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The Hopf category of Frobenius algebras

June 2024

DOI:10.48550/arXiv.2406.18499

Authors:

Paul Großkopf

University of Oxford

Joost Vercruysse

Université Libre de Bruxelles

Preprints and early-stage research may not have been peer reviewed yet.

We show that the universal measuring coalgebras between Frobenius algebras turn the category of Frobenius algebras into a Hopf category (in the sense of \cite{BCV}), and the universal comeasuring algebras between Frobenius algebras turn the category Frobenius algebras into a Hopf opcategory. We also discuss duality and compatibility results between these structures. Our theory vastly generalizes the well-known fact that any homomorphism beween Frobenius algebras is an isomorphism, but also allows to go beyond classical (iso)morphisms between Frobenius algebras, especially in finite characteristic, as we show by some explicit examples.

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arXiv:2406.18499v2 [math.QA] 11 Jul 2024

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS

PAUL GROSSKOPF AND JOOST VERCRUYSSE

Abstract. We show that the universal measuring coalgebras between Frobenius algebras

turn the category of Frobenius algebras into a Hopf category (in the sense of [3]), and the

universal comeasuring algebras between Frobenius algebras turn the category Frobenius al-

gebras into a Hopf opcategory. We also discuss duality and compatibility results between

these structures.

Our theory vastly generalizes the well-known fact that any homomorphism beween Frobe-

nius algebras is an isomorphism, but also allows to go beyond classical (iso)morphisms be-

tween Frobenius algebras, especially in ﬁnite characteristic, as we show by some explicit

examples.

The paper is concluded with some open questions and considerations about Topological

Quantum Field Theories.

Contents

1. Introduction 1

Notations and conventions 3

Acknowledgements 3

2. Preliminaries and ﬁrst results 3

2.1. Morphisms of Frobenius algebras 3

2.2. Hopf categories and Hopf opcategories 6

2.3. Measuring and comeasuring semi-Hopf categories for Ω-algebras 9

3. The Hopf category of Frobenius algebras 14

4. Dualities 23

4.1. Sweedler dual of a semi-Hopf opcategory 23

4.2. Duality for Frobenius algebras and the antipode of their universal Hopf category 25

5. Examples 30

6. Conclusions and outlook 39

6.1. Remaining questions 39

6.2. Motivation from topological quantum ﬁeld theory 39

References 41

1. Introduction

There exist many interesting connections between Hopf algebras and Frobenius algebras,

or more generally between Hopf structures and Frobenius structures. The most well-known

result in this spirit is probably the so-called Larson-Sweedler theorem [12], which tells that

any ﬁnite dimensional Hopf algebra has a non-zero integral and therefore is Frobenius. This

result has been reﬁned by Pareigis [14], who showed that a bialgebra is ﬁnite dimensional and

2 P. GROSSKOPF AND J. VERCRUYSSE

Hopf if and only if it is Frobenius and the Frobenius structure is in a suitable way compatible

with the bialgebra structure (roughly meaning that the Frobenius structure arises from an

integral). More generally, Kreimer and Takeuchi showed in [11] that Hopf-Galois objects, or

more generally ﬁnite Hopf-Galois extensions with free invariants, are Frobenius. To name

just one of the many other links between Hopf and Frobenius conditions, let us mention that

it was recently observed by Saracco that a bialgebra Bis Hopf if and only if the free functor

− ⊗ B:BMod →BModB

Bis Frobenius (i.e. has an isomorphic left and right adjoint), see [16]

and [17].

The aim of this paper is to show another relationship between Frobenius algebras and Hopf

structures, which is of a very diﬀerent nature. Our investigations start from the well-known

observation that any morphism between two Frobenius algebras (preserving both the algebra

and coalgebra structures) is necessarily an isomorphism. This turns the category of Frobenius

algebras into a groupoid. And as is also well-known, groupoids give rise to weak (multiplier)

Hopf algebras via linearisation. This is the link between ”Frobenius” and ”Hopf” that we

aim to explore, although in a more general setting.

To this end, we will make use of Sweedler’s theory of measuring coalgebras [18]. This

theory allows to treat homomorphisms between algebras Aand Bas well as derivations and

various other types of generalized morphisms at the same time. More precisely, one considers

a coalgebra Ptogether with a linear map ψ:P⊗A→Bsuch that the associated map

A→Hom(P, B) is an algebra morphism with respect to the convolution algebra structure on

the codomain. Moreover, Sweedler showed the existence of a universal measuring coalgebra

between any pair of algebras. This theory has been generalized in several ways, for example

to the setting of closed monoidal categories in [9] and to more general types of algebraic

structures, termed Ω-algebras in [1].

It was already implicitly observed in Sweedler that the existence of universal measuring

coalgebras allows the enrichement of the category of algebras over the category of coalgebras.

In the terminology of [3] this means that the category of algebras can be turned into a (k-

linear) semi-Hopf category. Following the work of [1], the same holds for any category of

Ω-algebras. A natural question arises whether it is also possible to make this category into a

Hopf-category, that is, to show the existence of an antipode for this semi-Hopf category. One

possibility is to extend the considered semi-Hopf category by a free or cofree construction.

The fact that this is indeed possible has been shown in [8]. However, more interestingly,

one could look for conditions such that the semi-Hopf category arising from the universal

measurings is Hopf. A ﬁrst result of this ﬂavour has been discussed in [9], where it was

shown that the universal measuring coalgebra between a cocommutative Hopf algebra and a

commutative bialgebra is itself a Hopf algebra in a natural way. Remark however, that this

Hopf algebra structure is “local” and hence this construction does not give rise to a Hopf

category.

The main aim of this paper is to show (see Theorem 3.2) that the semi-Hopf category of

universal measurings between Frobenius algebras is automatically Hopf, and that the antipode

is moreover invertible. By restricting to the grouplike elements in the measuring coalgebras,

one recovers from this the above mentioned result, that any morphism between Frobenius

algebras is invertible.

Dually to measuring coalgebras, one can consider comeasuring algebras. However, universal

comeasurings only exist under suitable ﬁniteness conditions (see [1]). Since Frobenius algebras

are necessarily ﬁnite dimensional, universal comeasurings exist as well between any pair of

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 3

Frobenius algebras. This, in turn, allows to endow a category of Frobenius algebras with a

semi-Hopf opcategory structure, which again turns out to be Hopf (see Theorem 3.5). As one

can expect, both theories are dual to each other, in the sense that the measuring coalgebras

arise as the Sweeder dual of comeasuring algebras. Furthermore, also the notion of Frobenius

algebra itself has a nice self-duality. This can be combined with the measurings in several

ways, which allows to recover the antipode of the measuring Hopf category, as we show in

Theorem 4.7.

To illustrate our theory, we explicitly compute the universal measuring coalgebra for some

small examples of Frobenius algebras arising from group algebras and matrix algebras. These

examples show some particular behaviour, that motivates us to formulate some open ques-

tions for future investigations. In particular, we discuss the remarkable behaviour of universal

measuring coalgebras, that existence of an (iso)morphism after base extension is already re-

ﬂected by the existence of non-zero measurings before base extension (see Proposition Propo-

sition 5.6). We ﬁnish the paper with some consideration about Topological Quantum Field

Theory that were in fact the initial motivation for our investigations.

Notations and conventions. Throughout this paper we work with algebras over a commu-

tative ﬁeld k, i.e. within the symmetric monoidal category of k-vector spaces that we denote

as Vectk. However, the results of this paper hold in a general symmetric monoidal base cat-

egory V, under the condition that universal (co)measuring objects exist in V(examples of

such categories are, apart from vector spaces over ﬁeld, representations of a group or graded

vector spaces over a commutative group, for more examples we refer for example to [2]).

We denote the identity morphism on an object Ain Vby A,IdAor just by Id.

Acknowledgements. We thank William Hautekiet for assistance to compute the dimen-

sions of universal measuring coalgebras by means of computer algebra. The ﬁrst author was

supported by a Fria fellowship of FNRS FNRS (Fonds National de la Recherche Scientiﬁque),

grant number FC41285, while working on this project. The second author would like to

thank the FWB (f´ed´eration Wallonie-Bruxelles) for support through the ARC project “From

algebra to combinatorics, and back”.

2. Preliminaries and first results

2.1. Morphisms of Frobenius algebras. We recall some deﬁnitions and known properties

of Frobenius algebras, as they will appear to be useful in the new results obtained later in

this paper. We refer to [5] and [10] for more details.

Deﬁnition 2.1. AFrobenius algebra is 5-tuple (A, µ, η, ∆, ν), with linear maps

µ:A⊗A→A, η :k→A,

∆ : A→A⊗A, ν :A→k,

such that:

•(A, µ, η) is an (associative, unital) algebra,

•(A, ∆, ν) is a (coassociative, counital) coalgebra,

•the Frobenius conditions are satisﬁed, meaning that the following diagram commutes

4 P. GROSSKOPF AND J. VERCRUYSSE

A⊗A



∆⊗A

ww♣♣

♣

♣♣

♣

♣♣

♣

♣♣A⊗∆

◆

A⊗A⊗A

A⊗µ''

◆

◆A

∆



A⊗A⊗A

µ⊗A

ww♣

♣

♣♣

♣

♣♣

♣

♣♣

♣

A⊗A.

Amorphism of Frobenius algebras is a linear map that is an algebra morphism as well as a

coalgebra morphism.

With Sweedler type notation ∆(a) = a(1) ⊗a(2), the Frobenius conditions tell us that for

all a, b ∈A:

(ab)(1) ⊗(ab)(2) =a(1) ⊗a(2)b=ab(1) ⊗b(2).

Consequently, the comultiplication is completely determined by its value in the unit, which

is given by the so-called Casimir element

∆(1) = 1(1) ⊗1(2) =: e1⊗e2=e∈A⊗A

which satisﬁes - because of the bilinearity of the comultiplication - the following Casimir

property

e1⊗e2a= ∆(1a) = ∆(a) = ∆(a1) = ae1⊗e2.(1)

In calculations one often has to deal with more than one copy of the Casimir element. Hence

we will use the notation

e1⊗e2=E1⊗E2=ε1⊗ε2.

By the Casimir property, we then have the following identities

E1e1⊗e2⊗E2=e1⊗e2E1⊗E2=e1⊗E1⊗E2e2.

The counitality of the comultiplication can be expressed in terms of the Casimir element in

the following way

ν(e1)e2= 1 = e1ν(e2).

It is well known that the coassociativity of the the comultiplication of a Frobenius algebra

can be deduced from its associativity (and vice versa). Moreover a Frobenius algebra Ais

called symmetric if ν(ab) = ν(ba) for all a, b ∈A, or equivalently the Casimir element satisﬁes

e1⊗e2=e2⊗e1.

Classical examples of Frobenius algebras are matrix algebras, ﬁnite group algebras and ﬁnite

dimensional Hopf algebras.

It turns out that the restrictions on Frobenius algebras are so strong, that every morphism

of Frobenius algebras is already an isomorphism. We give a detailed proof of this very well-

known result, since we will generalize the argument later.

Proposition 2.2. Every morphism of Frobenius algebras is an isomorphism.

Proof. Let Aand Bbe two Frobenius algebras. If h:A→Bis morphism of Frobenius

algebras, halso preserves the Casimir elements, since

h(e1)⊗h(e2) = (h⊗h)∆(1A) = ∆h(1A) = ∆(1B) =: f1⊗f2.

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 5

We can deﬁne an inverse g:B→Aby

g(b) = e1νB(h(e2)b).

Let us check that gand hare mutual inverses:

(hg)(b) = h(e1)νB(h(e2)b) = f1νB(f2b) = bf 1νB(f2) = b,

(gh)(a) = e1νB(h(e2)h(a)) = e1νB(h(e2a)) = e1νA(e2a) = ae1νA(e2) = a.

Therefore his an isomorphism of Frobenius algebras. 

Remark 2.3.As usual, we know that the linear inverse of an invertible morphism is a mor-

phism itself. In case of a morphism of Frobenius algebras, it can be veriﬁed explicitly that

the proposed inverse as in the above proof is a morphism of Frobenius algebras, even without

making explicit use of its invertibility. Let us provide the explicit computations here, as they

are the guideline for more general computations further in this paper.

The map gpreserves the comultiplication:

g(b(1))⊗g(b(2)) = g(f1)⊗g(f2b) = e1νB(h(e2)f1)⊗E1νB(h(E2)f2b)

=e1νB(h(e2)h(ε1)) ⊗E1νB(h(E2)h(ε2)b)

=e1νB(h(e2ε1)) ⊗E1νB(h(E2ε2)b)

=ε1e1νB(h(e2)) ⊗E1νB(h(E2ε2)b)

=ε1e1νA(e2)⊗E1νB(h(E2ε2)b)

=ε1⊗E1νB(h(E2ε2)b)

=ε1⊗ε2E1νB(h(E2)b)

= ∆(g(b)).

The map gpreserves the unit:

g(1B) = e1

AνB(h(e2)1B) = e1

AνA(e2) = 1A.

The map gpreserves the multiplication:

g(b)g(b′) = e1νB(h(e2)b)E1νB(h(E2)b′)

=e1E1νB(h(e2)b)νB(h(E2)b′)

=E1νB(h(e2)b)νB(h(E2e1)b′)

=E1νB(h(e2)b)νB(h(E2)h(e1)b′)

=E1νB(f2)b)νB(h(E2)f1b′)

=E1νB(f2))νB(h(E2)bf1b′)

=E1νB(h(E2)bf1νB(f2)b′)

=E1νB(h(E2)bb′)

=g(bb′).

The map gpreserves the counit:

νA(g(b)) = νA(e1)νB(h(e2)b) = νB(h(νA(e1)e2)b) = νB(h(1A)b) = νB(1Bb) = νB(b).

6 P. GROSSKOPF AND J. VERCRUYSSE

2.2. Hopf categories and Hopf opcategories. Hopf V-categories, where Vis a symmetric

(or braided) monoidal category were introduced in [3]. We recall the deﬁnitions and basic

properties for the reader’s convenience.

Deﬁnition 2.4. Asemi-Hopf V-category where (V,⊗, I, σ) is a braided category is a category

Aenriched over the monoidal category of coalgebras (or comonoids) in V. Explicitely, a

semi-Hopf V-category A= (A0, Axy) consists of a collection1of objects A0and for all objects

x, y ∈A0we have an object Axy ∈ V together with morphisms in V(for all x, y, z ∈A0):

mxyz :Axy ⊗Ayz →Axz , jx:I→Axx,

δxy :Axy →Axy ⊗Axy, ǫxy :Axy →I

satisfying the commutativity of the following diagrams.

Axy ⊗Ayz ⊗Azu

mxyz ⊗Id //

Id⊗myzu



Axz ⊗Azu

mxzu



Axy ⊗Ayu

mxyu //Axu

Axy ⊗Ayy

mxyy

❏

Axx ⊗Axy

mxxy

yyt

Axy ⊗I

Id⊗jy

∼

=//Axy I⊗Axy

jx⊗Id

∼

Axy

δxy //

δxy



Axy ⊗Axy

δxy ⊗Id



Axy ⊗Axy Id⊗δxy

//Axy ⊗Axy ⊗Axy

Axy

δxy

))

❘

∼

=//

∼



Axy ⊗I

I⊗Axy Axy ⊗Axy

Id⊗ǫxy

ǫxy⊗Id

Axy ⊗Ayz

δxy ⊗δxy //

mxyz



Axy ⊗Axy ⊗Ayz ⊗Ayz

Id⊗σ⊗Id



Axy ⊗Ayz ⊗Axy ⊗Ayz

mxyz ⊗mxyz



Axz δxz

//Axz ⊗Axz

I≃//



I⊗I

jx⊗jx



Axy ⊗Ayz

ǫxy⊗ǫy z

mxyz



I⊗I

≃



Id //I



Axx δxx

//Axx ⊗Axx Axz ǫxz //I Axx ǫxx //I

A morphism of semi-Hopf V-categories f:A→Bconsists of a map f0:A0→B0and

for each x, y ∈A0aV-morphism fxy :Axy →Bf xf y 2that respect the structure morphisms in

1This could be a proper class although to avoid set-theoretical issues we will mostly suppose that A0is a

set.

2In order not to overload notation, we denote f0(x) simply by f x for any x∈A0.

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 7

the following sense.

Axy

fxy //

δxy



Bfxf y

δfxf y



Axy ⊗Axy

fxy⊗fxy //Bf xfy ⊗Bf xf y

Axy

ǫxy

❆

fxy //Bfxf y

ǫfx f y

}}③

③

Axy ⊗Ayz

fxy⊗fy z //

mxyz



Bfxf y ⊗Bf yf z

mfxf y f z



Axz

fxz //Bfxf z

~~⑥

⑥

⑥jfx

❉

Axx

fxx //Bfxf x

In other words, fis a Coalg(V)-functor where Coalg(V) denotes the category of coalgebras in

V. We denote the category of semi-Hopf V-categories and their morphisms by V-sHopf.

From the deﬁnition it might be clear that A= ({∗}, A∗∗) is a semi-Hopf category with just

one object, if and only if A∗∗ is a bialgebra in V. Conversely, for any semi-Hopf category, A

and any object x∈A0, we have that Axx is a bialgebra in Vand Axy is an (Axx, Ayy )-bimodule

coalgebra. The transition from bialgebras to Hopf algebras leads in the “many-object case”

to the introduction of a Hopf category as in the next deﬁnition.

Deﬁnition 2.5. A semi-Hopf V-category His called Hopf V-category, if it is equipped with

a family of V-morphisms sxy :Hxy →Hyx , called the antipode, satisfying:

mxyx(Axy ⊗sxy)δxy =jxǫxy,

myxy (sxy ⊗Axy )δxy =jyǫxy .

The category of Hopf V-categories and morphisms of semi-Hopf categories between them is

denoted by V-Hopf.

In what follows, when V=Vectk, we will simply use the term “Hopf category” for a Hopf

Vectk-category. The reader should be warned that the name “Hopf category” is also used for

a diﬀerent notions, as considered in [6] or [15].

One can show (see [3]) that a morphism of semi-Hopf categories between Hopf categories

preserves the antipode, hence that the antipode for a semi-Hopf category is unique whenever

it exits. Furthermore, the antipode of a Hopf category Asatisﬁes the following properties:

sxymxyz =mzyxσ(sxy ⊗syx)δyxsxy =σ(sxy ⊗sxy)δxy (2)

sxxjx=jxǫyx sxy =ǫxy .(3)

In other words, scan be viewed as an identity-on-objects morphism of semi-Hopf categories

from Ato Aop,cop, where Acop is the semi-Hopf category with co-opposite comultiplications at

Axy for every x, y ∈A0and Aop is the semi-Hopf category with Hom-objects Aop

xy =Ayx for

all x, y ∈A0and composition Ayx ⊗Azy

σ−1

//Azy ⊗Ayx

mzyx //Azx .

Where (semi-) Hopf categories have a ”local” coalgebra structure and a ”global” algebra

structure, one can also consider dual objects with a ”local” algebra structure and ”global”

coalgebra structure. We term such objects ”(semi-)Hopf opcategories”, as is also done in [4]

and [8].

8 P. GROSSKOPF AND J. VERCRUYSSE

Deﬁnition 2.6. Asemi-Hopf V-opcategory A= (A0, Axy ) consists of a collection of objects

A0and for all objects x, y ∈A0we have an object Axy ∈ V together with morphisms in V

(for all x, y, z ∈A0):

µxy :Axy ⊗Axy →Axy , ηxy :I→Axy ,

dxyz :Axz →Axy ⊗Ayz , ex:Axx →I

satisfying the commutativity of the following diagrams.

Axu

dxyu //

dxzu



Axz ⊗Azu

dxyz ⊗Id



Axy ⊗Ayu

Id⊗dyzu //Axy ⊗Ayz ⊗Azu

Axy ⊗Ayy

Id⊗ey



Axx ⊗Axy

ex⊗Id



Axy ⊗I∼

=//Axy

dxyy

ee❏

❏

dxxy 99

tI⊗Axy

∼

Axy ⊗Axy ⊗Axy

Id⊗µxy //

µxy ⊗Id



Axy ⊗Axy

µxy



Axy ⊗Axy µxy //Axy

Axy ⊗Axy

µxy

))

❘

Axy ⊗I

Id⊗ηxy

I⊗Axy

ηxy ⊗Id

Axy

∼

Axz ⊗Axz

dxyz ⊗dxyz //

µxz



Axy ⊗Ayz ⊗Axy ⊗Ayz

Id⊗σ⊗Id



Axy ⊗Axy ⊗Ayz ⊗Ayz

µxy ⊗µyz



Axz dxyz

//Axy ⊗Ayz

I≃//

ηxz



I⊗I

ηxy⊗ηy z



Axx ⊗Axx

ex⊗ex//

µxx



I⊗I

≃



ηxx



Id //I



Axz δxyz

//Axy ⊗Ayz Axx ex//I Axx ex//I

A morphism of semi-Hopf V-opcategories f:A→Bconsists of a map f0:A0→B0and for

each x, y ∈A0aV-morphism fxy :Axy →Bf xf y that respect the structure morphisms in the

following sense.

Axz

fxz //

dxyz



Bfxf z

dfxf z



Axy ⊗Ayz

fxy⊗fy z //Bf xf y ⊗Bf yfz

Axx

❆

fxx //Bfxf x

efx

||③

③

Axy ⊗Axy

fxy⊗fxy //

µxy



Bfxf y ⊗Bf xf y

µfxf y



Axy

fxy //Bfxf y

ηxy

~~⑥

⑥

⑥ηfxf y

❉

Axy

fxy //Bfxf y

We denote the category of semi-Hopf V-opcategories and their morphisms by V-opsHopf.

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 9

Again, we have that A= ({∗}, A∗∗) is a semi-Hopf opcategory with just one object if and

only if A∗∗ is a bialgebra in V. Conversely, for any semi-Hopf opcategory, Aand any object

x∈A0, we have that Axx is a bialgebra in Vand Axy is an (Axx, Ayy )-bicomodule algebra.

The deﬁnition of a semi-Hopf opcategory is exactly dual to the deﬁnition of semi-Hopf

category, in the sense that a semi-Hopf V-opcategory is exactly a semi-Hopf Vop-category.

However, it is important to obsever that the notion of a morphism between semi-Hopf V-

opcategories is diﬀerent from a morphism of semi-Hopf Vop-categories ! Indeed, a morphism

of semi-Hopf V-opcategories f:A→Bconsists of a map f:A0→B0and a collection

of V-morphisms fxy :Axy →Bfxf y satisfying axioms. On the other hand, a morphism of

Vop-categories f:A→Bconsists of a collection of V-morphisms fxy :Bfxfy →Axy satisfying

axioms. Therefore the categories V-opsHopf and Vop-sHopf are truly diﬀerent, and the theory

of semi-Hopf opcategories can not be deduced from the theory of semi-Hopf categories by

direct dualization ! The latter category is what has been called ”dual (semi-)Hopf categories”

in [3]. For this reason, we have to treat semi-Hopf opcategories separately in what follows.

Deﬁnition 2.7. A semi-Hopf V-opcategory His called Hopf V-opcategory, if it is equipped

with a family of V-morphisms sxy :Hxy →Hyx , called the antipode satisfying:

µxy(Axy ⊗syx)dxyx =ηxy ex,

µyx(sxy ⊗Ayx)dxyx =ηyx ex.

The category of Hopf V-categories and morphisms of semi-Hopf categories between them is

denoted by V-opHopf.

2.3. Measuring and comeasuring semi-Hopf categories for Ω-algebras.

2.3.1. Measuring coalgebras. Measuring coalgebras and universal measuring coalgebras be-

tween algebras were introduced by Sweedler in [18], but can be considered for general algebraic

structures with structure maps of arbitrary arity, called Ω-algebras, see [1].

Deﬁnition 2.8. Let Ω be a set with two maps s, t : Ω →Ncalled respectively source and

target. An Ω-algebra is a vector space Aendowed with linear maps ωA:A⊗s(ω)→A⊗t(ω)for all

ω∈Ω. A morphism of Ω-algebras f:A→Bis a linear map satisfying f⊗t(ω)◦ωA=ωB◦f⊗s(ω)

for all ω∈Ω. We denote the category of Ω-algebras and morphisms of Ω-algebras by Ω-Alg.

Furthermore, given a class Xof Ω-algebras we deﬁne the full sub-category of Ω-algebras in

Xby Ω-Alg(X).

By considering suitable choices of (Ω, s, t), algebras, coalgebras, Frobenius algebras and

many other algebraic structures can be viewed as Ω-algebras. For example, in case of Frobe-

nius algebras, we have Ω = {µ, η, ∆, ν}. Remark that in the deﬁnition of Ω-algebra, we do not

suppose any coherence conditions (such as associativity of coassociativity) for the considered

structure maps. Therefore, it is often useful not to consider the category of all Ω-algebras

for a given Ω, but rather to consider a suitable (full) subcategory of it, that is, to consider a

suitable class Xof Ω-algebras, for example associative algebras with Ω = {µ, η}.

Deﬁnition 2.9. Let Pbe a coalgebra, Aand Btwo k-vector spaces. We call a linear map

ψ:P⊗A→Bameasuring from Ato B. By hom-tensor relations, a measuring is equivalent

to a linear map ˆ

ψ:P→Hom(A, B). Consequently, we will use the following notation for the

image under ψ:

ψ(p⊗a) = p(a).

10 P. GROSSKOPF AND J. VERCRUYSSE

Furthermore, for any n∈N0, we then deﬁne and denote ψn:P⊗A⊗n→B⊗nby

ψn(p⊗a) = p(a) := p(1)(a1)⊗ · · · ⊗ p(1)(an)∈B⊗n,

for any a=a1⊗ · · · ⊗ an∈A⊗n. We also deﬁne ψ0=ǫP:P∼

=P⊗A⊗0→B⊗0∼

=k.

Suppose now that Aand Bare Ω-algebras. Then a measuring ψ:P⊗A→Bis said to be

ameasuring of Ω-algebras if and only if for all p∈P,ω∈Ω and a∈A⊗s(ω)we have (with

notation introduced above):

p(ωA(a)) = ωB(p(a)) ∈B⊗t(ω).

A measuring coalgebra is denoted as a pair (P, ψ).

If (P, ψ) and (P′, ψ′) are two measurings from Ato B, then a morphism of measurings is

a coalgebra map f:P→P′such that ψ=ψ′◦(f⊗A). Measurings from Ato Band their

morphisms form a category Meas(A, B).

Lemma 2.10. (1) A linear map f:A→Bis a morphism of Ω-algebras if and only if the

canonically associated map f:k⊗A∼

=A→Bmakes kinto a measuring coalgebra from

Ato B.

(2) Let A,Band Cbe Ω-algebras, (P, ψ)be a measuring of Ω-algebras from Ato Band

(P′, ψ′)be a measuring of Ω-algebras from Bto C. Then (P′⊗P, ψ′′ =ψ′◦(P′⊗ψ)) is

a a measuring of Ω-algebras from Ato C.

(3) Let Aand Bbe Ω-algebras, let (P, ψ)be a measuring of Ω-algebras from Ato Band

f:P′→Pa morphism of coalgebras. Then (P′, ψ′=ψ◦(f⊗A)) is again a measuring

of Ω-algebras.

(4) If (P, ψ)is a measuring of Ω-algebras from Ato B, then for any grouplike element g∈P,

the map ψ(g⊗ −) : A→Bis a morphism of Ω-algebras.

Proof. (1). This follows directly from the deﬁnitions, taking into account the unique trivial3

coalgebra structure on k.

(2). For any ω∈Ω, any p⊗q∈P⊗P′and any a∈A⊗s(ω)we ﬁnd

(p⊗q)(ωA(a)) = p(q(ωA(a))) = p(ωB(q(a))) = ωC(p(q(a))) = ωC((p⊗q)(a)),

where the ﬁrst and last equality follow from the deﬁnition of ψ′′ as in the statement (and the

notation introduced above for the image under the measuring map) and the two intermediate

equalities express the measuring property of (P′, ψ)′and (P, ψ).

(3). This follows again by direct computation. For any ω∈Ω, any p′∈P′and any a∈A⊗s(ω)

we ﬁnd

p′(ωA(a)) = f(p′)(ωA(a)) = ωB(f(p′)(a)) = ωB(p′(a)),

where the ﬁrst and last equality follow from the deﬁnition of ψ′as in the statement and the

intermediate equality expresses the measuring property of (P, ψ).

(4). This follows by combining (2) with (3), taking into account that a g∈Pis a grouplike

element if and only if f:k→P, f (1k) = gis a morphism of coalgebras. 

Remarks 2.11.Similarly to the fact that then any grouplike element gin a coalgebra P

measuring from Ato Binduces a morphism of Ω-algebras from Ato B(see (4) in the above

lemma), each g-primitive element xof P(i.e. ∆(x) = x⊗g+g⊗x) induces what one could

call a g-derivation from Ato B.

3By this we mean the unique k-coalgebra structure on khaving the identity map as counit.

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 11

By item (2) of the above Lemma, given any class Xof Ω-algebras, we can build a new

category M(X) out of it, whose objects are the elements of X, and where for any pair

of objects A, B ∈ X , we deﬁne HomM(X)(A, B) to be the collection of all (isomorphism

classes of) measurings (P, ψ) from Ato B, where composition is deﬁned as in (2). By item

(1) of the Lemma, the one-dimensional measurings then correspond exactly to the classical

morphisms. However, this might lead to some set-theoretical issues, since it is not clear that

the collection of all measuring coalgebras from Ato B(even up to isomorphism) would form

a set. Therefore, we will follow another approach and use universal measurings to deﬁne a

coalgebra-enriched category, which basically contains the same information as the category

described in this remark.

Consider two Ω-algebras Aand B. Let C(Hom(A, B)) be the cofree coalgebra over the

k-linear Hom-space Hom(A, B), and denote the associated (universal) projection by p:

C(Hom(A, B)) →Hom(A, B). Then for any coalgebra Pthat is measuring from Ato

B, we can consider the associated map P→Hom(A, B), p 7→ ψ(p⊗ −). By the univer-

sal property of the cofree coalgebra, we therefore obtain a unique morphism of coalgebras

u:P→C(Hom(A, B)) such that ψ=p◦u. On the other hand, we can consider a measuring

map

ψAB ;C(Hom(A, B)) ⊗Ap⊗A//Hom(A, B)⊗Aev //B ,

where ev is the usual evaluation map. In general however, ψAB is not a measuring of Ω-

algebras from Ato B. Nevertheless, in view of the above, we can consider the sum of all

all (ﬁnite dimensional) sub-coalgebras of C(Hom(A, B)), for which the restriction of ψAB is

a measuring of Ω-algebras. The coalgebra obtained in this way is measuring from Ato Bas

Ω-algebras, and is moreover a terminal object in the category Meas(A, B), because the map

uabove factors through this object. Explicitly, the measuring C(Hom(A, B)) is universal in

the following sense.

Deﬁnition 2.12. Let Aand Bbe Ω-algebras. The universal measuring coalgebra from

Ato B, denoted (C(A, B), ψBA ) is a measuring coalgebra from Ato Bsuch that for any

(other) measuring coalgebra (P, ψ) from Ato B, there exists a unique coalgebra morphism

θ:P→ C(A, B) making the following diagram commutative

P⊗Aψ//

∃!θ⊗A



C(A, B)⊗A

ψBA

An immediate consequence of this deﬁnition is the following useful observation.

Lemma 2.13. Consider two Ω-algebras Aand Band let (C(A, B), ψBA )the universal mea-

suring coalgebra from Ato B. Then for any pair of limear maps f, g :P→ C(A, B), where

fis a coalgebra morphism, we have that f=g(hence gis a coalgebra morphism as well) if

and only if

ψBA ◦(f⊗A) = ψBA ◦(g⊗A)

Proof. By Lemma 2.10 we know that ψBA ◦(f⊗A) is a measuring, hence ψBA ◦(g⊗A)

is a measuring as well. The statement now follows directly by the universal property of

C(A, B). 

12 P. GROSSKOPF AND J. VERCRUYSSE

The existence of a universal measuring coalgebra for any given pair of Ω-algebras follows

from the reasoning above. This was proven in [1, Theorem 3.10] and already in [18] in case of

usual algebras. More general situations about the existence of universal measuring coalgebras

in general monoidal categories have been treated in [9] and [2], and allow to transfer also the

results of the present paper to such a more general setting. However, for clarity we prefer to

restrict to the k-linear case here. As observed in [9], and already implicitly present in [18], the

existence of universal measuring coalgebras between all objects of a given category, allows

in fact to enrich this category over the category of coalgebras. Reformulated in terms of

semi-Hopf categories, this is summarized in the following result.

Proposition 2.14. Let Xbe any class of Ω-algebras, then we can build a semi-Hopf category

C(X)as follows. The objects of C(X)are just the elements of X. Furthermore, for any pair

of objects (Ω-algebras) A,Bin X, we put CB ,A := C(A, B)the universal measuring coalgebra

from Ato B.

More explicitly, given any triple of Ω-algebras A, B, C ∈ C, there are coalgebra morphisms

mA,B,C :CA,B ⊗ CB,C → CA,C

and

jA:k→ CA,A

satisfying the associativity and unitality conditions as stated in the ﬁrst two diagrams of

Deﬁnition 2.4. In particular, CA,A is a bialgebra for any A∈ X .

The category Ω-Alg(X), whose morphisms are Ω-algebra morphisms between objects in X,

can be recovered from C(X)by restricting the Hom-objects to the grouplike elements in the

universal measuring coalgebras.

Proof. The existence of the coalgebra morphisms mA,B ,C and jA, as well as the associativ-

ity and unitality conditions for them, follow from the universal property of the universal

measuring coalgebras. Indeed, by Lemma 2.10(2), it follows that

ψABC :C(B, A)⊗ C(C, B)⊗CId⊗ψBC //C(B, A)⊗BψAB //A

is a measuring of Ω-algebras from Cto A. By the universal property of (C(C, A), ψAC )

there exists a coalgebra map mA,B ,C as in the statement of the proposition. Similarly, the

isomorphism k⊗A∼

=A, which is a measuring of Ω-algebras in a trivial way, implies the

existence of the coalgebra map k→ CA,A by the universal property of (C(A, A), ψAA). The

associativity and counitality axioms then following again by the unicity in the universal

properties of the universal measuring coalgebras.

The last statement is a consequence of Lemma 2.10(1). Indeed, by the universal property

of C(A, B), there is a bijection between measurings k⊗B→Aand coalgebra morphisms

k→ C(A, B). By Lemma 2.10(1), the former is exactly a morphism of Ω algebras B→A,

where as the latter corresponds to a grouplike element in C(A, B). 

Remark 2.15.As a consequence of the above proposition we deduce that C(A, A) is a bial-

gebra, given it is an endo-hom object in a semi-Hopf category. Furthermore, it acts on A

and this action is universal in the sense that for any other bialgebra Bacting on Avia some

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 13

ψ:B⊗A→A, there exists a bialgebra morphism θ:B→ CAA such that

B⊗Aψ//

∃!θ⊗A



C(A, A)⊗A

ψAA

This was shown in [1], by proving that the coalgebra morphism θgiven by the universal mea-

suring condition already preserves the algebra structure. We therefore call CAA the universal

acting bialgebra of A.

2.3.2. Comeasuring algebras. The dual notion of a measuring coalgebra is a comeasuring

algebra. We recall the notion here, however its existence requires some ﬁniteness conditions.

Deﬁnition 2.16. Let Qbe an algebra, Aand Btwo vector spaces. We call a linear map

ρ:A→B⊗Q, denoted by ρ(a) = a[0] ⊗a[1] ∈B⊗Q, a comeasuring from Ato B. For any

n∈N0, we then deﬁne and denote ρn:A⊗n→B⊗n⊗Qby

ρn(a) = a[0] ⊗a[1] := a[0]

1⊗ · · · ⊗ a[0]

n⊗(a[1]

1···a[1]

n)∈B⊗n⊗Q.

for any a=a1⊗ · · · ⊗ an∈A⊗n. Moreover, we deﬁne ρ0:A⊗0∼

=k→B⊗0⊗Q∼

=Qto be

just the unit map of Q.

Suppose now that Aand Bare Ω-algebras. Then a comeasuring ρ:A→B⊗Qis said

to be a comeasuring of Ω-algebras if and only if for all ω∈Ω and a∈A⊗s(ω)we have (with

notation introduced above):

ρt(ω)(ωA(a)) = (ωB⊗Q)(ρs(ω)(a)) ∈B⊗t(ω)⊗Q,

A comeasuring algebra is denoted as a pair (Q, ρ). Comeasurings from Ato Bform in a

natural way a category denoted by Comeas(A, B).

A comeasuring algebra of Ω-algebras (A(A, B), ρBA ) from Ato Bis said to be universal

if for any other comeasuring algebra (Q, ρ), there exists a unique algebra morphism φ:

A(A, B)→Qsuch that

Aρ//

ρBA %%

❑

❑B⊗Q

∃!B⊗φ



B⊗ A(A, B)

A univeral comeasuring algebra is an initial object in Comeas(A, B).

In general, a universal comeasuring algebra does not always exist, see for example [1,

Example 4.20]. However, in the same paper, suﬃcient conditions for the existence are given,

under a suitable restriction on the ”support” of the comeasuring. In particular, if Bis ﬁnite

dimensional, then a universal comeasuring algebra of the form A(A, B) always exists, and

can be constructed by quotienting the free algebra over B∗⊗A∼

=Hom(B, A) by an ideal

generated by the comeasuring relations (see Section 5) for more details.

The following result can be proven in a dual way as Lemma 2.10.

Lemma 2.17. (1) A linear map f:A→Bis a morphism of Ω-algebras if and only if the

canonically associated map f:A→B∼

=B⊗kmakes kinto a comeasuring algebra from

Ato B.

14 P. GROSSKOPF AND J. VERCRUYSSE

(2) Let A,Band Cbe Ω-algebras, (Q, ρ)be a comeasuring of Ω-algebras from Ato Band

(Q′, ρ′)be a comeasuring of Ω-algebras from Bto C. Then (Q′⊗Q, ρ′′ = (ρ′⊗Q)ρ)is a

a comeasuring of Ω-algebras from Ato C.

(3) Let Aand Bbe Ω-algebras, let (Q, ρ)be a comeasuring of Ω-algebras from Ato Band

f:Q→Q′a morphism of algebras. Then (Q′, ρ′= (A⊗f)◦ρ)is again a comeasuring

of Ω-algebras.

(4) If (Q, ρ)is a comeasuring of Ω-algebras from Ato B, then for an algebra morphism

f:Q→k(i.e. a character), the map ˆ

f:A→B, ˆ

f(a) = a[0]f(a[1] )is a morphism of

Ω-algebras.

Proposition 2.18. Let Xbe any class of Ω-algebras, such that for any pair of Ω-algebras

A, B in X, the universal comeasuring algebra (A(A, B), ρBA )exists.

Then we can build a semi-Hopf opcategory A(X)as follows. The objects of A(X)are

the elements of X. Furthermore, for any pair of objects (Ω-algebras) A,Bin X, we put

AA,B =A(B, A), the universal comeasuring algebra from Bto A.

More explicitly, given any triple of Ω-algebras A, B, C ∈ C, there are algebra morphisms

dA,B,C :AA,C → AA,B ⊗ AB ,C

and

eA:AA,A →k

satisfying the coassociativity and counitality conditions as stated in the ﬁrst two diagrams of

Deﬁnition 2.6. In particular, AA,A is a bialgebra for any Ain X.

The category Ω-Alg(X), whose morphisms are Ω-algebra morphisms between objects in X,

can be recovered from A(X)by replacing each Hom-object A(B, A)by its set of characters

(i.e. the set of algebra morphisms HomAlg (A(A, B), k)).

Remark 2.19.Similarly to the universal acting bialgebra C(A, A) the endo-hom objects of

A(X) are bialgebras A(A, A) with a universal coaction. For any other bialgebra Bcoacting

on Avia some ρ:A→A⊗B, there exists a bialgebra morphism φ:B→ AAA such that

Aρ//

ρAA %%

❑

❑A⊗B

∃!A⊗φ



A⊗ A(A, A)

as shown in [1]. We therefore call CAA the universal coacting bialgebra of A. For

Remark 2.20.Recall that when the considered Ω-algebras are ﬁnite dimensional, then A(A, B)

is constructed as a quotient of the free algebra over Hom(B, A). Consequently, there is a

canonical map Hom(B, A)→ A(A, B). Dualizing this map, we obtain a map A(A, B)∗→

Hom(B, A)∗∼

=Hom(A, B) (recall that Aand Bare ﬁnite dimesnional). If we restrict then to

those elements in A(A, B)∗that are algebra morphisms (i.e. characters), then the correspond-

ing images are exactly the Ω-algebra maps from Ato B, and this is exactly the correspondence

described in the last statement of the above proposition.

3. The Hopf category of Frobenius algebras

In this section, we prove the main result of this paper. In general, the semi-Hopf category

of universal measurings from Proposition 2.14 is not Hopf. In this case, one can consider the

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 15

Hopf envelope as constructed in [8]. However, we will now show that in the case of Frobenius

algebras, there exists already an antipode on the category semi-Hopf category of universal

measurings of Frobenius algebras, turning it into a Hopf category. Similarly, the semi-Hopf

opcategory of universal comeasurings from Proposition 2.18 will be shown to be Hopf.

Let us ﬁrst make the deﬁnition of measurings between Frobenius algebras explicit.

Deﬁnition 3.1. Let Aand Bbe Frobenius algebras and denote their Casimir elements

respectively by e=e1⊗e2and f=f1⊗f2. Then a measuring from Ato Bis a coalgebra

(P, δ, ǫ) endowed with a linear map P⊗A→B,p⊗a7→ p(a), satisfying the following

conditions for all a, a′∈Aand p∈P:

p(aa′) = p(1)(a)p(2) (a′), p(1A) = ǫ(p)1B(4)

p(a)(1) ⊗p(a)(2) =p(1)(a(1))⊗p(2)(a(2) ), νB(p(a)) = ǫ(p)νA(a).(5)

Remark that applying the ﬁrst equality of (5) to 1A, and combining it with the second equality

of (4) implies

ǫ(p)f1⊗f2=p(1)(e1)⊗p(2)(e2),(6)

As usual, a grouplike element g∈Pacts as an automorphism of Frobenius algebras:

g(aa′) = g(a)g(a′), g(1A) = 1B(7)

g(a)(1) ⊗g(a)(2) =g(a(1))⊗g(a(2)), νB(g(a)) = νA(a).(8)

On the other hand, ﬁxing a grouplike gin P, a g-primitive element x∈Pacts as a g-bi-

derivation:

x(aa′) = x(a)g(a′) + g(a)x(a′), x(1A) = 0B(9)

x(a)(1) ⊗x(a)(2) =x(a(1))⊗g(a(2)) + g(a(1) )⊗x(a(2)), νB(x(a)) = 0.(10)

Theorem 3.2. Let Xbe any class of Frobenius algebras, and consider the associated semi-

Hopf category C(X)as in Proposition 2.14. Then C(X)is Hopf and its antipode is bijective.

In particular, the universal acting bialgebra CAA on a Frobenius algebra Ais a Hopf algebra

with bijective antipode.

Proof. Let Aand Bbe two Frobenius algebras in X. We want to construct the components

SBA :CBA → CAB of the antipode. Denote by P=Ccop

BA the co-opposite coalgebra of CBA

and the universal measuring ψBA :CBA ⊗A→B. We denote the Casimir element of Aby

e1⊗e2=E1⊗E2=ε1⊗ε2and the Casimir element of Bby f1⊗f2. We show that Pis

equipped with a measuring from Bto Aby means of the map ¯

ψAB :P⊗B→Agiven for

all p∈Pand b∈Bby:

ψAB (p⊗b) = ¯p(b) := e1νB(ψBA(p⊗e2)b) = e1νB(p(e2)b),

16 P. GROSSKOPF AND J. VERCRUYSSE

where we make use of the notation introduced in Deﬁnition 2.9 for measurings. Let us check

that this is indeed a measuring of Frobenius algebras, i.e. that conditions (4)-(5) are satisﬁed.

¯p(2)(b(1) )⊗¯p(1)(b(2)) = ¯p(2) (f1)⊗¯p(1) (f2b) = e1νB(p(2)(e2)f1)⊗E1νB(p(1)(E2)f2b)

=e1νB(p(2)(e2)ǫ(p(3))f1)⊗E1νB(p(1)(E2)f2b)

=e1νB(p(2)(e2)p(3)(ε1)) ⊗E1νB(p(1)(E2)p(4) (ε2)b)

=e1νB(p(2)(e2ε1)) ⊗E1νB(p(1)(E2)p(3)(ε2)b)

=ε1e1νB(p(2)(e2)) ⊗E1νB(p(1)(E2)p(3)(ε2)b)

=ε1e1νA(e2)⊗E1νB(p(1)(E2)p(2)(ε2)b)

=ε1⊗E1νB(p(E2ε2)b)

=ε1⊗ε2E1νB(p(E2)b)

=ε1⊗ε2¯p(b)

= (¯p(b))(1) ⊗(¯p(b))(2) ,

where we used counitality of Pin line 2, (6) in line 3, the ﬁrst equation of (4) in line 4, the

Casimir property for e1⊗e2in line 5, the second equation of (5)and the conunitality of Pin

line 6, again (4) and e1ν(e2) = 1Ain line 7, the Casimir property for E1⊗E2in line 8 and

ﬁnally the deﬁnition of ¯pin line 9. Further for the counits we have

νA(¯p(b)) = νA(e1)νB(p(e2)b) = νB(p(νA(e1)e2)b) = νB(p(1A)b) = νB(ǫ(p)1Bb) = ǫ(p)νB(b),

hence ¯

ψAB satisﬁes (5). We show that it satisﬁes (4).

¯p(2)(b) ¯p(1)(b′) = e1νB(p(2)(e2)b)E1νB(p(1)(E2)b′)

=e1E1νB(p(2)(e2)b)νB(p(1)(E2)b′)

=E1νB(p(2)(e2)b)νB(p(1)(E2e1)b′)

=E1νB(p(3)(e2)b)νB(p(1)(E2)p(2)(e1)b′)

=E1νP(p(2))νB(f2b)νB(p(1)(E2)f1b′)

=E1νB(f2)νB(ǫ(p(2))p(1)(E2)bf1b′)

=E1νB(f2)νB(p(E2)bf1b′)

=E1νB(p(E2)bνB(f2)f1b′)

=E1νB(p(E2)bb′)

= ¯p(bb′),

where we used the Casimir property for E1⊗E2in line 3, the ﬁrst equation of (4) in line

4, (6) in line 5, the Casimir property for f1⊗f2in line 6, counitality of Pin line 7 and

νB(f2)f1= 1Bin line 9. Finally for the units we have

¯p(1B) = e1νB(p(e2)1B) = ǫ(p)e1νA(e2) = ǫ(p)1A.

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 17

Hence, ¯

ψAB is a measuring of Frobenius algebras and by universal property of CAB , there

exists a unique map SBA :P=Ccop

BA → CAB , such that

P⊗B¯

ψAB //

∃!SBA⊗B



CAB ⊗B

ψAB

✈

We show that the morphisms SAB satisfy the antipode conditions for Hopf categories as stated

in Deﬁnition 2.5. Let p∈ CBA , a ∈A, b ∈Band denote the coalgebra structure on CBA by

δBA and ǫBA . Then

ψAB (p(1) ⊗ψBA(p(2) ⊗a)) = e1νB(p(1)(e2)p(2)(a)) = e1νB(p(e2a))

=ae1νB(p(e2)) = ae1ǫBA (p)νA(e2) = ǫBA (p)aνA(e2)e1

=ǫBA(p)IdA(a)

and conversely

ψBA(p(1) ⊗¯

ψAB (p(2) ⊗b)) = p(1)(e1)νB(p(2)(e2)b) = ǫBA (p)f1νB(f2b) = ǫBA (p)IdB(b).

This means

ψAA(mABA ⊗A)(SBA ⊗ CBA ⊗A)(δB A ⊗A) = ψAB (CAB ⊗ψBA )(SBA ⊗ CB A ⊗A)(δBA ⊗A)

=ψAB (SBA ⊗B)(CBA ⊗ψB A)(δBA ⊗A)

=¯

ψAB (CBA ⊗ψBA )(δBA ⊗A)

=ψAA(jAǫBA ⊗A),

and similarly ψBB (mBAB ⊗B)(CB A ⊗SBA ⊗B)(δBA ⊗B) = ψBB (jBǫBA ⊗B). By Lemma 2.13,

we then obtain the antipode condition.

We now show that Sis bijective. Denote P=Ccop

AB and the Casimir element of Bby

f1⊗f2=F1⊗F2. Deﬁne ψ′

BA :P⊗A→Bby

ψ′

BA(p⊗a) := f2νA(aψAB (p⊗f1)) = f2νA(ap(f1)).

We will show that this is a measuring of Frobenius algebras. Firstly,

ψ′

BA (p(2) ⊗a(1))⊗ψ′

BA (p(1) ⊗a(2)) = f2νA(a(1)p(2)(f1)) ⊗F2νA(a(2) p(1)(F1))

=f2νA(ae1p(2)(f1)) ⊗F2νA(e2p(1) (F1))

=f2νA(ap(1)(F1)e1p(2)(f1)) ⊗F2νA(e2)

=f2νA(ap(1)(F1)p(2)(f1)) ⊗F2

=f2νA(ap(F1f1)) ⊗F2

=f2F1νA(ap(f1)) ⊗F2

=ψ′

BA (p⊗a)(1) ⊗ψ′

BA (p⊗a)(2),

where we used the characterisation of the comultiplication in Avia the Casimir element e1⊗e2

in line 2, the Casimir property of e1⊗e2in line 3, νA(e2)e1= 1Ain line 4, the ﬁrst equation

of (4) in line 5, the Casimir property of f1⊗f2in line 6 and the characterisation of the

comultiplication in Bvia the Casimir element F1⊗F2in line 7. Further we show

νB(ψ′

BA (p⊗a)) = νB(f2)νA(ap(f1)) = νA(ap(1B)) = νA(aǫ(p)1A) = ǫ(p)νA(a),

18 P. GROSSKOPF AND J. VERCRUYSSE

where we used νB(f2)f1= 1Band the second equation in (5). This shows that ψ′is a

measuring of coalgebras. Moreover,

ψ′

BA (p(2) ⊗a)ψ′

BA (p(1) ⊗a′) = f2νA(ap(2)(f1))F2νA(a′p(1 )(F1))

=f2νA(ap(2)(F2f1))νA(a′p(1)(F1))

=f2νA(a(p(f1))(2))νA(a′(p(f1))(1))

=f2νA(ae2p(f1))νA(a′e1)

=f2νA(ae2a′p(f1))νA(e1)

=f2νA(aa′p(f1))

=ψ′

BA (p⊗a′a),

where we used the Casimir property of f1⊗f2in line 2, the characterisation of the comul-

tiplication in Bvia the Casimir element F1⊗F2and the ﬁrst equation of (5) in line 3, the

characterisation of the comultiplication in Avia the Casimir element e1⊗e2in line 4, the

Casimir property of e1⊗e2in line 5 and νA(e1)e2= 1Ain line 6. Finally the second equation

in (5) and νB(f1)f2= 1Byield

ψ′

BA (p⊗1A) = f2νB(1Ap(f1)) = ǫ(p)f2νB(f1) = ǫ(p)1B,

therefore ψ′

BA is a measuring of Frobenius algebras. Hence there exists a map S′

AB :Ccop

AB →

CBA with

P⊗Aψ′

BA //

∃!S′

AB ⊗A



CBA ⊗A

ψBA

✈

Let p∈ CBA, a ∈A, then:

ψBA(S′

ABSBA ⊗A)(p⊗a) = ψ′

BA (SBA (p)⊗a)

=f2νA(aψAB (SBA ⊗B)(p⊗f1)) = f2νA(a¯

ψAB (p⊗f1))

=f2νA(ae1νB(p(e2)f1)) = f2νB(p(e2a)f1)νA(e1)

=f2νB(p(νA(e1)e2a)f2) = f2νB(p(a)f1)

=f2p(a)νB(f1)

=p(a) = ψBA(p⊗a)

and by universal property of CBA we ﬁnd S′

AB SBA =CBA . Analogously for p∈ CAB , b ∈B,

ψAB (SBAS′

AB ⊗B)(p⊗b) = ¯

ψAB (S′

AB(p)⊗b)

=e1νB(ψBA(S′

AB ⊗A)(p⊗e2)b) = e1νB(ψ′

BA(p⊗e2)b)

=e1νB(f2νA(e2p(f1))b) = e1νB(f2b)νA(e2p(f1))

=p(f1)e1νB(f2b)νA(e2) = p(f1)νB(f2b)

=p(bf1)νB(f2)

=p(b) = ψAB (p⊗b).

Therefore, SBAS′

AB =CAB by universal property of CAB.

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 19

Corollary 3.3. For A, B symmetric Frobenius algebras S−1

AB =SBA. Particularly CAA is a

Hopf algebra with involutive antipode.

Proof. For symmetric Frobenius algebras we have that ν(ab) = ν(ba) and dually e1⊗e2=

e2⊗e1. Hence the measuring ψ′

AB constructed in the proof of Proposition 3.2 is the same as

ψAB hence by universal property S−1

BA =SAB .

Let us now state the analogous result for comeasurings between Frobenius algebras.

Deﬁnition 3.4. Let Aand Bbe Frobenius algebras and denote their Casimir elements

respectively by e=e1⊗e2and f=f1⊗f2. Then a comeasuring from Ato Bis an algebra

Qendowed with a linear map A→B⊗Q, denoted ρ(a) = a[0] ⊗a[1], satisfying the following

conditions for all a∈A:

(aa′)[0] ⊗(aa′)[1] =a[0]a′[0] ⊗a[1]a′[1] (1A)[0] ⊗(1A)[1] = 1B⊗1Q(11)

a(1)[0] ⊗a(2)[0] ⊗a(1)[1]a(2)[1] =a[0](1) ⊗a[0](2) ⊗a[1], νA(a)1Q=νB(a[0] )a[1].(12)

Remark that applying the ﬁrst equality of (12) to 1Aand combining it with the second

equality of (11) leads to

e1[0] ⊗e2[0] ⊗e1[1]e2[1] =f1⊗f2⊗1Q.(13)

Theorem 3.5. Let Xbe any class of Frobenius algebras, and consider the associated semi-

Hopf opcategory A(X)as in Proposition 2.18. Then A(X)is Hopf and its antipode is bijective.

In particular, the universal coacting bialgebra AAA on a Frobenius algebra Ais a Hopf

algebra with bijective antipode.

For A, B symmetric Frobenius algebras we have that the antipode of the Hopf category A(X)

satisﬁes S−1

AB =SBA. Particularly, AAA is a Hopf algebra with involutive antipode if Ais a

symmetric Frobenius algebra.

Proof. We want to construct the components SAB :AAB →Aop

BA of the antipode for the

semi-Hopf opcategory A(X). Denote by Q=Aop

BA the opposite algebra of ABA. We show

that this algebra is equipped with a comeasuring ¯ρAB :B→A⊗Q, which for b∈Bdeﬁned

¯ρAB (b) := e1νB(e2[0]b)⊗e2[1] ∈A⊗Q

where ρBA(e2) = e2[0] ⊗e2[1] ∈B⊗Q.

Let us check that this is indeed a comeasuring. We denote the Casimir element of Aby

e1⊗e2=E1⊗E2=ε1⊗ε2and the Casimir element of Bby f1⊗f2. First we show that

20 P. GROSSKOPF AND J. VERCRUYSSE

¯ρAB :B→A⊗Qis an algebra morphism. Let b, b′∈B, then

¯ρAB (b)¯ρAB (b′) = (νB(E1[0]b)E2⊗E1[1])(νB(e1[0] b′)e2⊗e1[1])

=E2e2νB(E1[0]b)νB(e1[0]b′)⊗e1[1]E1[1]

=E2νB(e2[0]b)νB(E1[0]e1[0]b′)⊗E1[1] e1[1]e2[1]

=E2νB((1A))[0](2)b)νB(E1[0] (1A)[0](1)b′)⊗E1[1] 1[1]

=E2νB(1(2)

Bb)νB(E1[0]1(1)

Bb′)⊗E1[1]1Q

=E2νB(f2b)νB(E1[0]f1b′)⊗E1[1]

=E2νB(f2)νB(E1[0]bf 1b′)⊗E1[1]

=E2νB(E1[0]bνB(f2)f1b′)⊗E1[1]

=E2νB(E1[0]bb′)⊗E1[1]

= ¯ρAB (bb′)

where we build the product in the in B⊗ Aop

BA in line 2, we used the Casimir property for

e1⊗e2and the ﬁrst equation in (11) in line 3, we used (13) in line 4, the Casimir property

for f1⊗f2in line 5, νB(f2)f1= 1Bin line 7 and ﬁnally the deﬁnition of ¯ρAB in line 8. The

unitality follows from (12) applied on e2and νA(e1)e2= 1A:

¯ρAB (1B) = e2⊗νB(e1[0])e1[1] =νB(e1)e2⊗1Q= 1A⊗1Q

We show the comeasuring conditions for the coalgebras Aand Bfor b∈B. Denote the

multiplication in Qby mQ:Q⊗Q→Q, then:

(A⊗A⊗µQ)(A⊗σ⊗Q)(¯ρAB ⊗¯ρAB )∆B(b)

= (A⊗A⊗µQ)(A⊗σ⊗Q)(¯ρAB ⊗¯ρAB )(f1⊗f2b)

=νB(e1[0]f1)e2⊗νB(E1[0]f2b)E2⊗E1[1]e1[1]

=νB(e1[0]ε2[0])e2⊗νB(E1[0]ε2[0]b)E2⊗E1[1]e1[1] ε1[1]ε2[1]

=νB((e1ε1)[0])e2⊗νB(E1[0] ε2[0]b)E2⊗E1[1](e1ε1)[1]ε2[1]

=e2⊗νB(E1[0]ε2[0]b)E2⊗E1[1]νB((e1ε1)[0] )(e1ε1)[1]ε2[1]

=e2⊗νB(E1[0]ε2[0]b)E2⊗E1[1]νA(e1ε1)ε2[1]

=e2ε2⊗νB(E1[0]e1[0]b)E2⊗E1[1]νA(ε1)e1[1]

=e2νA(ε1)ε2⊗νB((E1e1)[0]b)E2⊗(E1e1)[1]

=e2⊗νB((E1e1)[0]b)E2⊗(E1e1)[1]

=e2E1⊗νB(e1[0]b)E2⊗e1[1]

=e2⊗νB(e1[0]b)e3⊗e1[1] = (∆A⊗Q)¯ρAB (b),

where we used the characterisation of comultiplication in Bvia the Casimir element f1⊗f2

in line 2, (13) in line 3, the ﬁrst equation of (11) in line 4, (12) in line 6, the Casimir property

of e1⊗e2in line 7, νA(e2)e1= 1Ain line 8 and the characterisation of comultiplication in A

via the Casimir element ε1⊗ε2in line 9. Further, using (12), (13) and νB(f1)f2= 1Bwe

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 21

have:

(νA⊗Q)¯ρAB (b) = νB(e1[0]b)νA(e2)1Ae1[1] =νB(e1[0] b)νB(e2[0])e2[1]e1[1]

=νB(f1b)νB(f2)1Q=νB(νB(f2)f1b)1Q

=νB(b)1Q

Hence ¯ρAB is a comeasuring of Frobenius algebras. By universal property we have an algebra

homomorphism SAB :AAB → Aop

BA =Q, such that

B¯ρAB

ρAB $$

■

■A⊗Q

A⊗ AAB

∃!A⊗SAB

commutes. So we show these morphisms satisfy the antipode condition of Hopf opcategories.

Denote the algebra structure on ABA by µBA and ηBA . Let b∈Band calculate

(B⊗µBA)(ρBA ⊗ ABA) ¯ρAB(b) = νB(e2[0]b)e1[0] ⊗e1[1]e2[1] =νB(f2b)f1⊗ηAB =IdB(b)⊗ηAB .

This shows, that:

(B⊗µBA)(B⊗ ABA ⊗SAB )(B⊗dBAB )ρBB

= (B⊗µBA)(B⊗ ABA ⊗SAB )(ρBA ⊗ AAB )ρAB

= (B⊗µBA)(ρBA ⊗ ABA) ¯ρAB

= (B⊗ηABeB)ρBB

Hence by universal property of ABB we obtain µBA(ABA ⊗SAB )dBAB =ηBAeB.Further

(A⊗µBA)(¯ρAB ⊗ ABA)ρBA(a) = e1νB(e2[0] a(0))⊗e2[1]a(1) =νB((e2a)[0])e1⊗(e2a)[1]

=e1⊗νA(e2a)ηBA =IdA(a)⊗ηBA ,

which proves

(A⊗µBA)(A⊗SAB ⊗ ABA )(A⊗dABA )ρAA

= (A⊗µBA)(A⊗SAB ⊗ ABA )(ρAB ⊗ ABA )ρBA

= (A⊗µBA)(¯ρAB ⊗ ABA)ρBA

= (A⊗ηBAeA)ρAA.

Therefore by universal property of AAA we have µBA(SAB ⊗ ABA )dABA =ηAB eA.

To show that SAB is invertible, denote Q:= Aop

BA and deﬁne ρ′AB :B→A⊗Qby

ρ′AB (b) := e2νB(be1[0])⊗e1[1],

22 P. GROSSKOPF AND J. VERCRUYSSE

where ρBA(e1) = e1[0] ⊗e1[1]. We show ρ′is a comeasuring from Bto A. Let therefore b, b′∈B

and calculate

ρ′AB (b)ρ′AB (b′) = e2νB(be1[0])E2νB(b′E1[0])⊗E1[1] e1[1]

=e2νB(b(E2e1)[0])νB(b′E1[0])⊗E1[1](E2e1)[1]

=e2νB(bE2[0] e1[0])νB(b′E1[0])⊗E1[1] E2[1]e1[1]

=e2νB(bf2e1[0])νB(b′f1)⊗e1[1]

=e2νB(bf2b′e1[0])νB(f1)⊗e1[1]

=e2νB(bb′e1[0])⊗e1[1]

=ρ′AB (bb′),

where we build the product in A⊗ Aop

BA in line 1, we use the Casimir property of e1⊗e2in

line 2, the ﬁrst equation of (12) in line 3, (13) in line 4, the Casimir property of f1⊗f2in

line 5 and ﬁnally νB(f1)f2= 1Bin line 6. For the units we have

ρ′AB (1B) = e2⊗νB(e1[0])e1[1] =νB(e1)e2⊗1Q= 1A⊗1Q,

using (12) for a=e1. We show that ρ′AB satisﬁes (12) for b∈Bby

(A⊗A⊗µQ)(A⊗σ⊗Q)(ρ′AB ⊗ρ′AB )dB(b)

=e2νB(f1e1[0])⊗E2νB(f2bE1[0])⊗E1[1]e1[1]

=e2νB(bE1[0] f1e1[0])⊗E2νB(f2)⊗E1[1]e1[1]

=e2νB(bE1[0] e1[0])⊗E2⊗E1[1]e1[1]

=e2νB(b(E1e1)[0])⊗E2⊗(E1e1)[1]

=νB(be1[0])e2E1⊗E2⊗e1[1]

= (∆A⊗Q)ρ′AB (b),

where we used that multiplication µQin Qis the opposite of the multiplication in ABA and

the characterisation of the comultiplication in Bby the Casimir element f1⊗f2in line 2,

the Casimir property for f1⊗f2in line 3, νB(f2)f1= 1Bin line 4, the ﬁrst equation in

(12) in line 5, Casimir property for e1⊗e2in line 6 and ﬁnally the characterisation of the

comultiplication in Aby the Casimir element E1⊗E2in line 7. For the counits we show

(νA⊗Q)ρ′AB (b) = νB(be1[0])e1[1] νA(e2) = νB(be1[0])νA(e2[0])e1[1] e2[1]

=νB(bf1)νB(f2)1Q=νB(b)1Q,

using (12) for a=e2and (13). Hence, ρ′AB is a comeasuring of Frobenius algebras and

therefore by universal property there exists an algebra homomorphism S′

AB :AAB → Aop

BA,

such that

Bρ′AB

ρAB $$

■

■A⊗Q

A⊗ AAB

∃!A⊗S′

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 23

commutes. Finally S′

AB is inverse to SBA, since for all b∈B

(A⊗SBAS′

AB )ρAB(b) = (A⊗SBA )ρ′AB (b)

= (A⊗SBA)(e2νB(be1[0] ⊗e1[1])

=e2νB(be1[0])⊗SBA(e1[1] )

=e2νB(bνA(f2[0]e1)f1)) ⊗f2[1]

=νA(e1)e2f2[0]νB(bf 1)⊗f2[1]

=νB(bf1)ρAB (f2) = νB(b(1) )ρAB (b(2))

=ρAB (b),

where we used (A⊗S′

AB )ρAB =ρ′AB in line 1 and (A⊗SBA )ρBA = ¯ρBA in line 4, furthermore

the Casimir property for e1⊗e2in line 5, νA(e1)e2= 1Aand the characterisation of the

comultiplication in Bby the Casimir element f1⊗f2in line 6 and ﬁnally counitality in B.

Conversely, for all a∈A

(B⊗S′

AB SBA)ρBA (a) = (B⊗S′

AB)¯ρBA (a)

= (B⊗S′

BA)(f1νA(f2[0]a)⊗f2[1]

=f1νA(f2[0]a)⊗S′

BA(f2[1])

=f1νA(νB(f2e1[0])e2a)⊗e1[1]

=νB(f2)e1[0]f1νA(e2a)⊗e1[1]

=νA(e2a)ρBA(e1) = νA(a(2))ρBA(a(1) )

=ρBA(a),

where we used (B⊗S′

BA )ρAB =ρ′AB and (A⊗SBA )ρBA = ¯ρBA, furthermore the Casimir

property for f1⊗f2in line 5, νA(e1)e2= 1Ain line 6. Hence by universal property of AAB

we ﬁnd that SBAS′

AB =AAB and similarly S′

ABSBA =ABA. Hence Sis bijective. 

4. Dualities

4.1. Sweedler dual of a semi-Hopf opcategory. Let Abe an algebra. Recall that the

Sweedler dual A◦of Ais the set of all linear functionals f∈A∗such that Kerfcontains a

two-sided ideal of ﬁnite codimension. Then A◦is a coalgebra in a natural way, where

∆(f) = f(1) ⊗f(2)

if and only if

f(ab) = f(1)(a)f(2)(b)

for all a, b ∈A. Moreover (−)◦:Alg →Coalg is a contravariant functor which is adjoint to

the contravariant functor (−)∗:Coalg →Alg, that sends a coalgebra Cto the convolution

algebra C∗. Then we have the following immediate result.

Lemma 4.1. Let Abe a semi-Hopf opcategory. Then there is a semi-Hopf category A◦

deﬁned as follows: objects of A◦are the same as objects of A, and for each pair of objects

x, y ∈A0= (A◦)0, we have

(A◦)xy = (Axy)◦

Moreover, if Ais a Hopf opcategory, then A◦is a Hopf category.

24 P. GROSSKOPF AND J. VERCRUYSSE

Proof. We already know that A◦

xy is a coalgebra for each pair of objects x, y. Let us show

that it also is a k-linear category. To this end, take any triple of objects x, y, z and deﬁne

mxyz :A◦

xy ⊗A◦

yz →A◦

as follows. For f∈A◦

xy and g∈A◦

yz , we have a functional (f⊗g)◦dxyz ∈A∗

xz. Now let Iand

Jbe ideals of ﬁnite codimension respectively in Kerfand Kerg. Let K⊂Axz be the inverse

image of I⊗Ayz +Axy ⊗Junder dxyz . Then Kis an ideal of ﬁnite codimension contained in

the kernel of (f⊗g)◦dxyz, hence the latter is in A◦

xz, which shows that mxyz is well-deﬁned

by mxyz(f⊗g) = (f⊗g)◦dxyz. Furthermore, mxyz is a coalgebra map as dxyz is an algebra

map. Finally, we deﬁne jx(1k) = ex, which clearly is in A◦

xx being an algebra map onto k.

If Sis an antipode for Athen their dual morphisms form an antipode for A◦.

Remark 4.2.The construction of Lemma 4.1 is not functorial. Indeed, if f:A→Bis a

morphism of semi-Hopf opcategories, then fsends objects of Ato objects of B, and for each

pair of objects x, y ∈A◦, we have that fxy :Axy →Bfxf y is an algebra morphism (satisfying

coherence). When we turn to Sweedler duals, we ﬁnd that each fxy induces a coalgebra

morphism f∗

xy :B◦

fxf y →A◦

xy, however on the level of objects, we still go from objects in A

(or A◦) to objects in B(or B◦).

We will now combine the above result with [1, Thm. 3.20, Thm. 4.14], which we state here

in the ﬁnite dimensional case (which is suﬃcient for our needs, since Frobenius algebras are

ﬁnite dimensional) and reformulate in terms of semi-Hopf categories.

Theorem 4.3. (1) Let Aand Bbe ﬁnite dimensional Ω-algebras and (ABA, ρBA)their

universal comeasring algebra. Let ˜

ψAB :A◦

BA ⊗A→Bthe measuring deﬁned by

ψBA(f⊗a) := (B⊗f)ρBA (a) = fa(1)a(0) ,∀f∈ A◦

BA, a ∈A.

Then there exist a coalgebra isomorphism θBA :A◦

BA → CBA such that

CBA ⊗AψBA //B

A◦

BA ⊗A

∃θBA ⊗A

ψBA

✈

commutes.

(2) If Xa class of ﬁnite dimensional Ω-algebras, C(X)the semi-Hopf category of univer-

sal measuring coalgebras as constructed in Proposition 2.14 and A(X)the semi-Hopf

opcategory of universal comeasuring algebras as constructed in Proposition 2.18, then

we have an isomorphism of semi-Hopf opcategories

θ:A(X)◦∼

=//C(X)

where A(X)◦is the Sweedler dual of A(X)as in Lemma 4.1, being the identity on

objects.

(3) If Xis a class of Frobenius algebras, then the isomorphism θ:A(X)◦∼

=//C(X)as

in part (2) is an isomorphism of Hopf categories. In particular, θAA :A◦

AA → CAA is

a Hopf algebra isomorphism.

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 25

Proof. Part (1) follows directly from [1, Thm. 3.20., Thm. 4.14].

By Lemma 4.1, we know that A(X)◦is indeed a semi-Hopf category, and Part (1) deﬁnes a

family of coalgebra morphisms θAB . We prove now the compatibility with the multiplicative

structure. Take A, B, C in X,f∈ A◦

AB,g∈ A◦

BC and c∈C. Then we ﬁnd

ψAC (θAC mA◦

ABC ⊗C)(f⊗g⊗c) = ˜

ψAC (mA◦

ABC (f⊗g)⊗c)

= (mA◦

ABC (f⊗g)⊗A)ρAC (c)

= (f⊗g⊗A)(dABC ⊗A)ρAC (c)

and

ψAC (mC

ABC ⊗C)(θAB ⊗θBC ⊗C)(f⊗g⊗c) = ψAB (CAB ⊗ψB C )(θAB(f)⊗θB C (g)⊗c)

=˜

ψAB (A◦

AB ⊗˜

ψBC )(f⊗g⊗c)

= (f⊗g⊗A)(ρAB ⊗ ABC )ρBC (c).

By deﬁnition of the opcategory structure of A, we have that

(ρAB ⊗ ABC )ρBC = (dAB C ⊗A)ρAC

and therefore the above computations imply that

ψAC (θAC mA◦

ABC ⊗ C) = ψAC (mC

ABC ⊗C)(θAB ⊗θBC ⊗C).

The universal property of the comeasuring (CAC , ψAC ) (see Lemma 2.13) then implies that

indeed

θAC mA◦

ABC =mC

ABC ◦(θAB ⊗θBC ).

In a similar way, one veriﬁes that θis compatible with the units. Indeed, by deﬁnition of the

V-category structure of A◦and the opcategory structure of Awe ﬁnd

ψAA(θAA jA◦

A⊗A) = ˜

ψAA(jA◦

A⊗A)) = ˜

ψAA(eA⊗A) = (A⊗eA)◦ρAA =A.

On the other hand, the deﬁnition of the V-category structure of Cguarantees that

ψAA(jC

A⊗A) = A

Again by Lemma 2.13 we then ﬁnd that θAAjA◦

A=jC

Part (3) follows directly from part (2) and the fact that a semi-Hopf category morphism

between Hopf categories is automatically preserving the antipodes. 

4.2. Duality for Frobenius algebras and the antipode of their universal Hopf cate-

gory. Recall that a Frobenius algebra Ais always ﬁnite dimensional and a dual base is given

by e1⊗ν(e2−)∈A⊗A∗(where as usual, we denote by e1⊗e2the Casimir element of Aand

by νits faithful functional or counit). Consequently, A∗is again a Frobenius algebra whose

structure is given by

(f·g)(a) := f(a(2))g(a(1) ) = f(e2a)g(e1) = f(e2)g(ae1),1A∗:= νA,

∆(f)(a⊗b) := f(1)(a)f(2)(b) = f(ba), νA∗(f) := f(1),

for all f, g ∈A∗, a, b ∈A. Remark that the Casimir element of A∗is given by

ν(e1−)⊗ν(e2−)∈A∗⊗A∗.

Moreover, the maps ι:A→A∗, ι(a)(b) = ν(ab)) and ι−1:A∗→A, f 7→ f(e1)e2are mutual

inverse isomorphisms of Frobenius algebras.

26 P. GROSSKOPF AND J. VERCRUYSSE

Lemma 4.4. Let A, B be Frobenius algebras, with duals A∗, B∗. Then Meas(A, B)and

Meas(A∗, B∗)are isomorphic as categories. Dually Comeas(A, B)and Comeas(A∗, B∗)are

isomorphic as categories.

Proof. Given a measuring ψ:P⊗A→Bwe can construct a measuring ψ∗:P⊗A∗→B∗

ψ∗(p⊗f) := ιB(ψ(p⊗ι−1

A(f))).

Since this is a composition of three coalgebra morphisms, it is a coalgebra morphism itself.

Moreover, ψ∗is a measuring of algebras since

ψ∗(p⊗fg) = ιB(ψ(p⊗ι−1

A(f)ι−1

A(g)) = ιB(p(1)(ι−1

A(f))p(2)(ι−1

A(g)))

=ιB(p(1)(ι−1

A(f)))ιB(p(2)(ι−1

A(g))) = ψ∗(p(1) ⊗f)ψ∗(p(2) ⊗g)

and

ψ∗(p⊗1A∗) = ιB(ψ(p⊗1A)) = ǫ(p)ιB(1B) = ǫ(p)1B∗.

Furthermore, let ψ′:P′⊗A→Bbe another measuring and φ:P→P′a coalgebra morphism

such that ψ=ψ′(φ⊗A) then

ψ′∗(φ⊗A) = ιBψ′(φ⊗ι−1

A) = ιBψ(P⊗ι−1

A) = ψ∗.

The assignment (P, ψ)7→ (P, ψ∗) and φ7→ φforms a functor Meas(A, B)→Meas(A∗, B∗).

Conversely, given a measuring ψ∗:P⊗A∗→B∗we can construct a measuring ψ:P⊗A→

Bby ψ(p⊗a) := ι−1

B(ψ∗(p⊗ιA(a))). We can again lift this to a functor Meas(A∗, B∗)→

Meas(A, B) by setting it to be the identity on morphisms. These two functors are inverse to

each other, hence Meas(A, B) and Meas(A∗, B∗) are isomorphic as categories.

Similarly any comeasuring ρ:A→B⊗Qgives rise to a comeasuring ρ∗:= (ιB⊗Q)ρι−1

A∗→B∗⊗Qand the assignment is bijective and functorial. 

Hence we obtain the following theorem.

Proposition 4.5. (1) Let A, B be Frobenius algebras and A∗, B∗their dual Frobenius alge-

bras. Then there is a unique coalgebra isomorphism

γAB :CAB

∼

=//CA∗B∗,

rendering the following diagram commutative

CAB ⊗B

ψAB



CAB ⊗ιB//CAB ⊗B∗γAB⊗B∗

//CA∗B∗⊗B∗

ψA∗B∗



AιA//A∗.

(2) Let then Xbe any class of Frobenius algebras and denote by X∗the class their dual

Frobenius algebras. Then the isomorphisms γAB as deﬁned in part (1), for all A, B ∈ X ,

form an isomorphism of Hopf categories

γ:C(X)→ C(X∗),

sending objects Ato their duals A∗. In particular,

γAA :CAA ≃ CA∗A∗

is a Hopf algebra isomorphism for any Frobenius algebra A.

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 27

Proof. Since the categories Meas(A, B) and Meas(A∗, B∗) are isomorphic by Lemma 4.4, so

are their ﬁnal objects. This induces the coalgebra isomorphism γAB as in part (1). For part

(2), we only show the compatibility with the multiplication and unit. We can compute

ψA∗C∗(mA∗B∗C∗⊗C∗)(γAB ⊗γBC ⊗C∗)

=ψA∗B∗(CA∗B∗⊗ψB∗C∗)(γAB ⊗γBC

=ψ∗

A∗B∗(CAB ⊗ψ∗

B∗C∗)

=ιAψAB (CAB ⊗ι−1

BιB)(CAB ⊗ψBC )(CAB ⊗ CBC ⊗ι−1

=ιAψAC (mABC ⊗C)(CAB ⊗ CBC ⊗ι−1

=ψ∗

A∗C∗(mABC ⊗C∗) = ψA∗C∗(γAC mABC ⊗C∗),

and

ψA∗A∗(γAAjA⊗A∗) = ψ∗

A∗A∗(jA⊗A∗) = ιAψAA(jA⊗ι−1

A) = ιAι−1

A=A∗=ψA∗A∗(jA∗⊗A∗)

by universal property of CA∗C∗and CA∗A∗we ﬁnd that γis a morphism of semi-Hopf categories.



Recall from [1, Thm. 4.16.] the existence of an ismorphism between the opposite universal

acting bialgebra of any Ω-algebra and the universal acting bialgebra of its dual. One can

show a similar duality for general measurings. We formulate this result here in the case of

Frobenius algebras.

Proposition 4.6. Let A, B be Frobenius algebras and A∗, B∗their duals. Denote their uni-

versal measuring by ψB∗A∗:CB∗A∗⊗A∗→B∗. Deﬁne ˆ

ψAB :Ccop

B∗A∗⊗B→Aby

ψAB (p⊗b) := X

ψB∗A∗(p⊗fi)(b)ei,∀p∈ CB∗A∗, b ∈B,

where {(ei, fi)} ⊂ A×A∗is a ﬁnite dual base for A. Then this induces a unique coalgebra

isomorphism πAB :CB∗A∗→ Ccop

AB such that

CB∗A∗⊗Bˆ

ψAB //

πAB⊗B



CAB ⊗B

ψAB

(14)

commutes. Furthermore, for Frobenius algebras A, B, C with duals A∗, B∗, C∗we have

πCA mA∗B∗C∗=mCBA σ(πBA ⊗πC B )

and πAAjA∗=jA.

In other words, if Xis a class of Frobenius algebras, then there is an isomorphism of Hopf

categories π:C(X∗)→ C(X)op,cop deﬁned component-wise as above. In particular,

πAA :CA∗A∗≃ Ccop,op

A,A

is a Hopf algebra isomorphism.

28 P. GROSSKOPF AND J. VERCRUYSSE

Proof. To check the conditions of a measuring, let b, c ∈B, p ∈ CB∗A∗and f∈A∗. Then we

ﬁnd

f(ˆ

ψAB(p⊗bc)) = ψB∗A∗(p⊗f)(bc) = ψB∗A∗(p⊗f)(1)(c)ψB∗A∗(p⊗f)(2)(b)

=ψB∗A∗(p(1) ⊗f(1))(c)ψB∗A∗(p(2) ⊗f(2))(b)

=f(1)(ˆ

ψAB (p(1) ⊗c))f(2)(ˆ

ψAB (p(2) ⊗b))

=f(ˆ

ψAB (p(2) ⊗b)ˆ

ψAB (p(1) ⊗c)),

where we used the deﬁnition of the comultiplication in B∗in line 1, (5) for ψB∗A∗in line 2

and the deﬁnition of the comultiplication in A∗in line 4. Moreover,

f(ˆ

ψAB (p⊗1B)) = ψB∗A∗(p⊗f)(1B) = νB∗(ψB∗A∗(p⊗f))

=ǫB∗A∗(p)νA∗(f) = ǫB∗A∗(p)f(1A),

where we used the deﬁnition of the counit in B∗in line 1, (5) for ψB∗A∗in line 2 and ﬁnally

the deﬁnition of the counit in A∗. Since these equations hold for arbitrary f∈A∗, we ﬁnd

that ˆ

ψAB is a measuring of algebras. To show that it is also a measuring of coalgebras let

b∈B, p ∈ C(A∗, B∗), f, g ∈A∗. Then:

(f⊗g)∆A(ˆ

ψAB (p⊗b))

=f(ˆ

ψAB (p⊗b)(1))g(ˆ

ψAB(p⊗b)(2))

= (gf )( ˆ

ψAB(p⊗b)) = ψB∗A∗(p⊗gf )(b)

= (ψB∗A∗(p(1) ⊗g)ψB∗A∗(p(2) ⊗f))(b)

= (ψB∗A∗(p(2) ⊗f))(b(1))(ψB∗A∗(p(1) ⊗g))(b(2))

=f(ˆ

ψAB (p(2) ⊗b(1)))g(ˆ

ψAB (p(1) ⊗b(2)))

= (f⊗g)( ˆ

ψAB(p(2) ⊗b(1))⊗ˆ

ψAB (p(1) ⊗b(2))),

where we used the deﬁnition of the multiplication in A∗in line 2, (4) for ψB∗A∗in line 3 and

the deﬁnition of the multiplication in B∗in line 4. Again this holds for arbitrary f, g ∈A∗

hence ˆ

ψAB satisﬁes the ﬁrst equation in (5) and for the counit we have:

νA(ˆ

ψAB (p⊗b)) = ψB∗A∗(p⊗νA)(b) = ǫB∗A∗(p)νB(b),

where we used (4) for ψB∗A∗and the fact that νAand νBare the units in A∗and B∗

respectively. This shows that ˆ

ψAB is a measuring of Frobenius algebras from Bto A

and hence there exists a unique coalgebra morphism πAB :CB∗A∗→ Ccop

AB , such that di-

agram (14) commutes. Now let A, B, C be Frobenius algebras with duals A∗, B∗, C ∗and

a∈A, p ∈ CA∗B∗, q ∈ CB∗C∗, f ∈C∗. Then:

f(ψCA (πCAmA∗B∗C∗⊗A)(p⊗q⊗a)) = f(ˆ

ψCA (pq ⊗a))

=ψA∗C∗(pq ⊗f)(a) = ψA∗B∗(p⊗ψB∗C∗(q⊗f))(a)

=ψB∗C∗(q⊗f)( ˆ

ψBA(p⊗a)) = f(ˆ

ψCB (q⊗ˆ

ψBA(p⊗a)))

=f(ψCB (πC B (q)⊗ψBA (πBA (p)⊗a)))

=f(ψCA (mCBA ⊗A)(πC B (q)⊗πBA(p)⊗a)

=f(ψCA (mCBA σ⊗C)(πBA ⊗πC B ⊗C)(p⊗q⊗a)),

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 29

where we used ψCA(πC A ⊗A) = ˆ

ψCA in line 1, the characterisation of mA∗B∗C∗in line 2, the

deﬁnition of ˆ

ψBA and ˆ

ψCB in line 3, ψC B (πCB ⊗B) = ˆ

ψCB and ψBA(πB A ⊗A) = ˆ

ψBA in line

4 and the characterisation of mCB A in line 5. Finally, for a∈Aand f∈A∗

f(ψAA(πAA jA∗⊗a)) = f(ˆ

ψAA(jA∗⊗a)) = ψA∗A∗(jA∗⊗f)(a) = f(a),

where we recall ψA∗A∗(jA∗⊗A∗) = A∗. Since fwas arbitrary and by universal property we

obtain the required compatibility with the multiplication and unit. We omit the proof of the

fact that πAB is an isomorphism, as this follows from Theorem 4.7 below. 

The previously obtained dualities γand πare related with the antipode as in the statement

of the following proposition.

Theorem 4.7. Let Xbe a class of Frobenius algebras and X∗the class of their dual Frobenius

algebras. We have the following commutative diagram of Hopf category isomorphisms:

C(X∗)π//C(X)op,cop

C(X)

dd❍

❍

❍S

Proof. Since πγ(A) = Awe know that the morphism agree on the index sets X. We show

πAB γBA =SBA . Therefore, let p∈ CAB , b ∈B, f ∈A∗and compute

f(ψAB (πAB γBA ⊗B)(p⊗b)) = f(ˆ

ψAB(γAB(p)⊗b))

=ψB∗A∗(γBA(p)⊗f)(b)

=ψ∗

B∗A∗(p⊗f)(b) = ιB(ψBA (p⊗ι−1

A(f))(b)

=νB(ψBA(p⊗f(e1)e2)b) = νB(f(e1)p(e2)b)

=f(e1νB(p(e2)b)) = f(ψAB(SB A(p)⊗b))

=f(ψAB(SBA ⊗B)(p⊗b)).

Since fwas arbitrary and by universal property of CAB we have πAB γBA =SBA . Since Sand

γare isomorphisms, it follows that πis an isomorphism as well. 

The results of this section can be dualized for comeasurings. We will avoid explicit proofs

and just state the concluding result in this case.

Theorem 4.8. Let Xbe a class of Frobenius algebras and X∗the class of the dual of algebras

in X. We have a commutative diagram of isomorphisms of Hopf opcategories as follows:

A(X∗)//A(X)op,cop

A(X)

dd■

■

■S

Remark 4.9.Let us illuminate how to deﬁne the isomorphisms αand dual to the above mor-

phisms γand πfor universal measuring coalgebras. The isomorphism αcan be constructed

by assigning to every comeasuring ρ:A→B⊗Qa comeasuring ρ∗:= (ιB⊗Q)ρι−1

A:A∗→

B∗⊗Q, where ιA:A→A∗and ιB:B→B∗are the isomorphisms of Frobenius algebras

30 P. GROSSKOPF AND J. VERCRUYSSE

recalled at the beginning of this section. Furthermore, given the universal comeasuring ρB∗A∗

from A∗to B∗, let ˆρAB :B→ Aop

B∗A∗⊗Bbe the comeasuring deﬁned by

((f⊗ A)ˆρAB )(b) := ρB∗A∗(f)(b⊗ AB∗A∗),∀b∈B, f ∈A∗.

Then this gives rise to the isomorphism .

5. Examples

The notion of a measuring between Frobenius algebras is rather restrictive. Let us illustrate

this by considering the trivial Frobenius algebra k(where all structure maps are identities),

and considering a non-zero comeasuring ρ:A→k⊗Q∼

=Q, where Ais an arbitrary Frobenius

algebra. By the second axiom of (12) we then already ﬁnd that

ρ(a) = ν(a)1Q,

so that ρis uniquely determined. Furthermore, the ﬁrst axiom of (11) then implies that

ν(aa′) = ν(a)ν(a′),

which means that the counit of Ais multiplicative. However, combining this with the counit

property and Frobenius property, we ﬁnd that

a=ν(ae1)e2=ν(a)ν(e1)e2=ν(a)1A

for any a∈A. In other words, we ﬁnd that no non-trivial comeasurings exist from Ato k,

unless Ais already isomorphic to k. Consequently, we ﬁnd that AA,k =Ak,A =CA,k =Ck,A =

0 whenever Ais non-isomorphic to k.

In this remaining section we compute some further examples of universal measuring coalge-

bras and comeasuring algebras. We follow the procedure described in the proof of [1, Theorem

3.16] to construct these objects. Throughout this section, we suppose that kis a ﬁeld (and

not just a commutative ring), which allows us to explicitly describe algebras and coalgebra

structures in terms of their base elements.

Since it is more convenient to work with the tensor algebra and factoring out ideals in the

comeasuring case than dealing with subcoalgebras of the cofree coalgebra in the measuring

case, we will, given two Frobenius algebras Aand B, ﬁrst construct AAB by means of gen-

erators and relations and then construct CAB =A◦

AB by duality. By specifying Aand Bto

particular small Frobenius algebras, we will provide an explicit description of the universal

measuring coalgebra in these cases. As it will turn out, all universal measuring coalgebras we

obtain are ﬁnite dimensional.

Another aim is to show that our Hopf category of Frobenius algebras is richer than the

ordinary category of Frobenius algebra, meaning there are examples where the universal

measuring coalgebra is not generated by grouplike elements (i.e. (iso)morphisms of Frobenius

algebras). Furthermore we will show that there are non-isomorphic Frobenius algebras Aand

Bfor which CBA is non zero.

Consider Frobenius algebras Aand Band choose a (ﬁnite) base (aα)α∈Iof Aand (bβ)β∈J

of B. Deﬁne the map ρ0:A→B⊗(B∗⊗A) given by

ρ0(aα) = X

β∈J

bβ⊗(b∗

β⊗aα),

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 31

where b∗

βare the dual base vectors for the chosen base for B. Now we denote the structure

constants aγ

m;α1,α2, aγ1,γ2

∆;α, aγ

1, aν;α∈kof Aas follows

aα1aα2=X

γ∈I

aγ

m;α1,α2aγ,1 = Pγ∈Iaγ

1aγ,

∆(aα) = X

γ1,γ2∈I

aγ1,γ2

∆;α(aγ1⊗aγ2), ν(aα) = aν;α.

for all α, α1, α2, γ, γ1, γ2∈I. Similarly we denote the structure constants bµ

m;β1,β2, aµ1,µ2

∆;β, aµ

1, aν;β∈

kof Bfor β, β1, β2, µ, µ1, µ2∈J. We now deﬁne ABA := T(B∗⊗A)/I where T(B∗⊗A) is

the tensor algebra over the vector space B∗⊗Aand Iis the ideal generated by the elements

γ∈I

aγ

m;α1,α2(b∗

µ⊗aγ)−X

β1,β2∈J

bµ

m;β1,β2(b∗

β1⊗aα1)(b∗

β2⊗aα2),

γ1,γ2∈I

aγ1,γ2

∆;α(b∗

µ1⊗aγ1)(b∗

µ2⊗aγ2)−X

β∈J

bµ1,µ2

∆;β(b∗

β⊗aα),

aγ

1(b∗

µ⊗aγ)−bµ

aν;α−X

bν;β(b∗

β⊗aα)

for all choices of α, α1, α2∈Iand µ, µ1, µ2∈J. It can now be shown that ρB A :A→B⊗ABA,

given by the composition

Aρ0//B⊗(B∗⊗A)//B⊗T(B∗⊗A)////B⊗ ABA

is the universal comeasuring from Ato B. By applying this coaction twice, we ﬁnd that the

cocomposition of the associated Hopf-opcategory is given on generators by the formula

δABC :AAC → AAB ⊗ ABC

δABC (a∗⊗c) = Pβ∈J(a∗⊗bβ)⊗(b∗

β⊗c).(15)

We will now apply the construction to the case of Frobenius algebras of the form k[G]

where Gis a ﬁnite group. The algebra structure is in this case given by the usual group

algebra and the comultiplication and counit on base elements (i.e. elements of G) is given by

∆G(g) = X

h∈G

gh−1⊗h,

ν(eG) = 1, ν(g) = 0, g 6=eG,

where eGis the unit element of G. Therefore the only non-zero structure constants are

axy

m;x,y = 1, ax,y

∆;xy = 1, a1G

1= 1, aν,1G= 1 for x, y ∈G.

Combining the above, we ﬁnd that the universal comeasuring algebra Ak[G],k[H]for some

ﬁnite groups Gand His given by the quotient of the free k-algebra over the set {g∗⊗h|g∈

32 P. GROSSKOPF AND J. VERCRUYSSE

G, h ∈H}, by the ideal generated by

((ab)∗⊗x)−X

h∈H

(a∗⊗xh−1)(b∗⊗h)a, b ∈G, x ∈H, (16)

(a∗⊗xy)−X

g∈G

(a∗(g−1)∗⊗x)(g∗⊗y)a∈G, x, y ∈H, (17)

(1∗

G⊗1H)−1,(18)

(1∗

G⊗x) 1H6=x∈H, (19)

(a∗⊗1H) 1G6=a∈G. (20)

Remark that all equations factored out are symmetric in the Gand Hcomponent. More

precisely, we have the following result.

Proposition 5.1. Let Gand Hbe ﬁnite groups and ka ﬁeld. Then the assignment h∗⊗g7→

g∗⊗hinduces an algebra isomorphism Ak[H],k[G]≃ Ak[G],k[H]. Therefore also Ck[H],k[G]≃

Ck[H],k[G]as coalgebras.

The previous proposition is in fact a reincarnation of the Hopf category isomorphism π

from Proposition 4.6. To see this, observe that the isomorphism ιk[G]:k[G]→k[G]∗, is given

by the formula ιk[G](g) = ν(g−) = (g−1)∗. Furthermore, following the proof of Theorem 3.5,

the antipode Sk[G],k[H]:Ak[G],k[H]→ Ak[H],k[G]is given by the formula

g∗⊗x7→ X

h∈H

νk[H](hx)h∗⊗g−1= (x−1)∗⊗g−1(21)

By combining the above formula, one then indeed recovers the commutative diagram from

Theorem 4.7.

Example 5.2. For the trivial group, the group algebra is just the base ﬁeld k, which is the

case that we already treated at the beginning of this section. The next simplest example is

the cyclic group with two elements G={e, g}. For any group Hnon-isomorphic to C2we

ﬁnd Ak[C2],k[H]= 0. If His the trivial group, then this follows from the reasoning in the

beginning of this section. Now, let |H| ≥ 3 and let hbe an arbitrary non trivial element in

H. Then there exists an element 1H6=k∈Hsuch that hk 6= 1H. Now we can use (17) and

(19) to show:

(g∗⊗h) = (g∗⊗hkk−1) = (g∗⊗hk)(e∗⊗k−1)+(e∗⊗hk)(g∗⊗k−1) = (g∗⊗hk)0+0(g∗⊗k−1) = 0

Since hwas arbitrary and (e∗⊗h) = 0 by (19), the only generator of Ak[C2],k[H]is (e∗⊗1H).

However using (18) and (17) we obtain

1 = (e∗⊗1H) = (e∗⊗hh−1) = (e∗⊗h)(e∗⊗h−1) + (g∗⊗h)(g∗⊗h−1) = 0

in Ak[C2],k[H]. Therefore, Ak[C2],k[H]=Ak[H],k[C2]=Ck[C2],k[H]=Ck[H],k [C2]= 0 for all groups

H6=C2.

Example 5.3. Let us now compute the universal coacting Hopf algebra of k[C2], which is

generated as an algebra by the element x:= (g∗⊗g) (remark that the other initial generators

of the free algebra become either 1 or 0 in the quotient). We can use (19), (17) and (18) to

show:

x2= (g∗⊗g)(g∗⊗g) = (g∗⊗g)(g∗⊗g) + (e∗⊗g)(e∗⊗g) = (e∗⊗gg) = (e∗⊗e) = 1.

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 33

Therefore we ﬁnd a representation of the universal coacting Hopf algebra

Ak[C2],k[C2]=k[x]/x2−1.

By (15), the comultiplication is then given by

δ(x) = δ(g∗⊗g) = (g∗⊗e)⊗(e∗⊗g) + (g∗⊗g)⊗(g∗⊗g) = x⊗x

and of course δ(1) = 1 ⊗1. Since both xand 1 are grouplike we have ǫ(1) = ǫ(x) = 1. The

antipode Sis the identity. We conclude that the universal coacting Hopf algebra in this case

is isomorphic to the group algebra k[C2].

It is well-known that the dual Hopf algebra of k[C2] is isomorphic to k[C2] if the characteris-

tic of kis diﬀerent from 2 and is generated by the two grouplike elements (1∗+x∗),(1∗−x∗).

These grouplike elements correspond to the identity on k[C2] and the Frobenius automorphism

sending e7→ eand g7→ −g.

For Char(k) = 2 the two grouplikes (and the corresponding two automorphisms of the Frobe-

nius algebra) coincide. However, in this case x∗is a primitive element:

δ′(x∗) = (1∗⊗x∗)+(x∗⊗1∗) = (1∗⊗x∗) +(x∗⊗1∗)+ 2(x∗⊗x∗) = (1∗+x∗)⊗x∗+x∗⊗(1∗+x∗).

This primitive element corresponds to a bi-derivation D:k[C2]→k[C2] (see beginning of

Section 3). Therefore in the case of characteristic 2 the universal acting Hopf algebra is

generated by the identity and a bi-derivation. Already in this small example we see therefore

that the universal acting Hopf algebra can be diﬀerent from a groupalgebra.

Example 5.4. Next in line is the group of order 3, C3:= {e, a, b}with a2=b, ab =eand

neutral element e. For the universal coacting algebra of k[C3] we ﬁnd by (19) and (20) that

the only generators diﬀerent from 0 and 1 are a∗⊗a,a∗⊗b,b∗⊗aand b∗⊗b. Furthermore,

for any x6=e6=yin C3we have

(a∗⊗x)(a∗⊗y) = (a∗⊗x)(a∗⊗y) + (e∗⊗x)(b∗⊗y) + (b∗⊗x)(e∗⊗y) = b∗⊗xy,

by using (19) and (17). Similar calculations for byields (b∗⊗x)(b∗⊗y) = a∗⊗xy. By symmetry

we also have (x∗⊗a)(y∗⊗a) = (xy)∗⊗band (x∗⊗b)(y∗⊗b) = (xy)∗⊗a. Therefore, the only

additional base elements for Ak[C3],k[C3]are (a∗⊗a)(b∗⊗b) and (b∗⊗a)(a∗⊗b). To simplify

notation we rename the elements as follows

A:= (a∗⊗a), B := (b∗⊗b), X := (a∗⊗a) (b∗⊗b)

C:= (a∗⊗b), D := (b∗⊗a), Y := (b∗⊗a) (a∗⊗b),

and with this notation, the multiplication table is presented below.

X A B Y C D

X X A B 000

A A B X 000

B B X A 000

Y000Y C D

C000C D Y

D000D Y C

One can notice that the multiplication is commutative and the unit is given by 1 = X+Y(=

e∗⊗e). Therefore we have a six dimensional algebra presented as the direct product of two 3

34 P. GROSSKOPF AND J. VERCRUYSSE

dimensional algebras: Ak[C3],k[C3]≃k[C3]×k[C3]. However the coalgebra structure intertwines

the two subalgebras. Explicitly, the comultiplication is given by the following formulas.

δ(A) = A⊗A+D⊗C δ(C) = C⊗A+B⊗C

δ(B) = B⊗B+C⊗D δ(D) = A⊗D+D⊗B

δ(X) = δ(A)δ(B) = X⊗X+Y⊗Y δ(Y) = δ(C)δ(D) = X⊗Y+Y⊗X

To obtain the formula for the counit ν, recall that νis an algebra morphism satisfying

(id ⊗ν)◦ρ=id from this, we can deduce that

ǫ(A) = ǫ(B) = ǫ(X) = 1, ǫ(C) = ǫ(D) = ǫ(Y) = 0.

For the antipode Swe use formula (21) and ﬁnd

S(A) = B, S(B) = A, S(X) = X, S(Y) = Y, S(C) = C, S(D) = D.

Now we dualize the Hopf algebra to ﬁnd the universal acting Hopf algebra Ck[C3],k[C3]. It

is generated by X∗, A∗, B∗, Y ∗, C∗, D∗. The coalgebra structure is given by the coalgebra

structure on the Frobenius algebra k[C3] with generators {X∗, A∗, B∗}and {Y∗, C∗, D∗}.

The multiplication table comes out as:

X∗A∗B∗Y∗C∗D∗

X∗X∗0 0 Y∗0 0

A∗0A∗0 0 0 D∗

B∗0 0 B∗0C∗0

Y∗Y∗0 0 X∗0 0

C∗0C∗0 0 0 B∗

D∗0 0 D∗0A∗0

The unit is given by X∗+A∗+B∗and the antipode is given by S∗(A∗) = B∗,S∗(B∗) = A∗

and the identity on the other generators. The universal action ψ:Ck[C3],k[C3]⊗k[C3]→k[C3]

is given by

ψ(X∗⊗e) = ψ(Y∗⊗e) = e

ψ(A∗⊗a) = ψ(D∗⊗b) = a

ψ(B∗⊗b) = ψ(C∗⊗a) = b

and zero everywhere else. Note that X∗and Y∗have the same action, but are distinguishable

by the coalgebra structure. We have two grouplike elements X∗+A∗+B∗, corresponding

to the identity of k[C3] and Y∗+C∗+D∗corresponding to the automorphism of Frobenius

algebras τ=ψ((Y∗+C∗+D∗)⊗ −) : k[C3]→k[C3] given by τ(a) = b, τ(b) = a. Remark

that, for example, the element X∗+A∗+B∗also acts as the identity on k[C3], although the

resulting element in Ck[C3],k[C3]is not grouplike.

If the characteristic of the ﬁeld kis diﬀerent from 3 and furthermore khas some primitive

third root of unity, then we ﬁnd additional grouplike elements. Indeed, suppose 1 6=ξ∈k

satisﬁes ξ2+ξ+ 1 = 0, then the following elements of Ck[C3],k[C3]are also grouplike:

X∗+ξA∗+ξ2B∗, X ∗+ξ2A∗+ξB∗, Y ∗+ξC∗+ξ2D∗, Y ∗+ξ2C∗+ξD∗.

As in this way, we found 6 grouplike elements, the Hopf algebra Ck[C3],k[C3]has a base of

grouplike elements. Since the comultiplication of Ak[C3],k[C3]was not cocommutative, the

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 35

multiplication of Ck[C3],k[C3]is not commutative, hence the grouplike elements form a non-

commutative group of order 6. We can conclude that

Ck[C3],k[C3]∼

=k[S3]

as Hopf algebra, and Ck[C3],k[C3]is completely determined by the set of Frobenius automor-

phisms of k[C3].

As in the example of the universal coaction of k[C2] above, we ﬁnd biderivations if the ﬁeld

is of characteristic 3. For example, we have in this case:

(X∗+A∗+B∗)⊗(A∗−B∗) + (A∗−B∗)⊗(X∗+A∗+B∗)

= (X∗⊗A∗) + (A∗⊗A∗) + (B∗⊗A∗)−(X∗⊗B∗)−(A∗⊗B∗)−(B∗⊗B∗)

+ (A∗⊗X∗) + (A∗⊗A∗) + (A∗⊗B∗)−(B∗⊗X∗)−(B∗⊗A∗)−(B∗⊗B∗)

+ (B∗⊗B∗) + (A∗⊗A∗)−(A∗⊗A∗)−(B∗⊗B∗)

=δ(A∗) + 0 −δ(B∗)−0 = δ(A∗−B∗)

Therefore A∗−B∗is a primitive element in Ck[C3],k[C3]and hence a biderivation. Similarly,

(C∗−D∗) is a biderivation too and further any linear combination of these. In particular, in

this case, Ck[C3],k[C3]is not a group algebra.

Example 5.5. Let Hbe a group non isomorphic to C3. Then we have that

Ak[C3],k[H]=Ak[H],k[C3]=Ck[C3],k[H]=Ck[H],k[C3]= 0.

Indeed, we use notation as above for the elements of C3and for the universal comeasuring

algebra. For any x6= 1H6=yin Hwe then have the following identity in Ak[C3],k[H]:

(a∗⊗xy) = (a∗⊗x)(e∗⊗y) + (e∗⊗x)(a∗⊗y) + (b∗⊗x)(b∗⊗y) = (b∗⊗x)(b∗⊗y).(22)

Similarly

(b∗⊗xy) = (a∗⊗x)(a∗⊗y).(23)

Suppose now |H| ≥ 4 and let z∈H. We can write z=xhh−1ysuch that x, h, y are all non

trivial as well as xh and h−1y. Then we ﬁnd

(b∗⊗z) = (a∗⊗x)(a∗⊗y) = (b∗⊗xh)(b∗⊗h−1)(b∗⊗h)(b∗⊗h−1y)

= (b∗⊗xh)(a∗⊗h−1h)(b∗⊗h−1y)

= (b∗⊗xh)(a∗⊗eH)(b∗⊗h−1y) = 0

using (22), (23) and (20). Since zwas arbitrary we obtain (b∗⊗z) = 0 for all z∈Hand

therefore (a∗⊗z) = (b∗⊗x)(b∗⊗y) = 0. Hence Ak[C3],k[H]is only generated by (e∗⊗1H).

But since (e∗⊗1H) = Px∈H(a∗⊗x−1)(b∗⊗x) = 0 we have that Ak[C3],k[H]= 0. The other

statements follow from Proposition 5.1 and Theorem 4.3.

The next groups to handle are those of order 4, the ﬁrst order for which we have two non-

isomorphic groups: C4and C2×C2. Although this example becomes already very large to

compute completely by hand, let us make some interesting observations in this case. Recall

that, given a ﬁeld kwith Char(k)6= 2 and with a primitive fourth root of unity i, the group

36 P. GROSSKOPF AND J. VERCRUYSSE

algebras of C4and C2×C2are isomorphic as Frobenius algebras. Denote C4:= {1, x, x2, x3}

and C2:= {e, g}then one can check that the linear map φ:k[C2×C2]→k[C4] given by

(e, e)7→ 1,

(g, e)7→ 1 + i

2x+1−i

2x3,

(e, g)7→ 1−i

2x+1 + i

2x3,

(g, g)7→ x2

is a morphism of algebras preserving the Frobenius structure, hence is an isomorphism. There-

fore, it follows that the universal measuring coalgebras Ck[C2×C2],k[C4]and Ck[C4],k[C2×C2]as well

as the universal comeasuring algebras Ak[C2×C2],k[C4]and Ak[C4],k[C2×C2]are non-trivial, since

they should contain at least the element associated to the above isomorphism. If now kis a

ﬁeld of characteristic diﬀerent from 2, but not containing a primitive fourth root of unity (e.g.

k=Q), then we know that k[C2×C2] and k[C4] are non-isomorphic, however also in this

case we can conclude that the universal measuring coalgebras Ck[C2×C2],k[C4]and Ck[C4],k[C2×C2]

as well as the universal comeasuring algebras Ak[C2×C2],k[C4]and Ak[C4],k[C2×C2]are still non-

trivial. To see this, let us ﬁrst state and prove the following useful observation, showing that

the universal measuring coalgebra already “detects” (iso)morphisms that arise only after base

extension.

Proposition 5.6. Let Aand Bbe two Frobenius algebras over a ﬁeld k. Let ℓbe any extension

of k. Consider the Frobenius algebras over ℓobtained by extension of scalars: ℓ⊗kAand

ℓ⊗kB. Then the following assertions hold.

(i) Aℓ⊗kA,ℓ⊗kB∼

=ℓ⊗kAA,B

(ii) The dimension of AA,B is at least the cardinality of the set of morphisms of Frobenius

algebras between ℓ⊗kBand ℓ⊗kA.

The same holds for arbitrary Ω-algebras.

Proof. (i). First remark that if we ﬁx a k-base {aα, α ∈I}for A, then {1ℓ⊗aα, α ∈I}is

an ℓ-base for ℓ⊗kA. Moreover the structure constants (belonging to k) for Awith respect

to the base {aα, α ∈I}are also the structure constants for ℓ⊗kAwith respect to the base

{1ℓ⊗aα, α ∈I}4.

Since the construction of the universal comeasuring algebra as described in the beginning

of this section is purely given by generators and relations, where the former are constructed

out of bases of the algebras and the latter depend on the structure constants for these bases.

As explained above, bases and structure constants are preserved under base extension, hence

the comeasuring algebra of base-extended algebras is the base extension of the comeasuring

algebra of the initial algebras.

(ii). It is well-known that grouplike elements in a coalgebra are linearly independent and

by Lemma 2.10 grouplike elements in the measuring coalgebra CAB correspond to morphisms

of Frobenius algebras from Bto A. Therefore the ℓ-dimension of Cℓ⊗kA,ℓ⊗kBis at least

the cardinality of the set of morphisms of Frobenius algebras between ℓ⊗kBand ℓ⊗kA.

4Hence the structure constants for ℓ⊗kAwith respect to the base {1ℓ⊗aα, α ∈I}belong to k. In fact, by

usual descent theory, an ℓ-algebra is obtained by extension of scalars from a k-algebra exactly if there exists

an ℓ-base whose structure constants belong to k.

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 37

Furthermore, since Cℓ⊗kA,ℓ⊗kB=A◦

ℓ⊗kA,ℓ⊗kB, also the ℓ-dimension of Aℓ⊗kA,ℓ⊗kBis at least

this cardinality. Finally, by part (i), we ﬁnd that the ℓ-dimension of Aℓ⊗kA,ℓ⊗kBequals the

k-dimension of AAB .

For the last statement, it suﬃces to observe that the above reasoning not particular for

Frobenius algebras, but also holds for arbitrary Ω-algebras. 

Let us now prove a general triviality result for universal comeasuring algebras between

group algebras.

Proposition 5.7. If Gand Hare ﬁnite groups and kis a ﬁeld such that Charkdoes not

divide |G| − |H|,

Ak[G],k[H]=Ak[H],k[G]=Ck[G],k[H]=Ck[H],k[G]= 0.

Proof. As before, it suﬃces to prove that Ak[G],k[H]= 0. With notation as above we can

compute

|G|1 = Pg∈G(1∗

G⊗1H) = Pg∈G((g∗)−1g∗⊗1H) = Pg∈GPh∈H((g∗)−1⊗h−1) (g∗⊗h)

=Ph∈H(1∗

G⊗h−1h) = Ph∈H(1∗

G⊗1H) = |H|1

If Charkdoes not divide |G| − |H|, then we can conclude that 0 = 1 in Ak[G],k[H]= 0. 

We will now look into another family of Frobenius algebras, namely matrix algebras.

Example 5.8. Fix n∈N0and consider the algebra Mn(k) of n×nmatrix algebras with

entries in k. We denote the canonical basis elements En

ij n

i,j=1, which are zero everywhere

except at the (i, j)-th position, where we have 1. The Frobenius structure is given by

µ(En

ij ⊗En

kl) = δjkEn

il In=PiEn

∆(En

ij ) = X

ik ⊗En

ki ν(En

ij ) = δij ,

where δij is the Kronecker-delta. Notice that just as group algebras, matrix algebras are

special Frobenius since µ◦∆ = n·Id and therefore the two behave fairly similar. The

universal comeasuring algebra is now given by AMm,Mn=T(M∗

m⊗ Mn)/I factoring out

the relations

δyz Em∗

uv ⊗En

iℓ =

p=1

(Em∗

uy ⊗En

ip)(Em∗

zv ⊗En

pℓ) (24)

δjk Em∗

uv ⊗En

iℓ =

w=1

(Em∗

uw ⊗En

ij )(Em∗

wv ⊗En

kℓ) (25)

δiℓ =

w=1

Em∗

ww ⊗Eiℓ (26)

δuv =

p=1

Em∗

uv ⊗En

pp (27)

for all i, j, k, ℓ = 1,...,n et u, v, y, z = 1,...,m.

Similarly to group algebras we ﬁnd that the existence of non-zero measurings between

matrix algebras puts severe restriction on their dimensions.

38 P. GROSSKOPF AND J. VERCRUYSSE

Proposition 5.9. Let kbe a ﬁeld and n, m ∈N0such that Charkdoes not divide n−m.

Then

AMn(k),Mm(k)=AMm(k),Mn(k)=CMn(k),Mm(k)=CMm(k),Mn(k)= 0.

Proof. Denote the canonical basis by Em

kl and En

ij . Then

m1 =

δkk =

((Em

kk)∗⊗En

ii) =

δii =n1

by using (26) and (27) and the deﬁnitions of the units in Mnand Mm. Therefore 1 = 0 in

AMm(k),Mn(k)if Charkdoes not divide n−m.

To ﬁnish this section we consider measurings between Frobenius group algebras and matrix

algebras.

Proposition 5.10. Let kbe a ﬁeld, Ga ﬁnite group and n∈N0such that Charkdoes not

divide n− |G|or n−1. Then

AMn(k),k[G]=Ak[G],Mn(k)=CMn(k),k[G]=Ck[G],Mn(k)= 0.

Proof. The universal comeasuring algebra is now given by Ak[G],Mn(k)=T(k[G]∗⊗ Mn)/I

factoring out the relations

((ab)∗⊗En

ij ) =

k=1

(a∗⊗En

ik)(b∗⊗En

kj ) (28)

δkℓ(a∗⊗En

ij ) = X

g∈G

((ag−1)∗⊗En

ik)(g∗⊗En

ℓj) (29)

δij = 1∗

G⊗Eij (30)

1 =

k=1

(1∗

G⊗En

kk ) (31)

for all a, b ∈Gand i, j, k, ℓ = 1,...,n.

Using relations (31) and (30) we ﬁnd:

1 =

k=1

(e∗

G⊗Ekk) =

k=1

δkk =n1.

So 1 = 0 whenever Charkdoes not divide n−1.

On the other hand, using (28), (29) and (31) we ﬁnd

|G|1 = X

g∈G

1 = X

g∈G

i=1

((gg−1)∗⊗En

ii)

g∈G

i=1

j=1

(g∗⊗En

ij )((g∗)−1⊗En

ji )

i=1

j=1

(1∗

G⊗En

ii) =

j=1

1 = n1

hence 1 = 0 whenever Charkdoes not divide n− |G|.

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 39

6. Conclusions and outlook

6.1. Remaining questions. In this paper, we studied measuring coalgebras and comeasur-

ing algebras between Frobenius algebras. As a main result we showed that the semi-Hopf

category (i.e. the coalgebra enriched category) of universal measuring coalgebras between

Frobenius algebras admits an invertible antipode. We discussed duality between measur-

ing and comeasurings and computed several explicit examples. Based on these results and

examples, we formulate a few remaining questions that require further investigations.

(1) Recall that if ℓis a ﬁeld extension of k, then a k-algebra Ais called a form of an

ℓ-algebra R, if and only if R∼

=ℓ⊗kA. So far, the only examples we have where CA,B

is non-zero for some Frobenius k-algebras Aand B, is when Aand Bare forms of the

same ℓ-algebra for some ﬁeld extension ℓof k. Therefore, and in view of Proposition 5.6

above, we pose the following question:

Is it true that CA,B is nonzero for some some Frobenius k-algebras Aand Bif

and only if Aand Bare forms of the same ℓ-algebra for some ﬁeld extension

ℓof k?

For the above to hold, it would be suﬃcient to see that there exists a base extension ℓ

of k, such that ℓ⊗kCA,B has a grouplike element, since such a grouplike element acts

as a morphism of Frobenius algebras, which is therefore an isomorphism. The results

we have proven so far (see Propositions 5.7, 5.10 and 5.9), already show that there

are strong dimension restrictions for group algebras and matrix algebras in order to

have existence of non-zero (co)measurings between them. Since both group algebras

and matrix algebras are so-called special Frobenius, meaning that the composition of

multiplication with comultiplicaton equals a scalar multiple of the identity, one could

wonder whether it is possible to generalize the triviality conditions to this setting.

(2) In all examples we computed, the universal measuring coalgebras and comeasuring

algebras were ﬁnite dimensional. However, there is a priori no reason why this should

always be the case. We therefore ask

Do there exist Frobenius algebras Aand B, whose universal measuring coal-

gebra CAB is inﬁnite dimensional ?

Furthermore, one could also wonder, if there is a formula to compute the dimension

of the universal measuring coalgebra. Based on the very small examples we have, one

already sees that the dimension of the universal acting Hopf algebra on a Frobenius

algebra grows more rapidly than the dimension of the Frobenius algebra itself (the

dimension of Ck[C2],k[C2]being 2 and the dimension of Ck[C3],k[C3]being 6, computer

algebra computations showed that the dimension of CC[C4],C[C4]is at least 96).

(3) One could wonder, if there exist other types of Ω-algebras such that the associated

measuring semi-Hopf category is Hopf ? Or even stronger: what are the conditions

on Ω and its Ω-algebras for this to be the case ? Based on Section 4.2 we expect that

this could be the case for Ω’s that are “self-dual”.

6.2. Motivation from topological quantum ﬁeld theory. Finally, let us mention some

possible application of our work to Topological Quantum Field Theory which was in fact the

initial motivation for our investigations. Recall that a Topological Quantum Field Theory

is nothing else than a symmetric monoidal functor Zfrom a suitably deﬁned category of

n-cobordisms to the category of vector spaces. One way to deﬁne the needed category of

40 P. GROSSKOPF AND J. VERCRUYSSE

cobordisms is as follows (many variations are possible). Objects in such a category are n−1-

dimensional oriented manifolds (with the possibility to restrict for example to closed ones or

open ones). A morphism between two such objects Mand N, is a diﬀeormorphism class Σ

of n-dimensional oriented manifolds, together with diﬀeomorphisms of oriented manifolds be-

tween Mand the in-boundary of Σ, and between Nand the out-boundary of Σ. Composition

is obtained from gluing, tensor product is just disjoint union.

In the one-dimensional case, where the cobordisms consist of strings (curved lines) between

(ordered, ﬁnite) families of (oriented) points, a TQFT is determined by a ﬁnite dimensional

vector space V. Indeed, possible cobordisms, generating the category are pictured below.

The functor Zis then completely determined by the image of the positive point, which is

the vector space V, and sends the negative point to its dual vector space. The cup and cap

strings are mapped to evaluation and coevaluation morphisms.

•

V∗

−•

V+

pqrs

//•

V+•

V∗

−

wvut

coev

//•

V+

•

V+

•

V∗

−

•

V∗

−



The “dual base property” then is a consequence of the following identities in the category of

cobordisms.

•

V∗

−

pqrs

•

V+

wvut

coev



•

V∗

−

•

V∗

−

•

V∗

−



•

V+

pqrs

•

V∗

−

wvut

coev

•

V+

•

V+

•

V+

In the two-dimensional open case, cobordisms are strips between (ordered, ﬁnite) families of

(oriented) line segments and a TQFT in this case is completely characterized by a symmetric

Frobenius algebra (recall that symmetric means that the multiplication composed with counit

is invariant under twisting of the arguments), see [13, Corollary 4.5]. The possible cobordisms

generating the category are again pictured below.

PQRS////

//WVUT////

////0123

//7654 //

//////

////

❉

❉ ❉

❉

③

③ ③

③

The following identity in the category of cobordisms visualizes the Frobenius property, the

functor Znow sends the oriented line segment to a Frobenius algebra A, and it sends the line

THE HOPF CATEGORY OF FROBENIUS ALGEBRAS 41

segement with reversed orientation to the dual Frobenius algebra A∗.

PQRS

WVUT

////

PQRS////

WVUT////

=PQRS

WVUT

Now, it is well-known that the endomorphisms of a ﬁnite dimensional vector space form

a symmetric Frobenius algebra (isomorphic to a matrix algebra). Hence, it is possible to

construct a 2-dimensional (open) TQFT out of a 1-dimensional one by moving from the

object characterizing the 1-dimensional TQFT (i.e. the ﬁnite dimensional vector space) to

its endomorphism object (i.e. the matrix algebra). Geometrically, this can be interpreted

as follows. If we consider a 2-cobordism, that is, a strip between dotted line segments as

in the example below, and we only look to the solid boundaries of this strip, those can be

viewed as 1-cobordisms between the endpoints of the line segments, to which we can apply

a 1-dimensional TQFT. In the example below, we see then indeed that the multiplication of

the algebra associated to the 2-dimensional TQFT constructed this way, arises by tensoring

the evaluation map on both sides with identities, which is exactly the multiplication of the

endomorphism algebra End(V)∼

=V⊗V∗of the vector space Vassociated to the 1-dimenional

TQFT.

pqrs

////

V·

V∗

V·

V∗

V·

V∗

The question which triggered us to start this work was if it would be possible to make a

similar construction as above one dimension higher. That is, whether the “endomorphisms

of a 2-dimensional TQFT” could give rise to a 3-dimensional one. Let us try to make this

a bit more precise. When we say “endomorphisms of a 2-dimensional TQFT”, we in fact

mean the “endomorphism object” of the algebraic structure associated to the 2-dimensional

TQFT, which is the (symmetric) Frobenius algebra. If we interpret the previously mentioned

“endomorphism object” as the universal measuring coalgebra (which is a bialgebra by Propo-

sition 2.14) from the considered Frobenius algebra to itself, then the results of our paper (see

Theorem 3.2) show that this is a Hopf algebra. Furthermore it is known (see [7]) that Hopf

algebras, or more precisely their representation categories, give indeed rise to 3-dimensional

TQFTs. So far however, we did not obtain a (satisfying) geometric interpretation of this

observation, but we hope that future investigations will lead to such.

References

[1] A. Agore, A. Gordienko and J. Vercruysse, V-universal Hopf algebras (co)acting on Ω-algebras,Commun.

Contemp. Math. 25 (2023), Paper No. 2150095, 40 pp. 2, 9, 12, 13, 14, 24, 25, 27, 30

42 P. GROSSKOPF AND J. VERCRUYSSE

[2] A. Agore, A. Gordienko and J. Vercruysse, Lifting of locally initial objects and universal (co)acting Hopf

algebras, preprint 2024, arXiv:2406.17677. 3, 12

[3] E. Batista, S. Caenepeel and J. Vercruysse, Hopf categories, Algebr. Represent. Theory 19 (2016), 1173–

1216. 1, 2, 6, 7, 9

[4] M. Buckley, T. Fieremans, C. Vasilakopoulou, and J. Vercruysse, A Larson-Sweedler Theorem for Hopf

V−categories, Advances in Mathematics,376 (2021) 7

[5] S. Caenepeel, G. Militaru and S. Zhu, Frobenius and separable functors for generalized module categories

and nonlinear equations,Lecture Notes in Mathematics 1787, Springer-Verlag, Berlin, 2002. 3

[6] L. Crane, I.B. Frenkel, Four-dimensional topological quantum ﬁeld theory, Hopf categories, and the

canonical bases, J. Math. Phys. 35 (1994), 5136–5154. 7

[7] C.L. Douglas, C. Schommer-Pries, N. Snyder, Dualizable tensor categories Mem. Amer. Math. Soc. 268

(2020), vii+88 pp. 41

[8] P. Großkopf, J. Vercruysse, Free and co-free constructions for Hopf categories, J. Pure Appl. Algebra

228 (2024), Paper No. 107704. 2, 7, 15

[9] M. Hyland, I. L´opez Franco, C. Vasilakopoulou, Hopf measuring comonoids and enrichment, Proc. Lond.

Math. Soc. (3) 115 (2017), 1118–1148. 2, 12

[10] Joachim Kock, Frobenius Algebras and 2D Topological Quantum Field Theories, Cambridge University

press, 2003. 3

[11] H.F. Kreimer, M. Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J.

30 (1981), 675–692. 2

[12] R.G. Larson, M.E. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Am. J. Math.

91 (1969) 75–94. 1

[13] A.D. Lauda, H. Pfeiﬀer, Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras

Topology Appl. 155 (2008), 623–666. 40

[14] B. Pareigis, When Hopf algebras are Frobenius algebras, J. Algebra 18 (1971) 588–596. 1

[15] H. Pfeiﬀer, 2-groups, trialgebras and their Hopf categories of representations, Adv. Math. 212 (2007),

62–108. 7

[16] P. Saracco, Antipodes, preantipodes and Frobenius functors, J. Algebra Appl. 20 (2021), Paper No.

2150124, 32 pp. 2

[17] P. Saracco, Hopf modules, Frobenius functors and (one-sided) Hopf algebras, J. Pure Appl. Algebra 225

(2021), Paper No. 106537, 19 pp. 2

[18] Sweedler, M. E. Hopf Algebras. W. A. Benjamin New York, 1969. 2, 9, 12

Paul Großkopf, D´

epartement de Math´

ematiques, Universit´

e Libre de Bruxelles, Belgium

Email address :paul.grosskopf@gmx.at

Joost Vercruysse, D´

epartement de Math´

ematiques, Universit´

e Libre de Bruxelles, Bel-

gium

Email address :joost.vercruysse@ulb.be

ResearchGate has not been able to resolve any citations for this publication.

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