis $\{v^{x}_{y}\}(y\in\Omega_{x})$ in each $H^{x}\ (x\in\Omega)$ choosing a norm one vector in every 1-dimensional vector subspace: $v^{g}_{h}\in Hom(h,(h-g)\otimes g)$, $v^{g}_{m}\in Hom(m,m\otimes g)$, $v^{m}_{g}\in Hom(m,g\otimes m)$, and $v^{m}_{\overline{g}}\in Hom(g,m\otimes m)$, where $g\in G$.

Lemma 4.1

Using notations (10) and 2.4, for all $g,h,k\in G$, $x\in\Omega$, one has:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof. For equations related to product $\circ$, these are computations made in [8] 2.1.5, where $\mathcal{H}_{x,y}$ must be replaced by $\mathcal{F}_{x,y}^{-1}$. Moreover, in the case of $\mathcal{T}\mathcal{Y}(G,\chi,\tau)$, the formulas of (8) imply that the isomorphisms $\Phi_{x}:H^{x}\to\overline{H^{x^{*}}}$ and $\Psi_{x}:\overline{H^{x}}\to H^{x^{*}}\ (x\in\Omega)$ of Theorem 2.3 are given, for all $g,h\in G$, by:

[TABLE]

[TABLE]

which implies, by (10) the formulas for involution $\sharp$. $\square$

Now the whole structure of a WHA $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$ is given by formulas (4), (5), (6), and (7). It was shown in [8] that this WHA is isomorphic to its dual whose $C^{*}$-algebra $\hat{B}\cong\underset{x\in\Omega}{\oplus}B(H^{x})$. This implies that $URep(\mathfrak{G}_{\mathcal{T}\mathcal{Y}})\cong UCorep(\mathfrak{G}_{\mathcal{T}\mathcal{Y}})$.

The isomorphisms $\Phi_{x}:H^{x}\to\overline{H^{x^{*}}}$ and $\Psi_{x}:\overline{H^{x}}\to H^{x^{*}}\ (x\in\Omega)$ of Theorem 2.3 are now given by:

[TABLE]

[TABLE]

which implies, for all $g,h,k\in G$:

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

[TABLE]

We have $B=\underset{g\in G}{\oplus}B^{g}\oplus B^{m}$, where $B^{g}\cong M_{|G|+1}(\mathbb{C}),\ \forall g\in G,\ B^{m}\cong M_{2|G|}(\mathbb{C})$

4.2 Classification of Indecomposable Finite Dimensional $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$-$C^{*}$-algebras

Let us first recall the following well known (see, for instance, [5], 7.4)

Lemma 4.2

Equivalence classes of left indecomposable $Vec_{G}$-module categories are parameterized by couples $(K,\phi)$, where $K$ is a stabilizing subgroup of $G$ and $\phi\in H^{2}(K,\mathbb{C}^{\times})$. The set of irreducibles of such a category $\mathcal{M}(K,\phi)$ is $\Lambda_{K}=G/K$ and $\phi$ defines the associativities. The corresponding fusion rule is $g\boxtimes\lambda:=g+\lambda,\ \forall g\in G,\ \lambda\in G/K$.

Although $\mathcal{C}=UCorep(\mathfrak{G}_{\mathcal{T}\mathcal{Y}})\cong\mathcal{T}\mathcal{Y}(G,\chi,\tau)$, these categories have different associativities, so we cannot apply directly the classification of module categories from [7], Section 9, however, we will use similar reasoning. The category $\mathcal{C}$ is $\mathbb{Z}_{2}$-graded, i.e., $\mathcal{C}=\mathcal{C}_{0}\oplus\mathcal{C}_{1}$, where $\mathcal{C}_{0}\cong Vec_{G}$ (both these categories have trivial associativities) and $\mathcal{C}_{1}$ is generated by a single simple object $U^{m}$. Indecomposable $\mathcal{C}_{0}$-module categories with trivial associativities are parameterized by their stabilizer subgroups $K<G$, they correspond to $Vec_{G}$-module categories of the form $\mathcal{M}(K,{\bf 1})$, where ${\bf 1}$ is the trivial cocycle. Let us denote them by $\mathcal{M}(K)$.

Then, according to [7], any indecomposable $\mathcal{C}$-module category $\mathcal{M}$ is either indecomposable over $\mathcal{C}_{0}$ (we say that it is of type (I), it is then of the form $\mathcal{M}(K)$) or equivalent to $\mathcal{M}(K_{0})\oplus\mathcal{M}(K_{1})$, where $K_{0}$ and $K_{1}$ are subgroups of $G$ (they can be equal) - a category of type (D).

Moreover, $\mathcal{C}_{1}$ is an invertible $\mathcal{C}_{0}$-bimodule category, so one can define an action of $\mathbb{Z}_{2}=<\sigma>$ on the set of (equivalence classes) indecomposable semisimple $\mathcal{C}_{0}$-module categories: $\sigma\cdot\mathcal{M}(K):=\mathcal{C}_{1}\boxtimes\mathcal{M}(K)$.

Notations 4.3

For $K<G,\ \rho\in\hat{K}$ denote $K^{\perp}_{\rho}=\{g\in G|\chi(g,k)=\rho(-k),\ \forall k\in K\}$. If $\rho={\bf 1}$ is trivial, denote $K^{\perp}_{\bf 1}$ by $K^{\perp}$. Note that $\hat{K}\cong G/K^{\perp}$.

Lemma 4.4

For any $K<G$, we have $\sigma\cdot\mathcal{M}(K)=\mathcal{M}(K^{\perp})$.

Proof. Adopting the strategy of the proof of [7], Lemma 30 to our context, let $A_{K}=\underset{k\in K}{\oplus}H^{k}$ be an algebra in the category $\mathcal{C}_{0}$ - the analog of the algebra $\mathbb{C}K$ in $Vec_{G}$. Viewed as a usual $C^{*}$-algebra, $A_{K}$ has the following minimal central orthoprojectors:

[TABLE]

So indecomposable right $A_{K}$-modules with support in $\mathcal{C}_{0}$ are: $V_{\lambda}=Vec\{v^{k}_{\lambda}|k\in K\}$ with the action $v^{k}_{\lambda}\circ H^{h}=v^{h+k}_{\lambda}\ (h,k\in K)$ and $V_{\rho}=\mathbb{C}\underset{k\in K}{\Sigma}\rho(k)v^{k}_{m}$ with the action $(\underset{k\in K}{\Sigma}\rho(k)v^{k}_{m})\circ H^{h}=\rho(-h)\underset{k\in K}{\Sigma}\rho(k)v^{k}_{m}\ (h\in H)$, where we denote $v^{x}_{\lambda}:=\underset{g\in\lambda}{\Sigma}v^{x}_{g}$ for any $x\in\Omega$. In both cases the stabilizer subgroup is $K$.

Then the category $\mathcal{C}_{1}\boxtimes\mathcal{M}(K)$ can be described as the category of right $A_{K}$-modules in $\mathcal{C}$ with support in $\mathcal{C}_{1}$ which are of the form $H^{m}\otimes_{R}V_{\lambda}=Vec\{v^{m}_{\overline{p}}|p\in\lambda\}$ with the action $v^{m}_{\overline{p}}\circ H^{h}=v^{m}_{\overline{p+h}}\ (p\in\lambda,h\in K)$ and $H^{m}\otimes_{R}V_{\rho}=Vec\{v^{m}_{r}|r\in K^{\perp}_{\rho}\}$ with the action $v^{m}_{r}\circ H^{h}=\chi(h,r)v^{m}_{r}\ (r\in H^{\perp}_{\rho},h\in H)$.

In order to determine the stabilizer of $H^{m}\otimes_{R}V_{\lambda}$, we calculate, as in the proof of [7], Lemma 30, for all $g\in G$ the modules $H^{g}\otimes_{R}(H^{m}\otimes_{R}V_{\lambda})=Vec\{\chi(g,p)v^{m}_{\overline{p}}|p\in\lambda\}$ with the action $\chi(g,p)v^{m}_{\overline{p}}\circ H^{h}=\chi(g,-h)\chi(g,p+h)v^{m}_{\overline{p+h}}\ (p\in\lambda,h\in K)$. Therefore, the stabilizer is $K^{\perp}$.

Similarly, we calculate for all $g\in G$ the modules $H^{g}\otimes_{R}(H^{m}\otimes_{R}V_{\rho})=Vec\{v^{m}_{r-g}|r\in K^{\perp}_{\rho}\}$, but $r-g\in K^{\perp}_{\rho}$ is equivalent to $g\in K^{\perp}$. $\square$

Thus, in case (I) necessarily $K=K^{\perp}$, so $|G|$ must be a square, and $\Lambda=G/K$. In case (D) $\mathcal{M}\cong\mathcal{M}(K)\oplus\mathcal{M}(K^{\perp})$ and $\Lambda=G/K\sqcup G/K^{\perp}$.

Corollary 4.5

The fusion rules for indecomposable $UCorep(\mathfrak{G}_{\mathcal{T}\mathcal{Y}})$-module categories are: $U^{g}\boxtimes M_{\lambda}=M_{g+\lambda}\ (\forall g\in G,\ M_{\lambda}\in Irr(\mathcal{M})$) in all cases and:

[TABLE]

For $\mathcal{M}=\mathcal{M}(K)\oplus\mathcal{M}(K^{\perp})$:

[TABLE]

where $M_{\lambda}\ (\lambda\in G/K)$ and $M_{\mu}\ (\mu\in G/K^{\perp})$ are in $Irr(\mathcal{M})$.

Proof. A priori, we have the following fusion rules with $U^{m}$:

[TABLE]

For $\mathcal{M}=\mathcal{M}(K)\oplus\mathcal{M}(K^{\perp})$:

[TABLE]

where $M_{\lambda}\ (\lambda\in G/K),\ M_{\mu}\ (\mu\in G/K^{\perp})$ are in $Irr(\mathcal{M})$ and $m^{\mu}_{\lambda},m^{\lambda}_{\mu}\in\mathbb{Z}_{+}$.

The relations of the type $(U^{g}\otimes U^{m})\boxtimes M_{\lambda}=U^{g}\boxtimes(U^{m}\boxtimes M_{\lambda})$, $(U^{m}\otimes U^{g})\boxtimes M_{\lambda}=U^{m}\boxtimes(U^{g}\boxtimes M_{\lambda})$ and similar relations with $M_{\mu}$ show that $n^{\mu}_{\lambda},m^{\mu}_{\lambda}$ and $m^{\lambda}_{\mu}$ do not depend on $\lambda$ and $\mu$. Then it remains to apply again $U^{m}$ to the above equalities and to use the last remark, the relation $U^{m}\otimes U^{m}=\underset{g\in G}{\Sigma}U^{g}$ and the fact that $|G|=|K||K^{\perp}|$. $\square$

Corollary 4.6

Any object $M=\underset{\lambda\in\Lambda}{\oplus}m_{\lambda}M_{\lambda}$ of an indecomposable semisimple $UCorep(\mathfrak{G}_{\mathcal{T}\mathcal{Y}})$-module category is a generator. Indeed, Corollary 4.5 shows that already any $\mathcal{M}_{\lambda}$ is a generator.

Therefore, the set of all couples $(\mathcal{M},M)$ is parameterized:

in case (I) by couples $(K,\{m_{\lambda}|\lambda\in G/K\})$, where $K=K^{\perp}<G$ and $m_{\lambda}\in\mathbb{Z}_{+}$ are such that at least one $m_{\lambda}>0$.

in case (D) by triples $(K,\{m^{0}_{\lambda}|\lambda\in G/K\},\{m^{1}_{\mu}|\mu\in G/K^{\perp}\})$, where $K<G$ and $m^{0}_{\lambda},m^{1}_{\mu}\in\mathbb{Z}_{+}$ are such that at least one of them is nonzero.

Lemma 4.7

The group $Aut(\mathcal{M})$ of autoequivalences of an indecomposable semisimple $UCorep(\mathfrak{G}_{\mathcal{T}\mathcal{Y}})$-module category $\mathcal{M}$ with trivial associativities is as follows:

(1) In case (I) for any $\phi\in Aut(\mathcal{M})$, there exists a unique $p\in G/K$ such that $\phi(M_{\lambda})=M_{p+\lambda}$, for all $\lambda\in G/K$, so $Aut(\mathcal{M})\cong G/K$.

(2) In case (D) and:

a) $K\neq K^{\perp}$, for all $\phi\in Aut(\mathcal{M})$, there exists a unique $(p_{0},p_{1})\in G/K\times G/K^{\perp}$ such that $\phi(M_{\lambda})=M_{p_{0}+\lambda}$ and $\phi(M_{\mu})=M_{p_{1}+\mu}$ for all $\lambda\in G/K,\mu\in G/K^{\perp}$, so $Aut(\mathcal{M}))\cong G/K\times G/K^{\perp}$.

b) $K=K^{\perp}$, $Aut(\mathcal{M})$, viewed as a bijection of $G/K\times G/K$ on itself, is generated by translations of irreducibles $(M_{\lambda},M_{\mu})$ by elements $(p_{0},p_{1})\in G/K\times G/K$ and the flip $(M_{\lambda},M_{\mu})\mapsto(M_{\mu},M_{\lambda})$. Therefore, $Aut(\mathcal{M})\cong(G/K\times G/K)\underset{\sigma}{\ltimes}\mathbb{Z}_{2}$, where $\sigma$ is the flip of $G/K\times G/K$.

Proof. (1) By definition of $\phi$, we must have $\phi(U^{g}\boxtimes M_{\lambda})=U^{g}\boxtimes\phi(M_{\lambda})$, for all $g\in G,\lambda\in G/K$. Then, putting $M_{p}=\phi(M_{K})$, we have the needed formula for $\phi$. Conversely, it is easy to check that for such a $\phi$ we have $\phi(U^{x}\boxtimes M_{\lambda})=U^{x}\boxtimes\phi(M_{\lambda})$, for all $x\in\Omega,\ \lambda\in G/K$.

(2a) As $\mathcal{M}=\mathcal{M}(K)\oplus\mathcal{M}(K^{\perp})$ and $\mathcal{M}(K)\ncong\mathcal{M}(K^{\perp})$, the above result applies to the corresponding restrictions of $\phi$.

(2b) Now the above mentioned components have equal rights, so $\phi$ can permute them and we are done. $\square$

Corollary 4.6 implies that any object $M=\underset{\lambda\in\Lambda}{\oplus}m_{\lambda}M_{\lambda}$ of a module category $\mathcal{M}$ as above can be identified either with a collection $\{m_{\lambda}|\lambda\in G/K\}$ or with a double collection $(\{m_{\lambda}|\lambda\in G/K\},\{m_{\mu}|\mu\in G/K^{\perp}\})$, where $m_{\lambda},m_{\mu}\in\mathbb{Z}_{+}$.

These considerations and Theorem 2.9 prove Theorem 1.2.

Remark 4.8

Let us compute the dimensions of the spectral subspaces of a finite dimensional $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$-$C^{*}$-algebra $(A,\alpha)$. By Theorem 2.9, given a $C^{*}$-module category $\mathcal{M}$ over $UCorep(\mathfrak{G}_{\mathcal{T}\mathcal{Y}})$ with a generator $M=\underset{\lambda\in\lambda}{\oplus}m_{\lambda}M_{\lambda}$, we have $A_{U^{x}}=F(U^{x})\otimes\overline{H}_{x}\ (\forall x\in\Omega)$, where $F:UCorep(\mathfrak{G}_{\mathcal{T}\mathcal{Y}})\to Corr_{f}(R)$ is the functor defined by $F(U^{x}):=Hom(M,U^{x}\boxtimes M)$, $R=End(M)$. Clearly,

[TABLE]

As $Hom(M_{\lambda},U^{g}\boxtimes M_{\rho})=\delta_{\lambda,g\cdot\rho}\mathbb{C},\ \forall\lambda,\rho\in\lambda$, we have $dim(X^{g})=\underset{\rho\in\lambda}{\Sigma}m_{\rho}m_{g\cdot\rho}$.

Now, in case (I), $Hom(M_{\lambda},U^{m}\boxtimes M_{\rho})\equiv\mathbb{C}$, so $dim(X^{m})=\underset{\lambda,\rho\in G/K}{\Sigma}m_{\lambda}m_{\rho}$.

And in case (D), $Hom(M_{\lambda},U^{m}\boxtimes M_{\rho})=0$ when $\lambda,\rho\in G/K$ or $\lambda,\rho\in G/K^{\perp}$, and $Hom(M_{\lambda},U^{x}\boxtimes M_{\rho})=\mathbb{C}$ otherwise. So, $dim(X^{m})=2\underset{\lambda\in G/K}{\Sigma}m_{\lambda}\times$ $\times\underset{\rho\in G/K^{\perp}}{\Sigma}m_{\rho}$. Therefore, in case (D), $dimX^{m}$ must be even.

5 Indecomposable Weak Coideals of $\mathfrak{G}_{TY}$

We begin the classification of indecomposable weak coideals of $\mathfrak{G}_{TY}$ by giving a canonical basis for them.

Notations 5.1

For all $g\in G$ and $X\subset G\sqcup\{m\}$, let us denote:

[TABLE]

Lemma 5.2

Let $A$ be a weak coideal of $B$. Then:

a) For any $g\in G$ such that X^{g}\neq\{0\}\, there exists a subset $I^{g}\subset I^{0}=\{\Gamma_{i}|i=1,2,...,k_{0}\}$ of cardinality $k_{g}$ and a set of vectors $\{v^{g}_{\Gamma_{i}}(\Theta^{g})|i\in I^{g}\}$ which is a basis of $X^{g}$, where $\Theta^{g}$ is a map from $\Gamma^{g}=\underset{i\in I^{g}}{\sqcup}\Gamma_{i}$ to $\mathbb{T}:=\{z\in\mathbb{C}||z|=1\}$.

b) If $X^{m}\neq\{0\}$, then $v^{0}_{m}\in X^{0}$, so we can chose $\{m\}\in I^{0}$, and there exists a subset $I^{m}\subset I^{0}\backslash\{m\}$ of cardinality $k_{m}$ and a basis of $X^{m}$ of the form $\{v^{m}_{\Gamma_{i}}(\Theta^{m}),v^{m}_{\overline{\Gamma_{i}}}(\Theta^{m})|i\in I^{m}\}$, where $\Theta^{m}$ is a map from $\Gamma^{m}=\underset{i\in I^{m}}{\sqcup}\Gamma_{i}$ to $\mathbb{T}:=\{z\in\mathbb{C}||z|=1\}$. If $k_{m}=k_{0}-1$, this weak coideal is indecomposable.

Proof. a) Let $v^{g}=\underset{x\in\Omega}{\Sigma}a_{x}v^{g}_{x}$ be a nonzero vector from $X^{g}$. Then $v^{g}=v^{g}\circ v^{0}_{\Gamma}=\underset{i\in I^{0}}{\Sigma}v^{g}(\Gamma_{i})$, where $v^{g}(\Gamma_{i})=\underset{x\in\Gamma_{i}}{\Sigma}a_{x}v^{g}_{x}$. Hence $X^{g}=\underset{i\in I^{g}}{\oplus}X^{g}_{\Gamma_{i}}$, where $X^{g}_{\Gamma_{i}}\ (i\in I^{g})$ are subspaces of $X^{g}$ containing $v^{g}(\Gamma_{i})\neq 0$. We have:

[TABLE]

Let $w^{g}(\Gamma_{i})=\underset{y\in\Gamma_{i}}{\Sigma}b_{y}v^{g}_{y}\in X^{g}_{\Gamma_{i}}$ be another vector with $|b_{y}|\equiv 1$, then:

[TABLE]

where $|D|=1$. Then $b_{x}=Da_{x}$ for all $x\in\Gamma_{i}$ which shows that any $X^{g}_{\Gamma_{i}}(i\in I^{g})$ is generated by a unique, up to a scalar $D\in\mathbb{T}$, vector as above. We fix such elements and denote them by $v^{g}_{\Gamma_{i}}(\Theta^{g})$, the map $\Theta^{g}$ being defined by the coefficients of the chosen elements.

b) Let $X^{m}\neq\{0\}$ and let $v^{m}=\underset{g\in G}{\Sigma}a_{g}v^{m}_{g}+\underset{h\in G}{\Sigma}b_{h}v^{m}_{\overline{h}}$ be its nonzero vector. Then $(v^{m})^{\sharp}:=\Psi_{m}(\overline{v}^{m})=|G|^{1/2}(\underset{g\in G}{\Sigma}\overline{a}_{g}v^{m}_{\overline{g}}+\underset{h\in G}{\Sigma}\overline{b}_{h}\tau^{-1}v^{m}_{h}$). Next, we compute:

[TABLE]

and similarly

[TABLE]

Hence, the components of index $p$ of these vectors are:

[TABLE]

and similarly

[TABLE]

In particular, the components of index [math] of these vectors are:

[TABLE]

and similarly

[TABLE]

Since at least one of $a_{g}$ or $b_{h}$ is nonzero, it follows that $v^{0}_{m}\in X^{0}$, so we can chose $\{m\}\in I^{0}$. Further:

[TABLE]

which shows that $v^{0}_{m}\notin Z(A)$ and that $X^{m}=X^{m}_{1}\oplus X^{m}_{2}$, where the subspaces $X^{m}_{1},X^{m}_{2}\subset X^{m}$ consist, respectively, of vectors of the form $\underset{g\in G}{\Sigma}a_{g}v^{m}_{g}$ and $\underset{h\in G}{\Sigma}b_{h}v^{m}_{\overline{h}}$. As $(X^{m}_{1})^{\sharp}=X^{m}_{2}$ and $(X^{m}_{2})^{\sharp}=X^{m}_{1}$, $dim(X^{m})$ must be even.

Now, the relations $v^{0}_{\Gamma_{i}}\circ v^{m}=\underset{g\in\Gamma_{i}}{\Sigma}a_{g}v^{m}_{g}:=w^{m}_{\Gamma_{i}}$ show that $X^{m}$ has a basis of the form $\{w^{m}_{\Gamma_{i}},(w^{m}_{\Gamma_{i}})^{\sharp}|i\in I^{m}\}$, and using the same reasoning as in part a), one can normalize: $w^{m}_{\Gamma_{i}}=v^{m}_{\Gamma_{i}}(\Theta^{m})$. Finally, if $k_{m}=k_{0}-1$, there is no a combination of $v^{0}_{\Gamma_{i}}$ which would commute with all $v^{m}_{\Gamma_{i}}(\Theta^{m})$, so $A$ is indecomposable. $\square$

Corollary 3.4 implies that for weak coideals we have $m_{\lambda}\in\{0,1\}$ for all $\lambda\in\Lambda$, so that the generator $M$ can be identified either with a nonempty subset $Z\subset G/K$ or with a couple of subsets $(Z_{0},Z_{1})\subset G/K\times G/K^{\perp}$, at least one of which is nonempty.

5.1 The case $A^{m}=\{0\}$

Remark 5.3

Let $A$ be an indecomposable weak coideal such that $dim(X^{m})=0$. Then either the set $I^{0}$ consists of only one subset $\tilde{\Gamma}\subset\Omega$ containing $\{m\}$ (so that $dim(X^{0})=1$) or does not contain subset $\tilde{\Gamma}\subset\Omega$ containing $\{m\}$.

Indeed, if $\tilde{\Gamma}\in I^{0}$, it suffices to show that $v^{0}_{\tilde{\Gamma}}$ commutes with any basis element $v^{g}_{\Gamma_{i}}(\Theta^{g})$ of $X$. Using the fact that no more than one element of $I^{0}$ can contain $\{m\}$ as well as Lemma 5.2, one can see that for any $\Gamma_{i}\in I^{0}\backslash\{\tilde{\Gamma}\}$ we have $v^{0}_{\tilde{\Gamma}}\circ v^{g}_{\Gamma_{i}}(\Theta^{g})=v^{g}_{\Gamma_{i}}(\Theta^{g})\circ v^{0}_{\tilde{\Gamma}}=0$ and $v^{0}_{\tilde{\Gamma}}\circ v^{g}_{\tilde{\Gamma}}(\Theta^{g})=v^{g}_{\tilde{\Gamma}}(\Theta^{g})\circ v^{0}_{\tilde{\Gamma}}=v^{g}_{\tilde{\Gamma}}(\Theta^{g})$. It follows that either the basis of $X$ consists only from vectors of the form $v^{g}_{\tilde{\Gamma}}(\Theta^{g})$ or does not contain such vectors at all.

The equality $dimX^{0}=1$ implies $M=M_{\lambda_{0}}$ for some $\lambda_{0}\in\Lambda$. Then $dimF(U^{g})=1$ if $g\in K$ and $dimF(U^{g})=0$ otherwise. This gives a unique, up to isomorphism of $\mathfrak{G}$-$C^{*}$algebras, connected coideal $I^{m}_{K}=\underset{k\in K}{\oplus}(\mathbb{C}v^{k}_{m}\otimes\overline{H}^{k})$.

Now suppose that $\Gamma_{i}\subset G,\ \forall i\in I^{0}$. As $dimF(U^{m})=0$, $M$ is supported only on $G/K$ or only on $G/K^{\perp}$. Let us consider the first of these cases, the second one is completely similar. Identify the generator $M$ with a nonempty subset $Z\subset G/K$. The following example shows that any such $Z$ gives rise to an indecomposable weak coideal of $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$.

Example 5.4

Let $Z$ be a nonempty subset of $G/K$, then Remark 4.8 gives $dimX^{g}=\underset{\lambda\in G/K}{\Sigma}m_{\lambda}m_{g+\lambda}=|Z\cap(g+Z)|$.

Put $X^{g}=Vec\{v^{g}_{\lambda}|\lambda\in Z\cap(g+Z)\}$ and $X^{m}=\{0\}$. For any $v^{g}_{\lambda}\in X^{g}\ (g\in G)$, we have $(v^{g}_{\lambda})^{\sharp}=v^{-g}_{\lambda-g}\in X^{-g}$. Indeed, as $\lambda\in Z\cap(g+Z)$, there is $\lambda^{\prime}\in Z$ such that $\lambda=g+\lambda^{\prime}$, so $\lambda-g=\lambda^{\prime}\in Z$. Clearly, $(\lambda-g)\in Z-g$, hence $(\lambda-g)\in Z\cap(Z-g)$. We also have:

[TABLE]

Indeed, as $\lambda\in Z\cap(g+Z),\ \mu\in Z\cap(h+Z)$, there are $\lambda^{\prime},\ \mu^{\prime}\in Z$ such that $\lambda=g+\lambda^{\prime},\ \mu=h+\mu^{\prime}$, so the above product is nonzero if and only if $\mu=h+\lambda=h+g+\lambda^{\prime}\in g+h+Z$. Since $\mu\in Z$, it follows that $\mu\in Z\cap(g+h+Z)$. Thus, Lemma 3.3, a) implies that the family $\{X^{x}|x\in\Omega\}$ generates a weak coideal $A\subset B$ with unit $1_{A}=v^{0}_{L}\otimes\overline{v}^{0}_{\Omega}$, where $L:=\underset{\lambda\in Z}{\bigcup}\lambda$.

Remark 5.5

$A$* is never a coideal but when $|Z|=1$ it is isomorphic to a connected coideal $I^{\Omega}_{K}=\underset{k\in K}{\oplus}(\mathbb{C}(\underset{x\in\Omega}{\Sigma}v^{h}_{x})\otimes\overline{H}^{k})$, which is the ”right sided” version of the left coideal $I_{K}$ from [8]. $I^{\Omega}_{K}$ is also isomorphic to $I^{m}_{K}$ above. If $|Z|>1$, $A$ is also indecomposable because for an arbitrary proper subset $Z_{0}\subset Z$, the element $\underset{\lambda\in Z_{0}}{\Sigma}v^{0}_{\lambda}$ does not commute with any $v^{g}_{\mu}\ (\mu\in Z_{0},\ g\notin K)$.*

It follows from Remark 4.8 that $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$-$C^{*}$-algebras $(A,\alpha)$ with $A^{m}=\{0\}$ can be only of type (D).

We can summarize the above considerations as follows:

Proposition 5.6

Isomorphism classes of indecomposable weak coideals $A$ of $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$ with $A^{m}=\{0\}$ are parameterized by couples $(K,Z^{orb})$, where $K<G$ and $Z^{orb}$ is the orbit of a nonempty subset $Z\subset G/K$ or $Z\subset G/K^{\perp}$ under the action of the group of the translations on $G/K$ (resp., on $G/K^{\perp}$). $A$ is isomorphic to a coideal if and only if $|Z|=1$.

5.2 The case $A^{m}\neq\{0\}$

Proposition 5.7

There is no weak coideals of $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$ corresponding to module categories $\mathcal{M}$ with $\Lambda=G/K$.

Proof. Let $A$ be such a weak coideal and $M$ be the corresponding generator identified with the subset $Z$ of $G/K$. Then $k_{0}=dim(X^{0})=|Z|$ and $dim(X^{m})=|Z|^{2}$. In terms of Lemma 5.2, b) we have $dim(X^{m})=2k_{m}$, where $k_{m}\leq k_{0}-1$, so that $|Z|^{2}\leq 2(|Z|-1)$ which is only possible if $|Z|=1$. But then $dim(X^{m})=1$ - contradicts to the fact that $dim(X^{m})$ must be even. $\square$

Proposition 5.8

Let $A$ be a weak coideal of $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$ corresponding to a module category $\mathcal{M}$ with $\Lambda=G/K\sqcup G/K^{\perp}$ and a generator $M$ defined by a nonempty subset $(Z_{0},Z_{1})\subset G/K\times G/K^{\perp}$. Then either $|Z_{0}|=1$ or $|Z_{1}|=1$.

Proof. We have $k_{0}=dim(X^{0})=|Z_{0}|+|Z_{1}|$ and $dim(X^{m})=2|Z_{0}||Z_{1}|$. In terms of Lemma 5.2, b) we have $dim(X^{m})=2k_{m}$, where $k_{m}\leq k_{0}-1$, so $|Z_{0}||Z_{1}|\leq|Z_{0}|+|Z_{1}|-1$ from where either $|Z_{0}|=1$ or $|Z_{1}|=1$. $\square$

The following example shows that any such set $(Z_{0},Z_{1})$ gives rise to an indecomposable weak coideal of $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$.

Example 5.9

Let $Z$ be a nonempty subset of $G/K$ and $\rho_{0}\in G/K^{\perp}$. For the generator corresponding to $Z\sqcup\rho_{0}$ we have $dimX^{m}=2|Z|$, $dimX^{g}=|Z\cap(g+Z)|$ if $g\notin K^{\perp}$ and $dimX^{g}=|Z\cap(g+Z)|+1$ if $g\in K^{\perp}$.

Put $X^{m}=Vec\{v^{m}_{\lambda},v^{m}_{\overline{\mu}}|\lambda,\mu\in Z\}$, $X^{g}=Vec\{v^{g}_{m},v^{g}_{\lambda}|\lambda\in Z\cap(g+Z)\}$ if $g\in K^{\perp}$ and $X^{g}=Vec\{v^{g}_{\lambda}|\lambda\in Z\cap(g+Z=\}$ if $g\notin K^{\perp}$. The next relations, where $g,h\in G,k,l\in K^{\perp},\lambda,\mu\in Z,u(\lambda)$ is a representative of the coset $\lambda$, show that the family $\{X^{x}|x\in\Omega\}$ satisfies the conditions a) of Lemma 3.3:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and finally, using the fact that $\underset{k\in K}{\Sigma}\chi(g,k)=|K|$ if $g\in K^{\perp}$ and is [math] otherwise:

[TABLE]

So, this family generates an indecomposable weak coideal $A\subset B$, $1_{A}=(v^{0}_{m}+v^{0}_{L})\otimes\overline{v}^{0}_{\Omega}$, where $L=\underset{\lambda\in Z}{\bigcup}\lambda$. $A$ is a coideal if and only if $L=G$ in which case it is the analog of the left connected coideal $J_{K}$ constructed in [8].

Now we can summarize the above considerations as follows:

Proposition 5.10

Isomorphism classes of indecomposable weak coideals $A$ of $\mathfrak{G}_{\mathcal{T}\mathcal{Y}}$ with $A^{m}\neq\{0\}$ are parameterized by pairs $(K,(Z_{0},Z_{1})^{orb})$, where $K<G$ and $(Z_{0},Z_{1})^{orb}$ is the orbit of a subset $(Z_{0},Z_{1})\subset G/K\times G/K^{\perp}$ such that $min\{|Z_{0}|,|Z_{1}|\}=1$ under the action of:

a) the group $G/K\times G/K^{\perp}$ by translations, if $K\neq K^{\perp}$;

b) the semi direct product $(G/K\times G/K)\underset{\sigma}{\ltimes}\mathbb{Z}_{2}$ generated by the group $G/K\times G/K$ acting by translations and the flip $\sigma:(Z_{0},Z_{1})\mapsto(Z_{1},Z_{0})$ if $K=K^{\perp}$.

$A$* is isomorphic to a coideal if and only if either $Z_{0}=G/K$ or $Z_{1}=G/K^{\perp}$.*

Finally, Theorem 1.3 follows from Propositions 5.6 and 5.10.