# Bosonization for dual quasi-bialgebras and preantipode

**ABSTRACT** In this paper, we associate a dual quasi-bialgebra, called bosonization, to

every dual quasi-bialgebra $H$ and every bialgebra $R$ in the category of

Yetter-Drinfeld modules over $H$. Then, using the fundamental theorem, we

characterize as bosonizations the dual quasi-bialgebras with a projection onto

a dual quasi-bialgebra with a preantipode. As an application we investigate the

structure of the graded coalgebra $grA$ associated to a dual quasi-bialgebra

$A$ with the dual Chevalley property (e.g. $A$ is pointed).

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**ABSTRACT:**In this paper, the theory to construct quantum lines for general dual quasi-bialgebras is developed followed by some specific examples where the dual quasi-bialgebras are pointed with cyclic group of points.12/2013;

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arXiv:1111.4325v1 [math.QA] 18 Nov 2011

BOSONIZATION FOR DUAL QUASI-BIALGEBRAS AND PREANTIPODE

ALESSANDRO ARDIZZONI AND ALICE PAVARIN

Abstract. In this paper, we associate a dual quasi-bialgebra, called bosonization, to every

dual quasi-bialgebra H and every bialgebra R in the category of Yetter-Drinfeld modules over

H.Then, using the fundamental theorem, we characterize as bosonizations the dual quasi-

bialgebras with a projection onto a dual quasi-bialgebra with a preantipode. As an application

we investigate the structure of the graded coalgebra grA associated to a dual quasi-bialgebra A

with the dual Chevalley property (e.g. A is pointed).

Contents

1.

2.

2.1.

2.2.

2.3.

3.

3.1.

4.

5.

6.

6.1.

6.2.

Appendix A.

A.1.Example: the group algebra

References

Introduction

Preliminaries

The category of bicomodules for a dual quasi-bialgebras

An adjunction betweenHMH

The notion of preantipode

Yetter-Drinfeld modules over a dual quasi-bialgebra

The restriction of the equivalence (F,G)

Monoidal equivalences

The main results: bosonization

Applications

The associated graded coalgebra

On pointed dual quasi-bialgebras

The weak right center

1

3

4

5

5

8

9

HandHM

15

21

28

29

30

31

31

32

1. Introduction

Let H be a bialgebra. Consider the functor T := (−)⊗H : M → MH

spaces to the category of right Hopf modules. It is well-known that T determines an equivalence

if and only if H has an antipode i.e. it is a Hopf algebra. The fact that T is an equivalence is

the so-called fundamental (or structure) theorem for Hopf modules, which is due, in the finite-

dimensional case, to Larson and Sweedler, see [LS, Proposition 1, page 82]. This result is crucial

in characterizing the structure of bialgebras with a projection as Radford-Majid bosonizations (see

[Ra]). Recall that a bialgebra A has a projection onto a Hopf algebra H if there exist bialgebra

maps σ : H → A and π : A → H such that π ◦ σ = IdH. Essentially using the fundamental

theorem, one proves that A is isomorphic, as a vector space, to the tensor product R ⊗ H where

R is some bialgebra in the categoryH

HYD of Yetter-Drinfeld modules over H. This way R ⊗ H

inherits, from A, a bialgebra structure which is called the Radford-Majid bosonization of R by H

and denoted by R#H. It is remarkable that the graded coalgebra grA associated to a pointed

Hopf algebra A (here ”pointed” means that all simple subcoalgebras of A are one-dimensional)

Hfrom the category of vector

1991 Mathematics Subject Classification. Primary 16W30; Secondary 16S40.

Key words and phrases. Dual quasi-bialgebras, preantipode, Yetter-Drinfeld modules, bosonization, projections.

This paper was written while the authors were members of GNSAGA.

1

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2 ALESSANDRO ARDIZZONI AND ALICE PAVARIN

always admits a projection onto its coradical. This is the main ingredient in the so-called lifting

method for the classification of finite dimensional pointed Hopf algebras, see [AS].

In 1989 Drinfeld introduced the concept of quasi-bialgebra in connection with the Knizhnik-

Zamolodchikov system of partial differential equations. The axioms defining a quasi-bialgebra are

a translation of monoidality of its representation category with respect to the diagonal tensor

product. In [Dr], the antipode for a quasi-bialgebra (whence the concept of quasi-Hopf algebra) is

introduced in order to make the category of its flat right modules rigid. If we draw our attention

to the category of co-representations of H, we get the concepts of dual quasi-bialgebra and of dual

quasi-Hopf algebra. These notions have been introduced in [Maj3] in order to prove a Tannaka-

Krein type Theorem for quasi-Hopf algebras.

A fundamental theorem for dual quasi-Hopf algebras was proved by Schauenburg in [Sch4] but

dual quasi-Hopf algebras do not exhaust the class of dual quasi-bialgebras satisfying the funda-

mental theorem. It is remarkable that the functor T giving the fundamental theorem in the case

of ordinary Hopf algebras must be substituted, in the “quasi“ case, by the functor F := (−) ⊗ H

between the categoryHM of left H-comodules and the categoryHMH

H-bicomodules (essentially this is due to the fact that, unlike the classical case, a dual quasi-

bialgebra H is not an algebra in the category of right H-comodules but it is still an algebra in the

category of H-bicomodules). In [AP, Theorem 3.9], we showed that, for a dual quasi-bialgebra H,

the functor F is an equivalence if and only if there exists a suitable map S : H → H that we called

a preantipode for H. Moreover for any dual quasi-bialgebra with antipode (i.e. a dual quasi-Hopf

algebra) we constructed a specific preantipode, see [AP, Theorem 3.10].

Hof right dual quasi-Hopf

The main aim of this paper is to introduce and investigate the notion of bosonization in the

setting of dual quasi-bialgebras. Explicitly, we associate a dual quasi-bialgebra R#H (that we call

bosonization of R by H) to every dual quasi-bialgebra H and bialgebra R inH

fundamental theorem, we characterize as bosonizations the dual quasi-bialgebras with a projection

onto a dual quasi-bialgebra with a preantipode. As an application, for any dual quasi-bialgebra A

with the dual Chevalley property (i.e. such that the coradical of A is a dual quasi-subbialgebra

of A), under the further hypothesis that the coradical H of A has a preantipode, we prove that

there is a bialgebra R inH

HYD such that grA is isomorphic to R#H as a dual quasi-bialgebra.

In particular, if A is a pointed dual quasi-Hopf algebra, then grA comes out to be isomorphic to

R#kG(A) as dual quasi-bialgebra where R is the diagram of A and G(A) is the set of grouplike

elements in A. We point out that the results in this paper are obtained without assuming that the

dual quasi-bialgebra considered are finite-dimensional.

HYD. Then, using the

The paper is organized as follows.

Section 2 contains preliminary results needed in the next sections. Moreover in Theorem 2.16,

we investigate cocommutative dual quasi-bialgebras with a preantipode and in Corollary 2.20, we

provide a Cartier-Gabriel-Kostant type theorem for dual quasi-bialgebras with a preantipode. In

the connected case such a result was achieved in [Hu1, Theorem 4.3].

Section 3, is devoted to the study of the categoryH

dual quasi-bialgebra H. Explicitly, we consider the pre-braided monoidal category?H

Yetter-Drinfeld modules over a dual quasi-bialgebra H and we prove that the functor F, as above,

induces a functor F :H

H(that is an equivalence in case H has a preantipode, see

Proposition 3.8).

In Section 4, we prove that the equivalence between the categoriesH

monoidal if we equipH

Hwith the tensor product ⊗H (or ?H) and unit H (see Lemma 4.4

and Lemma 4.8). As a by-product, in Lemma 4.11, we produce a monoidal equivalence between

(H

H,?H,H).

Section 5 contains the main results of the paper. In Theorem 5.2, to every dual quasi-bialgebra

H and bialgebra R inH

HYD we associate a dual quasi-bialgebra structure on the tensor product

R⊗H that we call the bosonization of R by H and denote by R#H. Now, let (A,H,σ,π) be a dual

quasi-bialgebra with projection and assume that H has a preantipode S. In Lemma 5.6, we prove

that such an A is an object in the categoryH

H. Therefore the fundamental theorem describes

HYD of Yetter-Drinfeld modules over a

HYD,⊗,k?of

HYD →H

HMH

HMH

HandH

HYD becomes

HMH

HMH

H,⊗H,H) and (H

HMH

HMH

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BOSONIZATION FOR DUAL QUASI-BIALGEBRAS AND PREANTIPODE3

A as the tensor product R ⊗ H of some vector space R by H. Indeed, in Theorem 5.7, we prove

that the dual quasi-bialgebra structure inherited by R ⊗ H through the claimed isomorphism is

exactly the bosonization of R by H. The analogous of this result for quasi-Hopf algebras, anything

but trivial, has been established by Bulacu and Nauwelaerts in [BN], but their proof can not be

adapted to dual quasi-bialgebras with a preantipode.

In Section 6 we collect some applications of our results. Let A be a dual quasi-bialgebra with

the dual Chevalley property and coradical H. Since A is an ordinary coalgebra, we can consider

the associated graded coalgebra grA. In Proposition 6.3, we prove that grA fits into a dual quasi-

bialgebra with projection onto H. As a consequence, in Corollary 6.4, under the further assumption

that H has a preantipode, we show that there is a bialgebra R inH

to R#H as a dual quasi-bialgebra. When A is a pointed dual quasi-Hopf algebra it is in particular

a dual quasi-bialgebra with the dual Chevalley property and its coradical has a preantipode. Using

this fact, in Theorem 6.10 we obtain that grA is of the form R#kG(A) as dual quasi-bialgebra,

where R is the so-called diagram of A.

HYD such that grA is isomorphic

2. Preliminaries

In this section we recall the definitions and results that will be needed in the paper.

Notation 2.1. Throughout this paper k will denote a field. All vector spaces will be defined over

k. The unadorned tensor product ⊗ will denote the tensor product over k if not stated otherwise.

2.2. Monoidal Categories. Recall that (see [Ka, Chap. XI]) a monoidal category is a category

M endowed with an object 1 ∈ M (called unit), a functor ⊗ : M × M → M (called tensor

product), and functorial isomorphisms aX,Y,Z : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z), lX : 1 ⊗ X → X,

rX : X ⊗ 1 → X, for every X,Y,Z in M. The functorial morphism a is called the associativity

constraint and satisfies the Pentagon Axiom, that is the equality

(U ⊗ aV,W,X) ◦ aU,V ⊗W,X◦ (aU,V,W⊗ X) = aU,V,W⊗X◦ aU⊗V,W,X

holds true, for every U,V,W,X in M. The morphisms l and r are called the unit constraints and

they obey the Triangle Axiom, that is (V ⊗ lW) ◦ aV,1,W= rV ⊗ W, for every V,W in M.

The notions of algebra, module over an algebra, coalgebra and comodule over a coalgebra can

be introduced in the general setting of monoidal categories. Given an algebra A in M one can

define the categoriesAM, MAandAMAof left, right and two-sided modules over A respectively.

Similarly, given a coalgebra C in M, one can define the categories of C-comodulesCM,MC,CMC.

For more details, the reader is refereed to [AMS1].

Let M be a monoidal category. Assume that M is abelian and both the functors X ⊗ (−) :

M → M and (−) ⊗ X : M → M are additive and right exact, for any X ∈ M. Given an algebra

A in M, there exist a suitable functor ⊗A:AMA×AMA→AMAand constraints that make the

category (AMA,⊗A,A) monoidal, see [AMS1, 1.11]. The tensor product over A in M of a right

A-module (V,µr

W) is defined to be the coequalizer:

V) and a left A-module (W,µl

(V ⊗ A) ⊗ W

µr

V⊗W

??

aV,A,W

?

???

?

?

?

?

?

?

?

?

?

?

V ⊗ W

???

?

?

?

?

?

?

?

?

?

?

AχV,W

??V ⊗AW

??0

V ⊗ (A ⊗ W)

V ⊗µl

W

Note that, since ⊗ preserves coequalizers, then V ⊗AW is also an A-bimodule, whenever V and

W are A-bimodules.

Dually, given a coalgebra (C,∆,ε) in a monoidal category M, abelian and with additive and left

exact tensor functors, there exist a suitable functor ?C:CMC×CMC→CMCand constraints

that make the category (CMC,?C,C) monoidal. The cotensor product over C in M of a right

Page 4

4 ALESSANDRO ARDIZZONI AND ALICE PAVARIN

C-comodule (V,ρr

V) and a left C-comodule (W,ρl

W) is defined to be the equalizer:

0

??V ?CW

CςV,W

??V ⊗ W

V ⊗ρl

W

??

ρr

V⊗W

???

?

?

?

?

?

?

?

?

?

?

V ⊗ (C ⊗ W)

???

?

?

?

?

?

?

?

?

?

?

?

(V ⊗ C) ⊗ W

aV,C,W

Note that, since ⊗ preserves equalizers, then V ?CW is also a C-bicomodule, whenever V and W

are C-bicomodules.

Definition 2.3. A dual quasi-bialgebra is a datum (H,m,u,∆,ε,ω) where

• (H,∆,ε) is a coassociative coalgebra;

• m : H ⊗ H → H and u : k → H are coalgebra maps called multiplication and unit

respectively; we set 1H:= u(1k);

• ω : H ⊗ H ⊗ H → k is a unital 3-cocycle i.e. it is convolution invertible and satisfies

ω (H ⊗ H ⊗ m) ∗ ω (m ⊗ H ⊗ H)=mk(ε ⊗ ω) ∗ ω (H ⊗ m ⊗ H) ∗ mk(ω ⊗ ε) (1)

andω(h ⊗ k ⊗ l)=ε(h)ε(k)ε(l)whenever1H∈ {h,k,l};(2)

• m is quasi-associative and unitary i.e. it satisfies

m(H ⊗ m) ∗ ω = ω ∗ m(m ⊗ H), (3)

m(1H⊗ h) = h, for all h ∈ H,(4)

m(h ⊗ 1H) = h, for all h ∈ H.(5)

ω is called the reassociator of the dual quasi-bialgebra.

A morphism of dual quasi-bialgebras (see e.g. [Sch1, Section 2])

α : (H,m,u,∆,ε,ω) → (H′,m′,u′,∆′,ε′,ω′)

is a coalgebra homomorphism α : (H,∆,ε) → (H′,∆′,ε′) such that

m′(α ⊗ α) = αm, αu = u′,ω′(α ⊗ α ⊗ α) = ω.

It is an isomorphism of quasi-bialgebras if, in addition, it is invertible.

A dual quasi-subbialgebra of a dual quasi-bialgebra (H′,m′,u′,∆′,ε′,ω′) is a quasi-bialgebra

(H,m,u,∆,ε,ω) such that H is a vector subspace of H′and the canonical inclusion α : H → H′

yields a morphism of dual quasi-bialgebras.

2.1. The category of bicomodules for a dual quasi-bialgebras. Let (H,m,u,∆,ε,ω) be

a dual quasi-bialgebra. It is well-known that the category MHof right H-comodules becomes

a monoidal category as follows. Given a right H-comodule V , we denote by ρ = ρr

V ⊗ H,ρ(v) = v0⊗ v1, its right H-coaction. The tensor product of two right H-comodules V and

W is a comodule via diagonal coaction i.e. ρ(v ⊗ w) = v0⊗ w0⊗ v1w1. The unit is k, which is

regarded as a right H-comodule via the trivial coaction i.e. ρ(k) = k ⊗ 1H. The associativity and

unit constraints are defined, for all U,V,W ∈ MHand u ∈ U,v ∈ V,w ∈ W,k ∈ k, by

V: V →

aH

U,V,W((u ⊗ v) ⊗ w) := u0⊗ (v0⊗ w0)ω(u1⊗ v1⊗ w1),

lU(k ⊗ u) := kuandrU(u ⊗ k) := uk.

The monoidal category we have just described will be denoted by (MH,⊗,k,aH,l,r).

Similarly, the monoidal categories (HM,⊗,k,Ha,l,r) and (HMH,⊗,k,HaH,l,r) are introduced.

We just point out that

HaU,V,W((u ⊗ v) ⊗ w) := ω−1(u−1⊗ v−1⊗ w−1)u0⊗ (v0⊗ w0),

HaHU,V,W((u ⊗ v) ⊗ w) := ω−1(u−1⊗ v−1⊗ w−1)u0⊗ (v0⊗ w0)ω(u1⊗ v1⊗ w1).

Page 5

BOSONIZATION FOR DUAL QUASI-BIALGEBRAS AND PREANTIPODE5

Remark 2.4. We know that, if (H,m,u,∆,ε,ω) is a dual quasi-bialgebra, we cannot construct the

category MH, because H is not an algebra. Moreover H is not an algebra in MHor inHM. On

the other hand ((H,ρl

with ρl

H-modules inHMH. Hence, we can set

H,ρr

H),m,u) is an algebra in the monoidal category (HMH,⊗,k,HaH,l,r)

H= ∆. Thus, the only way to construct the categoryHMH

H= ρr

His to consider the right

HMH

H:= (HMH)H.

The categoryHMH

2.3].

His the so-called category of right dual quasi-Hopf H-bicomodules [BC, Remark

Remark 2.5. [AMS1, Example 1.5(a)] Let (A,m,u) be an algebra in a given monoidal category

(M,⊗,1,a,l,r). Then the assignments M ?−→ (M ⊗A,(M ⊗m)◦aA,A,A) and f ?−→ f ⊗A define

a functor T : M → MA. Moreover the forgetful functor U : MA→ M is a right adjoint of T.

2.2. An adjunction betweenHMH

HMH

HandHM that will be crucial afterwards.

HandHM. We are going to construct an adjunction between

2.6. Consider the functor L :

upper empty dot denotes the trivial right coaction while the upper full dot denotes the given

left H-coaction of V. The functor L has a right adjoint R :

by R(•M•) :=•McoH, where McoH:= {m ∈ M | m0⊗ m1 = m ⊗ 1H} is the space of right

H-coinvariant elements in M.

By Remark 2.5, the forgetful functor U :HMH

namely the functor T :HMH→HMH

which tensor factors we have a codiagonal coaction and the lower dot indicates where the action

takes place. Explicitly, the structure of T (•M•) is given as follows:

HM →

HMHdefined on objects by L(•V ) :=

•V◦where the

HMH→HM defined on objects

H→HMH,U (•M•

•) :=•M•has a right adjoint,

•. Here the upper dots indicate on

H,T (•M•) :=•M•⊗•H•

ρl

ρr

M⊗H(m ⊗ h):= m−1h1⊗ (m0⊗ h2),

= (m0⊗ h1) ⊗ m1h2,

(m ⊗ h)l := ω−1(m−1⊗ h1⊗ l1)m0⊗ h2l2ω(m1⊗ h3⊗ l3).

M⊗H(m ⊗ h):

µr

M⊗H[(m ⊗ h) ⊗ l]=

Define the functors F := TL :HM →HMH

•McoHand F(•V ) :=•V◦⊗•H•

Hand G := RU :HMH

•so that, for every v ∈ V,h,l ∈ H,

H→HM. Explicitly G(•M•

•) =

ρl

ρr

V ⊗H(v ⊗ h) = v−1h1⊗ (v0⊗ h2),

V ⊗H(v ⊗ h) = (v ⊗ h1) ⊗ h2,

V ⊗H[(v ⊗ h) ⊗ l] = (v ⊗ h)l = ω−1(v−1⊗ h1⊗ l1)v0⊗ h2l2.µr

Remark 2.7. By the right-hand version of [Sch4, Lemma 2.1], the functor F :HM →HMH

left adjoint of the functor G, where the counit and the unit of the adjunction are given respectively

by ǫM : FG(M) → M,ǫM(x ⊗ h) := xh and by ηN : N → GF(N),ηN(n) := n ⊗ 1H, for every

M ∈HMH

F is fully faithful.

His a

H,N ∈HM. Moreover ηN is an isomorphism for any N ∈HM. In particular the functor

2.3. The notion of preantipode. Next result characterizes when the adjunction (F,G) is an

equivalence of categories in term of the existence of a suitable map τ.

Proposition 2.8. [AP, Proposition 3.3] Let (H,m,u,∆,ε,ω) be a dual quasi-bialgebra. The fol-

lowing assertions are equivalent.

(i) The adjunction (F,G) is an equivalence.

(ii) For each M ∈HMH

H, there exists a k-linear map τ : M → McoHsuch that:

τ(mh)=ω−1[τ(m0)−1⊗ m1⊗ h]τ(m0)0, for all h ∈ H,m ∈ M,

τ(m0)−1m1⊗ τ(m0)0, for all m ∈ M,

m ∀m ∈ M.

(6)

m−1⊗ τ(m0)

τ(m0)m1

= (7)

=(8)