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

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Classifying (Weak) Coideal Subalgebras of Weak Hopf C

¦ -Algebras

Leonid Vainerman, Jean-Michel Vallin

To cite this version:

Leonid Vainerman, Jean-Michel Vallin. Classifying (Weak) Coideal Subalgebras of Weak Hopf C ¦

-Algebras. Journal of Algebra, Elsevier, 2020, 550, pp.333-357. �hal-02146994�

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

Leonid Vainerman Jean-Michel Vallin Dedicated to the Memory of Etienne Blanchard

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.

Contents

1 Introduction 3

2 Preliminaries 5

2.1 Weak Hopf C

-algebras . . . . 5 2.2 Unitary representations and corepresentations of a weak Hopf

C

-algebra . . . . 7 2.3 The Hayashi’s fiber functor and reconstruction theorem. . . . 8 2.4 Coactions. . . . . 11 2.5 Categorical duality. . . . . 12 3 Classifying Indecomposable Weak Coideals 14

4 Weak Hopf C

-Algebras related to Tambara-Yamagami cat-

egories 17

4.1 Tambara-Yamagami categories . . . . 17 4.2 Classification of Indecomposable Finite Dimensional G

-C

-

algebras . . . . 19

5 Indecomposable Weak Coideals of G

22 5.1 The case A

t 0 u . . . . 24 5.2 The case A

t 0 u . . . . 26

1 2

1AMS Subject Classification [2010]: Primary 16T05, Secondary 18D10, Tertiary 46L05.

2 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 rep- resentation 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 [7] 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 [7], [15], [17] 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 [14]. The main difference between weak and usual Hopf algebra is that in the former the coproduct ∆ is not necessarily unital.

Apart from tensor categories, weak Hopf algebras have interesting ap- plications to the subfactor theory. In particular, for any finite index and finite depth II

-subfactor N € M , there exists a weak Hopf C

-algebra G such that the corresponding Jones tower can be expressed in terms of crossed products of N and M with G and its dual. Moreover, there is a Galois cor- respondence between intermediate subfactors in this Jones tower and coideal C

-subalgebras of G - see [13]. 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 G-C

-algebra, which is, by definition, a unital C

algebra A equipped with a coaction a of a WHA G p B, ∆, S, ε q . More exactly, we will use the following

Definition 1.1 A weak right coideal C

-subalgebra of B is a right G-C

- algebra p A, a q with a C

-algebra inclusion i : A ÞÑ B (not necessarily unital) satisfying ∆ p i b id

q a. One can think of A as of a C

-subalgebra of B such that a ∆. 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 G is a usual Hopf C

-algebra, then one can prove that necessarily 1

1

, so weak and usual coideals coincide.

It was shown in [20] that any G-C

-algebra p A, a q corresponds to a pair

p M, M q , where M is a module category with a generator M over the category

of unitary corepresentations of G.

In Preliminaries we recall definitions and facts needed for the exact formu- lation of this result expressed in Theorem 2.9. Note that similar categorical duality for compact quantum group coactions was obtained earlier in [4], [11].

Section 3 is devoted to necessary conditions which a pair p M, M q satisfies if p A, a q is an indecomposable (weak) coideal.

In Sections 4 and 5 the above mentioned general approach is applied to the problem of classification of G-algebras and weak coideals of WHA’s associated with a concrete class of fusion categories - Tambara-Yamagami categories T Y p G, χ, τ q [18].

Recall that simple objects of T Y p G, χ, τ q are exactly the elements of a finite abelian group G and one separate element m satisfying the fusion rule g h gh, g m m g m, m

Σ

gPG

g, g

g, m m

p g, h P G q . These categories are parameterized by non degenerate symmetric bicharacters χ : G G Ñ Czt 0 u and τ | G |

. For any subset K € G, we shall denote K

: t g P G | χ p k, g q 1, @ k P K u .

The Hayashi’s reconstruction theorem allows to construct a WHA G

associated with T Y p G, χ, τ q - see [10]. We recall this construction in slightly different form in Subsection 4.1. Then, using the methods elaborated in [6], [8], [9], we classify in Subsection 4.2 indecomposable module categories over representations of G

, 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 G

-C

-algebras:

(i) those parameterized by pairs p K, t m

u

q , where K   G and t m

u

is the orbit of a nonzero collection t m

P Z | λ P G { K u under the action of the group of translations on G { K.

(ii) those parameterized by pairs p K, pt m

u , t m

uq

q , where K   G and pt m

u , t m

uq

is the orbit of a nonzero double collection pt m

P Z | λ P G { K u , t m

P Z | µ P G { K

uq under the action of:

a) the group of translations on G { K G { K

if K K

; b) the semi-direct product p G { K G { K q

σ

Z

generated by translations on G { K G { K and the flip σ : pt m

u , t m

uq Ø pt m

u , t m

uq if K K

.

Finally, Section 5 is devoted to the classification of indecomposable (weak)

coideals of G

. Their classification is given by the following

Theorem 1.3 Isomorphism classes of indecomposable weak coideals of G

are parameterized by pairs p K, p Z

, Z

q

q , where K is a subgroup of G and p Z

, Z

q

is the orbit of a nonempty subset p Z

, Z

q € G { K G { K

such that either | Z

| ¤ 1 or | Z

| ¤ 1, under the action of:

a) the group of translations on G { K G { K

if K K

; b) the semi-direct product p G { K G { K q

σ

Z

generated by translations on G { K G { K and the flip σ : p Z

, Z

q Ø p Z

, Z

q if K K

.

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

when K  K

, to the four orbits tp λ, Hq{ λ P G { K u , tpH , µ q , { µ P G { K

u , tp G { K, µ q{ µ P G { K

u , tp λ, G { K

q{ λ P G { K u ,

when K K

, to the two orbits tp λ, Hq Y pH , λ q , { λ P G { K u and tp G { K, λ q Y p λ, G { K q{ λ P G { K u .

In fact, we give an explicit construction of representatives of all isomor- phism classes of indecomposable finite dimensional G

-C

-algebras and in- decomposable (weak) coideals of G

.

Our references are: to [5] for tensor categories, to [12] for C

-tensor cat- egories and to [14] for weak Hopf algebras (finite quantum groupoids).

2 Preliminaries

2.1 Weak Hopf C

-algebras

A weak Hopf C

-algebra (WHA) G p B, ∆, S, ε q is a finite dimensional C

- algebra B with the comultiplication ∆ : B Ñ B b B, counit ε : B Ñ C , and antipode S : B Ñ B such that p B, ∆, ε q is a coalgebra and the following axioms hold for all b, c, d P B :

(1) ∆ is a (not necessarily unital) -homomorphism :

∆ p bc q ∆ p b q ∆ p c q , ∆ p b

q ∆ p b q

,

(2) The unit and counit satisfy the identities (we use the Sweedler leg notation ∆ p c q c

b c

, p ∆ b id

q ∆ p c q c

b c

b c

etc.):

ε p bc

q ε p c

d q ε p bcd q ,

p ∆ p 1 q b 1 qp 1 b ∆ p 1 qq p ∆ b id

q ∆ p 1 q ,

(3) S is an anti-algebra and anti-coalgebra map such that m p id

b S q ∆ p b q p ε b id

qp ∆ p 1 qp b b 1 qq , m p S b id

q ∆ p b q p id

b ε qpp 1 b b q ∆ p 1 qq , where m denotes the multiplication.

The right hand sides of two last formulas are called target and source counital maps ε

and ε

, respectively. Their images are unital C

-subalgebras of B called target and source counital subalgebras B

and B

, respectively.

The dual vector space ˆ B has a natural structure of a weak Hopf C

- algebra ˆ G p B, ˆ ∆, ˆ S, ˆ ε ˆ q given by dualizing the structure operations of B:

  ϕψ, b ¡   ϕ b ψ, ∆ p b q ¡ ,   ∆ ˆ p ϕ q , b b c ¡   ϕ, bc ¡ ,

  S ˆ p ϕ q , b ¡   ϕ, S p b q ¡ ,   φ

, b ¡   ϕ, S p b q

¡ ,

for all b, c P B and ϕ, ψ P B. The unit of ˆ ˆ B is ε and the counit is 1.

The counital subalgebras commute elementwise, we have S ε

ε

S and S p B

q B

. We say that B is connected if B

X Z p B q C (where Z p B q is the center of B), coconnected if B

X B

C , and biconnected if both conditions are satisfied.

The antipode S is unique, invertible, and satisfies p S q

id

. We will only consider regular quantum groupoids, i.e., such that S

|

id. In this case, there exists a canonical positive element H in the center of B

such that S

is an inner automorphism implemented by G HS p H q

, i.e., S

p b q GbG

for all b P B . The element G is called the canonical group-like element of B, it satisfies the relation ∆ p G q p G b G q ∆ p 1 q ∆ p 1 qp G b G q . There exists a unique positive functional h on B, called a normalized Haar measure such that

p id

b h q ∆ p ε

b h q ∆, h S h, h ε

ε, p id

b h q ∆ p 1

q 1

.

We will dehote by H

the GNS Hilbert space generated by B and h and by

Λ

: B Ñ H

the corresponding GNS map.

2.2 Unitary representations and corepresentations of a weak Hopf C

-algebra

Let G p B, ∆, S, ε q be a weak Hopf C

-algebra. We denote by ε

, ε

its target and source counital maps, by B

and B

its target and source subalge- bras, respectively, and by G its canonical group-like element. We also denote by h the normalized Haar measure of G.

Any object of the category U Rep p G q of unitary representations of G is a left B-module of finite rank such that the underlying vector space is a Hilbert space H with a scalar product   , ¡ satisfying

  b v, w ¡  v, b

w ¡ , for all v, w P H, b P B.

U Rep p G q 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

, H

P U Rep p G q as the Hilbert subspace ∆ p 1

q p H

b H

q of the usual tensor product together with the action of B given by ∆. Here we use the fact that ∆ p 1

q 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 P U Rep p G q is automatically a B

-bimodule via z v t : zS p t q v, @ z, t P B

, v P E, and that the above tensor product is in fact b

, moreover the B

-bimodule structure for H

b

H

is given by z ξ t p z b S p t qq ξ, @ z, t P B

, ξ P H

b

H

. The above tensor product is associative, so the associativity isomorphisms are trivial. The unit object of U Rep p G q is B

with the action of B given by b z : ε

p bz q , @ b P B, z P B

and the scalar product   z, t ¡ h p t

z q .

For any morphism f : H

Ñ H

, let f

: H

Ñ H

be the adjoint linear map:   f p v q , w ¡  v, f

p w q ¡ , @ v P H

, w P H

. Clearly, f

is B-linear, f

f, p f b

g q

f

b

g

, and End p H q is a C

-algebra, for any object H. So U Rep p G q is a finite C

-multitensor category (1 can be decomposable).

The conjugate object for any H P U Rep p G q is the dual vector space ˆ H naturally identified (v ÞÑ v) with the conjugate Hilbert space H with the action of B defined by b v G

S p b q

G

v, where G is the canonical group-like element of G. Then the rigidity morphisms defined by

R

p 1

q Σ

p G

e

b

e

q , R

p 1

q Σ

p e

b

G

e

q , (1)

where t e

u

is any orthogonal basis in H, satisfy all the needed properties -

see [3], 3.6. Also, it is known that the B -module B

is irreducible if and only

if B

X Z p B q C 1

, i.e., if G is connected. So that, we have

Proposition 2.1 U Rep p G q is a rigid finite C

-multitensor category with trivial associativity constraints. It is C

-tensor if and only if G is connected.

Definition 2.2 A right unitary corepresentation U of G on a Hilbert space H

is a partial isometry U P B p H

q b B such that:

(i) p id b ∆ qp U q U

U

. (ii) p id b ε qp U q id.

If U and V are two right corepresentations on Hilbert spaces H

and H

, respectively, a morphism between them is a bounded linear map T P B p H

, H

q such that p T b 1

q U V p T b 1

q . We denote by U Corep p G q the category whose objects are unitary corepresentations on finite dimensional vector spaces with morphisms as above.

If G is coconnected (i.e., if B

X B

C 1

), U Corep p G q is a rigid C

- tensor category with trivial associativities isomorphic to U Rep p G ˆ q . Namely, any H

is a right B-comodule via v ÞÑ U p v b 1

q , therefore, automatically a p B

, B

q -bimodule. Then tensor product U j V : U

V

acts on H

b

H

, the unit object U

P B p B

q b B is defined by z b b ÞÑ ∆ p 1

qp 1

b zb q , @ z P B

, b P B, and the rigidity morphisms related to the conjugate U of an object U which acts on the conjugate Hilbert space H

of H

, are

R

p 1

q Σ

p G ˆ

e

b

e

q , R

p 1

q Σ

p e

b

G ˆ

e

q , (2) where t e

u

is any orthogonal basis in H

. We denote by Ω an exhaustive set of representatives of the equivalence classes of irreducibles in U Corep p G q .

Denote H

by H

, then U

`

m

b U

, where m

are the matrix units of B p H

q with respect to some orthogonal basis t e

u P H

and U

are the corresponding matrix coefficients of U

. Recall that B `

B

, where B

Span p U

q .

2.3 The Hayashi’s fiber functor and reconstruction theorem.

Let C be a rigid finite C

-tensor category and Ω Irr p C q be an exhaustive set of representatives of equivalence classes of its simple objects. Let R be the C

-algebra R C

À

xPΩ

C p

, where p

p

are mutually orthogonal

idempotents: p

p

δ

p

, for all x, y P Ω. Let us define a functor H from C

to the category Corr

p R q of finite dimensional Hilbert R-bimodules (called R-correspondences) by:

H p x q H

à

y,zPΩ

Hom p z, y b x q , for every x P Ω,

where Hom p x, y q is the vector space of morphisms x Ñ y. The R-bimodule structure on H

is given by:

p

H

p

Hom p z, y b x q , for all x, y, z P Ω.

If f P Hom p x, y q , then H p f q : H

Ñ H

is defined by:

H p f qp g q p id

b f q g, for any z, t P Ω and g P p

H

p

.

The tensor structure of H is a family of natural isomorphisms H

: H

b H

Ñ H

b

R

H

defined by:

H

p v b w q a

p v b id

q w P p

H

p

, (3) for all v P p

H

p

, w P p

H

p

, z, s, t P Ω. Here a

are the associativity isomorphisms of C.

We define the scalar product on H

as follows. If x, y, z P Ω and f, g P Hom p z, y b x q , then g

P Hom p y b x, z q and g

f P End p z q C , so one can put   f, g ¡

g

f . The subspaces Hom p z, y b x q are declared to be orthogonal, so H

P Corr

p R q . Dually, H

P Corr

p R q via z

v z

z

v z

, for all z

, z

P R, v P H

. Now one can check that H is a unitary tensor functor in the sense of [12] 2.1.3.

Theorem 2.3 (a C

-version of the Hayashi’s theorem -see [7], [16])

Let C be a rigid finite C

-tensor category, Ω Irr p C q and H : C Ñ Corr

p R q be the Hayashi’s functor, where R C

| . Then the vector space

B à

xPΩ

H

b H

, (4)

has a regular biconnected weak Hopf C

-algebra structure G such that C

U Corep p G q as rigid C

-tensor categories.

Explicitly, if v, w P H

, g, h P H

and t e

u is an orthonormal basis in H

, for all x, y P Ω, then:

p w b v q

p g b h q

p H

p w b g q b H

p v b h qq

P H

b H

(5)

∆ p w b v q à

j

p w b e

q

b p e

b v q

, (6) ε p w b v q   w, v ¡

. (7) Now define an antipode and an involution. Consider the natural isomor- phisms Φ

: H

Ñ H

and Ψ

: H

Ñ H

, where x

is the dual of x P Ω:

Φ

p v q p id

b R

q a

p v b id

q , Ψ

p v q p v b id

q a

p id

b R

q , (8) where x, y, z P Ω, we identify y with y b 1, v P p

H

p

, R

and a

are, respectively, the rigidity morphisms and the associativities in C . Then:

S p w b v q Ψ

p v q b Φ

p w q , (9) p w b v q

w

b v

, where w

Ψ

p w q , v

Φ

p v q . (10) Any H

is a right B-comodule via

a

p v q Σ

j

e

b e

b v, where v P H

,

one checks that it is unitary which gives the equivalence C U Corep p G q . The algebra of the dual quantum groupoid ˆ G is

B ˆ à

xPΩ

B p H

q , (11)

the duality is given, for all x P Ω, A P B p H

q , v, w P H

by:

  A, w b v ¡  Aw, v ¡

.

B ˆ is clearly a C

-algebra with the obvious matrix product and involution, Notations 2.4 For all x, y P Ω and all v P H

, w P H

, we denote:

v w H

p v b

w q

Remark 2.5 Let 0 be the unit element of C, and H

: `

xPΩ

Hom p x, x q , then using (3) and (8) it is easy to check that p H

, , 7q is a commutative C

- algebra and if, for all x P Ω, v

is a normalized vector in Hom p x, x q , then p v

q

is a basis of mutually orthogonal projections in H

.

Remark 2.6 Let C be a rigid finite C

-tensor category and F : C Ñ Corr

p R q be a unitary tensor functor, where R is a finite dimensional unital C

-algebra.

Then there exists [17] a regular biconnected finite quantum groupoid G with B

B

R such that C U Corep p G q as C

-tensor categories. For any fixed C, the set of C

-algebras R for which the above mentioned functor F exists, contains at least R C

(where Ω Irr p C q ), then F 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 F : C Ñ Hilb

, i.e., R 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 G on a unital -algebra A, is a -homomorphism a : A Ñ A b B such that:

1) p a b i q a p id

b ∆ q a.

2) p id

b ε q a id

. 3) a p 1

q P A b B

.

One also says that p A, a q is a G- -algebra.

If A is a C

-algebra, then a is automatically continuous, even an isometry.

There are -homomorphism α : B

Ñ A and -antihomomorphism β : B

Ñ A with commuting images defined by α p x q β p y q : p id

b ε qrp 1

b x q a p 1

qp 1

b y qs , for all x, y P B

. We also have a p 1

q p α b id

q ∆ p 1

q ,

a p α p x q aβ p y qq p 1

b x q a p a qp 1

b y q , (12) and

p α p x q b 1

q a p a qp β p y q b 1

q p 1

b S p x qq a p a qp 1

b S p y qq . (13)

The set A

t a P A | a p a q a p 1

qp a b 1

qu is a unital -subalgebra of A (it

is a unital C

-subalgebra of A when A is a C

-algebra) commuting pointwise

with α p B

q . A coaction a is called ergodic if A

C 1

.

Definition 2.8 A G C

-algebra p A, a q is said to be indecomposable if it cannot be presented as a direct sum of two G C

-algebras.

It is easy to see that p A, a q is indecomposable if and only if Z p A qX A

C 1

. Clearly, any ergodic G C

-algebra is indecomposable.

For any p U, H

q P U Corep p G q , we define the spectral subspace of A cor- responding to p U, H

q by

A

: t a P A | a p a q P a p 1

qp A b B

qu . Let us recall the properties of the spectral subspaces:

(i) All A

are closed.

(ii) A `

A

.

(iii) A

A

€ `

A

, where z runs over the set of all irreducible direct summands of U

j U

.

(iv) a p A

q € a p 1

qp A

b B

q and A

p A

q

. (v) A

is a unital C

-algebra.

2.5 Categorical duality.

Let us recall the main result of [20]:

Theorem 2.9 Given a regular coconnected WHA G, the following two cat- egories are equivalent:

(i) The category of unital G-C

-algebras with unital G-equivariant - homomorphisms as morphisms.

(ii) The category of pairs p M, M q , where M is a left module C

-category with trivial module associativities over the C

-tensor category U Corep p G q and M is a generator in M, with equivalence classes of unitary module func- tors respecting the prescribed generators as morphisms.

In particular, given a unital G-C

-algebra A, one constructs the C

-category M D

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 p B

, A q -bimodule, and the bifunctor U b X : H

b

X P D

, for all U P U Corep p G q and X P D

, turns D

into a left module C

- category over U Corep p G q with generator A and trivial associativities.

Vice versa, if a pair p M, M q is given, the construction of a G-C

-algebra

p A, a q contains the following steps. First, denote by R the unital C

-algebra

End p M q and consider the functor F : C Ñ Corr p R q defined on the objects by F p U q Hom

p M, U b M q @ U P C. Here X F p U q is a right R-module via the composition of morphisms, a left R-module via rX p id b r q X, the R-valued inner product is given by   X, Y ¡ X

Y , the action of F on morphisms is defined by F p T q X p T b id q X. The weak tensor structure of F (in the sense of [11]) is given by J

p X b Y q p id b Y q X, for all X P F p U q , Y P F p V q , U, V P U Corep p G q .

Then consider two vector spaces:

A à

xPΩ

A

: à

xPΩ

p F p U

q b H

q (14) and

A ˜ à

UP}U CoreppGq}

A

: à

UP}U CoreppGq}

p F p U q b H

q , (15) where F p U q À

i

F p U

q corresponds to the decomposition U À

U

into irreducibles, and } U Corep p G q} is an exhaustive set of representatives of the equivalence classes of objects in U Corep p G q (these classes constitute a count- able set). ˜ A is a unital associative algebra with the product

p X b ξ qp Y b η q p id b Y q X b p ξ b

η q , @p X b ξ q P A

, p Y b η q P A

, and the unit

1

id

b 1

.

Note that p id b Y q X J

p X b Y q P F p U j V q . Then, for any U P U Corep p G q , choose isometries w

: H

Ñ H

defining the decomposition of U into irreducibles, and construct the projection p : ˜ A Ñ A by

p p X b ξ q Σ

p F p w

q X b w

ξ q , @p X b ξ q P A

, (16) which does not depend on the choice of w

. Then A is a unital -algebra with the product x y : p p xy q , for all x, y P A and the involution x

: p p x

q , where p X b ξ q

: p id b X

q F p R

q b G ˆ

ξ, for all ξ P H

, X P F p U q , U P U Corep p G q . Here R

is the rigidity morphism from (2). Finally, the map

a p X b ξ

q X b Σ

p ξ

b U

q , (17)

where t ξ

u is an orthogonal basis in H

and p U

are the matrix elements of

U

in this basis, is a right coaction of G on A. Moreover, A admits a unique

C

-completion A such that a extends to a continuous coaction of G on it.

Remark 2.10 1) We say that a U Corep p G q -module category is indecompos- able if it is not equivalent to a direct sum of two nontrivial U Corep p G q -module subcategories. Theorem 2.9 implies that a G C

-algebra p A, a q is indecom- posable if and only if the U Corep p G q -module category M is indecomposable.

2) Let I be a unital right coideal -subalgebra of B. Then I

I X B

and F p U

q can be identified with a Hilbert subspace of H

p@ x P Ω q and the coaction is the restriction of ∆.

Example 2.11 The C

-algebra B with coproduct ∆ viewed as G-C

-algebra, corresponds to the U Corep p G q -module C

-category C orr

p B

q with genera- tor M B

: for any element U P U Corep p G q and N P Corr

p B

q , one defines U b N : F p U q b

N , where the functor F : U Corep p G q Ñ Corr

p B

q p F p U q H

q is the forgetful functor. Indeed, identifying M p B

, H

q with H

, we get an isomorphism of the algebra A ˜ constructed from the pair p M, M q onto B ˜ À

U

p H

b H

q and then an isomorphism A B À

xPGˆ

p H

b H

q such that p : ˜ A Ñ A turns into the map B ˜ Ñ B sending ξ b η P H

b H

into the matrix coefficient U

.

3 Classifying Indecomposable Weak Coideals

If dim p A q   8 , we have the following remarks.

Remark 3.1 If p A, a q is a finite dimensional G C

-algebra, then M D

is a semisimple C

-category. Indeed, dim p Hom

p E , E qq   8 , for any E P D

which is finitely generated. Then the proof of [4], Proposition 3.9 applies. As A is a generator of M, the set t M

| λ P Λ u of its (classes of ) simple objects is finite and we have the corresponding fusion rule

U

b M

Σ

µ

n

M

, where x P Ω, n

dim p Hom

p U

b M

, M

qq P Z . (18) The associativity and the unit object conditions mean, respectively, that

z

Σ

c

n

Σ

µPΛ

n

n

, and n

δ

, @ x, y P Ω, ρ, λ P Λ, (19)

where c

are the fusion coefficients of C U Corep p G q . Proposition 7.1.6

of [5] gives n

n

, for all λ, µ P Λ, x P Ω.

Remark 3.2 If A is a coideal of B, then, due to [19], Theorem 1.1, there is an inclusion j : M ÞÑ C such that

j p M q `

xPΩ

U

, (20)

where M is the left C-module category with generator M coming from p A, ∆ |

q and C U Corep p G q is viewed as a C-module category with generator the

x

`

U

.

If Λ is the set of irreducibles of M (we denote them by M

), we can write j p M

q Σ

xPΩ

a

U

, for all λ P Λ, where a

P Z . Writing M Σ

λPΛ

m

M

, we must have:

λ

Σ

m

a

1, for all x P Ω. (21) Recall that due to the reconstruction theorem for G, any H

p x P Ω q is the direct sum of 1-dimensional subspaces Hom p z, y b x q , where y, z P Ω are such that z € p y b x q . In particular, H

`

zPΩ

Hom p z, z q (where 0 denotes the trivial corepresentation of G); we will denote by v

a norm one vector generating Hom p z, z q viewed as a subspace of H

.

The following lemma allows to select weak coideals of B from all G C

- algebras.

Lemma 3.3 Let us fix a U Corep p G q -module category M and a generator M in it, and let p A, a q be a G-algebra constructed from this data using the weak tensor functor p F, J

q . Then:

a) p A, a q is a weak coideal of B if and only if each F p U

q can be identified with a subspace X

€ H

such that the map ζ ÞÑ ζ

Ψ

p ζ q sends X

onto X

F p U

q and J

H

, for all x, y P Ω.

b) X

is a C

-subalgebra of H

. The unit of X

is v

: `

xPΓ

v

, where Γ € Ω is some nonempty subset. A `

xPΩ

p X

b H

q is a coideal if and only if Γ Ω.

c) A weak coideal A `

xPΩ

p X

b H

q is decomposable if and only if Z p A q contains an element of the form p v

b v

, where Γ

is a proper nonempty subset of Γ.

d) For any two identifications, F p U

q X

and F p U

q X ˜

, @ x P Ω, satisfying the above mentioned conditions, the corresponding weak coideals

x

`

p X

b H

q and `

xPΩ

p X ˜

b H

q are isomorphic as G-C

-algebras.

Proof. a) If p A, a q is a weak coideal of B, then A

€ B

, for any U P U Corep p G q . Indeed, by [20], Proposition 3.17 A

t a P A | ∆ p a q P

∆ p 1

qp A b B

qu , but ∆ p 1

q ∆ p 1

qp 1

b 1

q , hence ∆ p a q P ∆ p 1

qp A b B

q € ∆ p 1

qp B b B

q , so that A

€ B

. It follows from [20], Theorem 4.12 and Theorem 2.21, respectively, that A

F p U

qb H

and B

H

b H

, so the above inclusions mean that F p U

q € H

, for all x P Ω.

The multiplication in A is the restriction of that in B , therefore, compar- ing the formulas (15) and [20], (16) and using the relation J

p X b

Y q p id b Y q X p@ X P F p U

q , Y P F p U

qq , we have J

H

.

The involution in A sends X b η onto X

bp η q

(see Subsection 2.4) and is the restriction of that in B, the last one is defined by ζ b η ÞÑ ζ

bp η q

@ ζ, η P H

, x P Ω. Then for X P F p U

q € H

we have X

X

.

Conversely, suppose that F p U

q € H

and J

H

, for all x, y P Ω. It follows from the argument above that the multiplication in A is the restriction of that in B. Next, compare the formulas [20], (29) for a and [20], (14) for ∆. Since B

H

b H

, for any U

- see [20], (12), the matrix coefficient U

with respect to a basis t ζ

u of H

can be identified with ζ

b η

, for all x P Ω. Now it is clear that a is the restriction of

∆. Finally, putting X

X

for any X P F p U

q and using the fact that F p U

q

F p U

q , one checks that p A, a q is a coideal of B .

b) By Remark 2.5, H

`

xPΩ

C v

is a commutative unital C

-algebra, v

p x P Ω q are mutually orthogonal projections, and if A `

xPΩ

p X

b H

q is a weak coideal of B, then X

is a C

-subalgebra of H

. Its spectral mutually orthogonal projectors are v

i

, where Γ

€ Ω p i 1, ..., k

dim p X

qq are disjoint subsets of Ω, the unit of X

, i.e., the image of F p id

q , is v

, where Γ \

Γ

. As 1

v

b v

and 1

v

b v

, A is a coideal if and only if Γ Ω.

c) One checks that B

H

b v

and that any nontrivial orthoprojector p P r Z p A q X B

s gives a decomposition A pA ` p 1 p q A into the direct sum of two weak coideals of B . As 1

v

b v

, p must be of the form v

0

b v

, where Γ

is a proper nonempty subset of Γ.

d) The two G-C

-algebras are isomorphic because they correspond to the

same couple p M, M q .

Corollary 3.4 It follows from the definition of the functor F that X

F p U

q End

p M q . This finite dimensional C

-algebra is commutative due

to the statement b) which is only possible if m

P t 0, 1 u for all λ P Λ.

4 Weak Hopf C -Algebras related to Tambara- Yamagami categories

4.1 Tambara-Yamagami categories

These categories denoted by T Y p G, χ, τ q (G is a finite group; we consider them only over C ) are Z

-graded fusion categories whose 0-component is V ec

- the category of finite dimensional G-graded vector spaces with trivial associativities (its simple objects are g P G) and 1-component is generated by single simple object m. The Grothendieck ring of T Y p G, χ, τ q is isomorphic to the Z

-graded fusion ring T Y

Z G ` Zt m u such that g m m g m, m

Σ

gPG

g, m m

. These categories exist if and only if G is abelian, they are parameterized by non degenerate symmetric bicharacters χ : G G Ñ Czt 0 u and τ | G |

- see [18], [5], Example 4.10.5. The associativities φ p U, V, W q : p U b V q b W Ñ U b p V b W q are

φ p g, h, k q id

, φ p g, h, m q id

, φ p m, g, h q id

, φ p g, m, h q χ p g, h q id

, φ p g, m, m q `

hPG

id

, φ p m, m, g q `

hPG

id

, φ p m, g, m q `

hPG

χ p g, h q id

, φ p m, m, m q p τ χ p g, h q

id

q

,

where g, h, k P G. The unit isomorphisms are trivial. T Y p G, χ, τ q becomes a C

-tensor category when χ : G G Ñ T t z P C|| z | 1 u , from now on we assume that this is the case. The dual objects are: g

g, for all g P G, and m

m. The rigidity morphisms are defined by R

: 0 Ñ

g

b g, R

: 0

Ñ

g b g

, R

τ | G |

ι, and R

| G |

ι, where ι : 0 Ñ m b m is the inclusion. Then dim

p g q 1, for all g P G, and dim

p m q a

| G | . Let us apply Theorem 2.3 to the category T Y p G, χ, τ q in order to con- struct a biconnected regular WHA G

p B, ∆, S, ε q with U Corep p G

q T Y p G, χ, τ q . The Hayashi’s functor H : T Y p G, χ, τ q Ñ Corr

p R q , where C

-algebra R : End p `

xPΩ

x q C

, was constructed in [10]. Denoting Ω

Ω : G \ t m u and Ω

: G \ G, where g P G and G is the second copy of G, one easily computes that H

C

, for all g P G and H

: C

.

Let us fix a basis t v

up y P Ω

q in each H

p x P Ω q choosing a norm one

vector in every 1-dimensional vector subspace: v

P Hom p h, p h g q b g q ,

v

P Hom p m, m b g q , v

P Hom p m, g b m q , and v

P Hom p g, m b m q ,

where g P G.