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Communications in Algebra®, 36: 4341–4367, 2008Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870802182424

ON HOPF ALGEBRAS AND THEIR GENERALIZATIONS

Gizem KaraaliDepartment of Mathematics, Pomona College, Claremont, California, USA

We survey Hopf algebras and their generalizations. In particular, we compare andcontrast three well-studied generalizations (quasi-Hopf algebras, weak Hopf algebras,and Hopf algebroids), and two newer ones (Hopf monads and hopfish algebras). Eachof these notions was originally introduced for a specific purpose within a particularcontext; our discussion favors applicability to the theory of dynamical quantum groups.Throughout the note, we provide several definitions and examples in order to make thisexposition accessible to readers with differing backgrounds.

Key Words: Hopf algebra; Hopf algebroid; Hopfish algebra; Hopf monad; Quantum group;Quasi-Hopf algebra; Weak Hopf algebra.

2000 Mathematics Subject Classification: Primary 16W30; Secondary 17B37, 18D10, 20G42, 81R50.

1. INTRODUCTION: GOALS AND MOTIVATION

The purpose of this note is to provide a historical and comparative study ofthe several notions of generalized Hopf algebras that have been introduced andstudied in recent years. We start with a brief discussion of Hopf algebras. We thenconsider the three main contenders for the correct notion of a generalized Hopfalgebra: quasi-Hopf algebras, weak Hopf algebras, and Hopf algebroids. We thendiscuss two newer notions that generalize Hopf algebras: Hopf monads and hopfishalgebras. As these are newer concepts, our study of them is necessarily rathercursory.

In our pursuit we trace the steps of many researchers, both mathematicallyand philosophically. Hence, this is an expository note, with no claim of introducingnew mathematics. Its sole purpose is to collect together several pieces of interestingmathematics and present them in one comprehensive historical narrative, so as toprovide a broader perspective on the current status of research involving variousgeneralizations of Hopf algebras.

The technical details and basic examples provided are intended to make thisnote accessible to a beginner in the theory of Hopf algebras, while we hope thatthose with more experience will still enjoy reading the discussions.

Received August 19, 2007; Revised October 21, 2007. Communicated by M. Cohen.Address correspondence to Gizem Karaali, Department of Mathematics, Pomona College, 610

North College Avenue, Claremont, CA 91711, USA; Fax: (909)607-1247; E-mail: gizem.karaali@pomona.edu

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4342 KARAALI

2. HOPF ALGEBRAS

Naturally, we begin with the definition of a Hopf algebra.

Definition 2.1. A Hopf algebra over a commutative ring � is a �-module Hsuch that:

(1) H is an associative unital �-algebra (with m � H ⊗H → H as the multiplicationand u � � → H as the unit) and a coassociative counital �-coalgebra (with� � H → H ⊗H as the comultiplication and � � H → � as the counit);

(2) The comultiplication � and the counit � are both algebra homomorphisms;(3) The multiplication m and the unit u are both coalgebra homomorphisms;(4) There is a bijective �-module map S � H → H , called the antipode, such that for

all elements h ∈ H :

S�h�1��h�2� = �u � ���h� = h�1�S�h�2��� (2.1)

Actually, the bijectivity of the antipode is not a part of the standard definition.In fact Hopf algebras with non-bijective antipodes have been constructed andstudied as early as in 1971, see Takeuchi (1971). In this note, we will disregardthese very interesting examples, and for simplicity focus on Hopf algebras and othergeneralizations with bijective antipodes. We refer the reader to Skryabin (2006) forthe implications of this bijectivity assumption on the structure of the Hopf algebra.Quite briefly, this corresponds to a finiteness condition.

Next we should point out that in the defining formula for the antipode, we aremaking use of the famous Sweedler notation,1 which is quite well-established in theHopf algebra literature. In short, the Sweedler notation is a generalization of theEinstein notation (in that it intrinsically demands a summation). More specificallyfor any h ∈ H , we write

��h� = h�1� ⊗ h�2��

This presentation itself is purely symbolic; the terms h�1� and h�2� do not stand forparticular elements of H . The comultiplication � takes values in H ⊗H , and so weknow that

��h� = �h1�1 ⊗ h1�2�+ �h2�1 ⊗ h2�2�+ �h3�1 ⊗ h3�2�+ · · · + �hN�1 ⊗ hN�2�

for some elements hi�j of H and some integer N . The Sweedler notation is just a wayto separate the hi�1 from the hj�2. In other words, one can say that the notation h�1�

stands for the generic hi�1, and the notation h�2� stands for the generic hj�2. Howeverthe summation is inherent in the notation: Whenever one sees a term like S�h�1��h�2�

(the left-hand side of (2.1)), one has to realize that this stands for a sum of the form

S�h1�1�h1�2 + S�h2�1�h2�2 + S�h3�1�h3�2 + · · · + S�hN�1�hN�2�

1In the introduction to Sweedler (1969), Sweedler remarks that this notation was developedduring years of joint work with Robert Heyneman. However, it is much more common to seereferences to the Sweedler notation than to the Heyneman–Sweedler notation. We follow thetradition here.

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ON HOPF ALGEBRAS AND THEIR GENERALIZATIONS 4343

Another remark that should be made here involves commutative diagrams.In most texts introducing the notion of a Hopf algebra, one will come acrosscommutative diagrams similar to the following one.

H ⊗HS⊗id−−−−→ H ⊗H

�

� �m

H −−−−→u�� H

H ⊗Hid⊗S−−−−→ H ⊗H

�

� �m

H −−−−→u�� H�

In fact, these two diagrams are equivalent to the single-line Eq. (2.1), but for thosecomfortable with commutative diagrams, they provide a much more visual way tounderstand it. Incidentally, an even more compact way to describe S would be asthe convolution inverse of the identity homomorphisn on H .

For the sake of completeness, we include here the diagrams representing theproperties of the counit (the so-called counit axiom):

(where we use H � H ⊗� � �⊗H), and the comultiplication (the coassociativity):

H�−−−−→ H ⊗H

�

���⊗1

H ⊗H −−−−→1⊗�

H ⊗H ⊗H�

The second and third conditions of Definition 2.1 are equivalent to oneanother and, together with the first condition, give us the following:

Definition 2.2. A bialgebra over a commutative ring � is a �-module B such that:

(1) B is both an associative unital�-algebra (withm � B ⊗ B → B as themultiplicationand u � � → B as the unit) and a coassociative counital �-coalgebra (with� � B → B ⊗ B as the comultiplication and � � B → � as the counit);

(2) The comultiplication � and the counit � are both algebra homomorphisms.

Then we can define a Hopf algebra to be a bialgebra with an antipode, in otherwords, a bialgebra B with an antihomomorphism S � B → B satisfying Eq. (2.1).

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4344 KARAALI

Hopf algebras were first introduced in the early 1940s by Hopf (1941), wherehe was working on homology rings of certain compact manifolds. (For a modernexposition of his results using the language of Hopf algebras, we refer the readerto McCleary, 2001). Later on, more examples were discovered and basic referencesstarted appearing, see for instance the early classics like Sweedler (1969) and Abe’s(1980). For a more modern approach emphasizing actions of Hopf algebras, seeMontgomery (1993).

For a simple example of Hopf algebras, consider the group algebra ��G ofa group G over a field �. Here we define the coalgebra structure by defining, onthe elements g of G, the comultiplication and the counit by setting ��g� = g ⊗ gand ��g� = 1. (More generally, for an arbitrary coalgebra C, the elements g of Csatisfying ��g� = g ⊗ g and ��g� = 1 are called group-like). The antipode is definedon the group elements by S�g� = g−1.

Dually, we can look at the algebra F��G� of �-valued functions on a groupG.2 There is a natural (commutative) multiplication m on this algebra given by

m�f1� f2��g� = f1�g�f2�g� for all g ∈ G�

The unit maps elements of the field � to the associated constant functions.To define a comultiplication, we simply use the natural embedding3 of F��G�⊗F��G� into F��G×G�. Then the group multiplication mG � G×G → G on Ginduces a comultiplication as follows:

��f��g1� g2� = f�mG�g1� g2���

The counit is the map taking f ∈ F��G� to f�eG� ∈ �, where eG is the identity elementof the group G. Finally, the antipode is the map S defined by S�f��g� = f�g−1�.

Function algebras give us a large collection of examples of commutative Hopfalgebras, (which are just Hopf algebras with commutative multiplications). In fact,one can show that any finite dimensional commutative Hopf algebra over a field �of characteristic zero is the function algebra of an affine algebraic group G over �.We will come back to this in Section 8.

Another family of examples is given by universal enveloping algebrasU��� of Liealgebras �. In this case, the coalgebra structure is defined on the elements x of � by

��x� = x⊗ 1+ 1⊗ x ��x� = 0�

(More generally, for an arbitrary coalgebra C, elements x of C satisfying ��x� =x⊗ 1+ 1⊗ x and ��x� = 0 are called primitive). The antipode S is defined on theLie algebra elements as S�x� = −x.

2We are intentionally being vague, in order to include several classes of algebras here; if, forinstance, G is an (affine) algebraic group, we could be thinking of the coordinate ring of G; if G is aPoisson group, we could be looking at the Poisson algebra C��G�.

3Without additional conditions on G and F��G�, the range of the most natural comultiplicationmap defined as dual to the multiplication will not be contained in F��G�⊗ F��G�, hence the need forthis embedding.

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ON HOPF ALGEBRAS AND THEIR GENERALIZATIONS 4345

This last class of examples provides us with a large collection of cocommutativeHopf algebras. Cocommutativity is the property of the comultiplication � describedby the following commutative diagram:

H ⊗HT←−−−− H ⊗H

�

���

H ←−−−−id

H

which is exactly the diagram one obtains by switching the directions of the arrowsand replacing m by � in the diagram describing commutativity of a product m:

H ⊗HT−−−−→ H ⊗H

m

��m

H −−−−→id

H�

In both of these diagrams T � H ⊗H → H ⊗H is the usual twist map: T�a⊗ b� =b ⊗ a.

After seeing the result above about commutative Hopf algebras, one mayexpect an analogous result regarding cocommutative Hopf algebras. In fact onecan show that a cocommutative Hopf algebra generated by its primitive elementsis indeed the universal enveloping algebra of a Lie algebra. This result will bementioned in Section 8 as well.

Ever since their introduction, Hopf algebras have been studied by manymathematicians. In the early 1970s, Hochschild, while developing the theory ofalgebraic groups, translated much of representation theory into the language ofHopf algebras. (See for instance Hochschild, 1981 for a classic written with thisperspective; Ferrer Santos and Rittatore, 2005 is a more modern text using the sameapproach). In some sense, we can say that Hopf algebras provided mathematiciansthe ultimate framework to do representation theory.

The development of the theory of quantum groups brought a fresh revivalof interest in Hopf algebras. Since the early 1980s, many mathematicians havebeen working on structures which today we loosely call quantum groups. In thefollowing paragraphs, we will provide only a brief sketch of the ideas of quantumgroup theory and refer the more interested reader to one of many textbooks andmonographs in the subject (eg., Brown and Goodearl, 2002; Chari and Pressley,1995; Etingof and Schiffman, 1998; Hong and Kang, 2002; Kassel, 1995; Lambe andRadford, 1997; Lusztig, 1993; Majid, 1995; Shnider and Sternberg, 1993).

The earliest examples of quantum groups were particular deformations ofuniversal enveloping algebras of simple Lie algebras and function algebras of simplealgebraic groups. In the mid 1980s, Drinfeld showed that the correct framework touse when studying quantum groups was that of Hopf algebras; see, for instance,his ICM address (Drinfeld, 1987). Thus the known world of Hopf algebras wassignificantly expanded to include all these new examples, and results from quantumgroups began to add more spice and flavor to the classical theory.

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Broadly speaking, a quantum group is a special type of noncommutativenoncocommutative Hopf algebra. One obtains such Hopf algebras by deforming themultiplication or the comultiplication of a commutative or a cocommutative Hopfalgebra. Since the end result of such a deformation is not commutative, it cannotproperly be associated to a group and be its function algebra. Similarly, since itis not cocommutative, it is by no means the enveloping algebra of a Lie algebra.However, we can still view it as if it were the function algebra or the envelopingalgebra of some phantom group or Lie algebra, and thus we have the term quantumgroup.4

For the sake of completeness, we briefly describe the most well-knownquantum group here: Quantum sl2���, denoted by Uh�sl2����. The notation is moreilluminating than the name; this is a particular (Hopf algebra) deformation of theuniversal enveloping algebra U�sl2���� of sl2���.

As an algebra, it is generated by E� F�H subject to the following relations:

�H� E = 2E� �H� F = −2F� �E� F = qH − q−H

q − q−1

where h is viewed as a formal parameter, q = eh = ∑�n=0

hn

n! ∈ ���h, and qH� q−H

should be interpreted in a similar manner. The coalgebra structure is defined on thegenerators by:

��E� = E ⊗ qH + 1⊗ E ��E� = 0

��F� = F ⊗ 1+ q−H ⊗ F ��F� = 0

��H� = H ⊗ 1+ 1⊗H ��H� = 0�

Finally, we define the antipode on the generators as

S�E� = −Eq−H S�F� = −qHF S�H� = −H�

Extended linearly, these give us a Hopf algebra structure, the famous quantum sl2.Currently, there is no consensus on what the precise definition of a

quantum group should be. Mainly, there are several well-known examples and anexponentially growing literature investigating their properties. For the purposes ofthis note, it will suffice to identify the term quantum group with quasitriangularor co-quasitriangular Hopf algebras, i.e., noncommutative noncocommutative Hopfalgebras associated to solutions of a certain equation, the quantum Yang–BaxterEquation, which we will describe in the next section.5

4This may remind some readers the philosophy of noncommutative geometry a la Connes.A similar approach to quantum groups would involve viewing them as symmetry objects of somequantum space; see Manin (1988).

5More precisely, a quasitriangular Hopf algebra is a Hopf algebra H together with an invertibleelement � ∈ H ⊗H such that:

(1) �op�h� = ���h��−1 for all h ∈ H ;(2) ��⊗ id���� = �13�23 and �id⊗ ����� = �13�12.

A coquasitriangular Hopf algebra is defined in a dual manner, see Majid (1995) and Schauenburg(1992b).

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ON HOPF ALGEBRAS AND THEIR GENERALIZATIONS 4347

3. LONG INTERLUDE: WHY GENERALIZE?

We believe that the previous section provides a sufficient overview of Hopfalgebras, and prepares the reader for the discussion on the generalizations of thisclassical theory. For the reader who wishes to learn more about Hopf algebras, werecommend the references already mentioned above, as well as any textbook onquantum groups. Chari and Pressley (1995), Etingof and Schiffman (1998), Kassel(1995), Majid (1995), and Shnider and Sternberg (1993) are a few of the many bookson quantum groups which provide a detailed exposition of Hopf algebras. Anotherquick introduction to Hopf algebras, focusing on some applications to physics,may be found in Hazewinkel (1991). Nichols and Sweedler (1982) and Aguiar andMahajan (2006) emphasize combinatorial applications of Hopf algebras.

Before we move on to the comparative study and technical details of thevarious generalizations, we would like to philosophize a bit about why we areinterested in any generalization. To those who view mathematics as an intellectualpursuit merely interested in pure abstractions, the answer will be clear: Why not?However, for those who may need more motivation, we will provide one, whichcomes from the theory of quantum groups.

The study of quantum groups goes back to the well-known quantum Yang–Baxter equation, which in its simplest form is as follows:

R12R13R23 = R23R13R12� (3.1)

Here we can view R as a map R � G×G → G×G for some factorizable Poisson–Lie group. Then

R12�g� h� k� = �R�g� h�� k��

and R13 and R23 can be defined likewise. In the realm of Hopf algebras, the quantumYang–Baxter equation (henceforth referred to as the QYBE) is the same Eq. (3.1);however, this time its solutions, the so-called quantum R-matrices, live in H ⊗H .Jimbo (1989, 1990) are surveys on the QYBE, displaying the several connections tothe physics literature.

The QYBE and its solutions give rise to quantum groups: in the geometric(Poisson) picture, they give a (Hopf algebra) deformation of the function algebraof the relevant group G; in the Lie algebra picture, they give a (Hopf algebra)deformation of the associated universal enveloping algebra. In the case of quantumsl2 as presented above, the relevant quantum R-matrix is

R = q12H⊗H

∑n≥0

qn�n−1�

2�q − q−1n

�nq!En ⊗ Fn�

Here we are using the q-notation common in the literature:

�nq =qn − q−n

q − q−1= qn−1 + qn−2 + · · · + q−n+1 and �nq! =

n∏k=1

�kq

In 1984, a modified version of the QYBE appeared in the mathematical physicsliterature (Gervais and Neveu, 1984). This new equation, later named the quantum

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4348 KARAALI

dynamical Yang–Baxter equation, (henceforth labeled QDYBE), lives in V ⊗ V ⊗ Vfor a semisimple module V of an abelian Lie algebra �:

R12��− h�3��R13���R23��− h�1�� = R23���R13��− h�2��R12���� (3.2)

Here, the term h�i� is to be substituted by �i if �i is the weight of the ith tensorcomponent, i = 1� 2� 3. A solution R � �∗ → End��V ⊗ V� to the QDYBE is calleda quantum dynamical R-matrix. For an introduction to the dynamical Yang–Baxterequations, we refer the reader to Felder (1994). A more geometric exposition can befound in Etingof and Varchenko (1998).

Studying the QDYBE and its solutions, we get into the realm of dynamicalquantum groups. The first examples of dynamical quantum groups that appeared inthe literature are Felder’s elliptic quantum groups, which were introduced in Felder(1994, 1995). A standard example that is studied in much detail is E ���sl2�, whichis an algebra over � generated by two kinds of generators: meromorphic functionsof a single variable h, and matrix elements of a matrix L���w� ∈ End��2�. The twosubscripts � � are nonzero complex numbers with Im� � > 0.

Without going into much detail, we note that the 4× 4 matrix solution to theQDYBE associated with E ���sl2� is in the following form:

R���w� � �� = E1�1 ⊗ E1�1 + E2�2 ⊗ E2�2

+ ���� w� � ��E1�1 ⊗ E2�2 + ����w� � ��E1�2 ⊗ E2�1

+ ���� w� � ��E2�1 ⊗ E1�2 + ���� w� � ��E2�2 ⊗ E1�1�

Here the functions �� �� �� � are defined in terms of the theta function:

��z� � = −�∑

j=−�exp

(�i

(j + 1

2

)2

+ 2�i(j + 1

2

)(z+ 1

2

))

by

���� w� � �� = ��w����+ 2����w − 2������

� ���� w� � �� = ��−w − ����2����w − 2������

���� w� � �� = ��w − ����2����w − 2������

� ���� w� � �� = ��w����− 2����w − 2������

�

For more on E ���sl2�, including an explicit presentation in terms of generators andrelations, we refer the reader to Felder and Varchenko (1996).

The objects, like Felder’s elliptic quantum groups, associated to solutions ofthe QDYBE turn out to be quite Hopf-like in many respects, but they are not allnecessarily Hopf algebras. Or in other words, we can say that, just as in the caseof quantum groups, some Hopf-like structures come into play when one studiesthe solutions of the QDYBE. And herein lies our motivation for the purpose ofthis particular note. A need for the correct Hopf-like object that will provide theframework for the theory of dynamical quantum groups leads us to the study ofvarious generalizations of Hopf algebras.

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ON HOPF ALGEBRAS AND THEIR GENERALIZATIONS 4349

4. CANDIDATE 1: QUASI-HOPF ALGEBRAS

Our first candidate in our search for the correct generalization of Hopfalgebras is the quasi-Hopf algebra, first introduced by Drinfeld (1989, 1990), andused to give a natural proof of Kohno’s theorem relating the monodromy of theKnizhnik–Zamolodchikov equations to a representation of the braid group arisingfrom a quantum group. In order to define quasi-Hopf algebras, we first need tointroduce the notion of a quasi-bialgebra.

Definition 4.1. A quasi-bialgebra B over a commutative ring � is a unitalassociative �-algebra equipped with two algebra homomorphisms � � B → � (thecounit) and � � B → B ⊗ B (the comultiplication) together with an invertible element� of B ⊗ B ⊗ B. Furthermore, we require the following to hold:

(1) � satisfies the pentagon relation

��⊗ 1⊗ 1�� · �1⊗ 1⊗ ��� = ��⊗ 1� · �1⊗ �⊗ 1�� · �1⊗��

(2) � is quasi-coassociative

�1⊗ ����b� = �−1��⊗ 1���b�� for all b ∈ B

(3) � is compatible with the counit � in the following sense:

�1⊗ �⊗ 1�� = 1

(4) � satisfies the counit axiom.6

From (1), (3), and (4), we get:

��⊗ 1⊗ 1�� = �1⊗ �⊗ 1�� = �1⊗ 1⊗ ��� = 1�

� of this definition is alternatively called an associator or a co-associator.We will call it the associator, merely to ease our typing efforts. However we shouldnote that both versions have merit. The term “co-associator” makes sense because� gives information on how far � is from coassociativity. The term “associator”makes sense because using �, one can define associativity isomorphisms in therepresentation category of B which makes it into a (not necessarily strict) monoidalcategory. The pentagon relation (1) satisfied by � in the above definition translatesautomatically to the pentagon relations of the associativity isomorphisms in therepresentation category of B.

Comparing Definitions 2.2 and 4.1, it is clear that this generalization has costus only the coassociativity of the comultiplication. However, this price is perfectlyacceptable to many researchers, as quasibialgebras and their Hopf relatives fit verynicely into several theories. We will discuss these shortly.

6We presented the commutative diagram version of the counit axiom earlier; in closed formit reads:

��⊗ 1�� = 1 = �1⊗ ����

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Here is the definition of the Hopf version:7

Definition 4.2. A quasi-Hopf algebra H over a commutative ring � is aquasi-bialgebra over� equipped with an (algebra and coalgebra) antihomomorphismS � H → H and two canonical elements �� � ∈ H such that:

(1) m · �S ⊗ ����b� = ��b�� for all b ∈ B;(2) m · �1⊗ �S���b� = ��b�� for all b ∈ B;(3) m�m⊗ 1� · �S ⊗ �⊗ �S�� = 1;(4) m�m⊗ 1� · �1⊗ �S ⊗ ���−1 = 1.

Here (1) and (2) are the modified versions of the antipode axiom (2.1).If � = � = 1, and � = 1⊗ 1⊗ 1, H becomes a Hopf algebra.

Drinfeld introduced quasi-Hopf algebras in his work relating the Knizhnik–Zamolodchikov equations to the theory of quantum groups. However, thesestructures proved to be a lot more than mere tools in one proof. The large interestthey invoked in the mathematicians in the field can easily be deduced, for example,from J. Stasheff’s enthusiastic MathSciNet review MR1091757 of Drinfeld’s (1989).

One general reason for this excitement can be explained, in category-theoreticterms, as follows: Quasi-Hopf algebras have the right kind of representationcategories. In particular, their representation categories are (not necessarily strict)rigid monoidal categories. In the following, we digress briefly and venture intocategory theory. The classic reference for the terms we use is MacLane (1998).Kassel (1995) provides ample background and many details within the context ofHopf algebras and quantum groups.

We begin with a comment on terminology. The monoidal categories inthis note are precisely the “categories with multiplication” of Bénabou (1963) andMacLane (1963). (See the historical notes at the end of Chapter VII in MacLane,1998). They are, in some instances, also called tensor categories, see for instanceJoyal and Street (1991b) and Kassel (1995). This multiplicity of terminology isperhaps unfortunate because while some use them interchangeably, others (e.g.,Bakalov and Kirillov, 2001; Etingof and Schiffman, 1998) reserve the term tensorcategory for a specific kind of monoidal category. Here we avoid possible confusionby avoiding the term tensor category altogether.

A monoidal category is strict if its associativity isomorphism is canonical.Any (nice enough) strict monoidal category is the representation category of abialgebra, and any (nice enough) rigid strict monoidal category is the representationcategory of a Hopf algebra.8

7We note here that our definitions in this section are not quite as general as the original ones inDrinfeld (1989). Drinfeld’s original definitions involved two invertible elements l and r, or equivalentlya left unit constraint l and a right unit constraint r, satisfying the so-called Triangle Axiom. However,Drinfeld showed, also in Drinfeld (1989), that he could always reduce his quasi-Hopf algebras intoquasi-bialgebras where r = l = 1. Therefore, we will not be worried much about our more restrictivedefinitions.

8Obviously the phrase “nice enough” needs to be explained for this vague sentence to becomea precise (and correct) mathematical statement. For instance, for a Tannaka–Krein type construction(cf. Section 8), we need a fiber functor from our category into the category of vector spaces. We donot attempt this here.

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The well-known Coherence Theorem of MacLane asserts that every nonstrictmonoidal category is in fact tensor equivalent to a strict one. So it is natural to lookat algebraic objects whose representation categories are nonstrict monoidal becausethese have in essence the same type of representation theory as that of bialgebrasand Hopf algebras.

It turns out that quasi-Hopf algebras are just the right structures inthis perspective! More specifically, quasi-bialgebras are precisely those algebras,equipped with counit and comultiplication, whose representation categories aremonoidal. In fact in Kassel (1995), quasi-bialgebras are defined to be thosestructures which have monoidal representation categories. (This is immediatelyfollowed by a proof of the equivalence of this definition to the definition of Drinfeld,1989). Likewise quasi-Hopf algebras have rigid monoidal representation categories.

When the first examples of dynamical quantum groups began to appear,people noticed that they were not Hopf algebras, but still looked very much likethem. Therefore, several researchers focused on quasi-Hopf algebras, as structuresalready under serious inspection for the reasons mentioned earlier, with theexpectation that these could possibly provide the right Hopf-like theory to describedynamical quantum groups. Indeed, it turns out that Felder’s elliptic quantumgroups naturally fit into the framework of quasi-Hopf algebras, see Enriquez andFelder (1998).

Besides Felder’s elliptic quantum groups, the fundamental example of anontrivial quasi-Hopf algebra (i.e., one that is not a Hopf algebra) is the oneconstructed by Drinfeld. This construction involves the monodromy of the Knizhnik–Zamolodchikov equations and is beyond the scope of this note.We refer the interestedreader to the original work of Drinfeld (1989) and the more pedagogical exposition inEtingof and Schiffman (1998).

The representation theory of quasi-Hopf algebras is quite exciting. Category-theoretic arguments ensure that quasi-objects will generally be only a twist awayfrom their nonquasi counterparts.9 MacLane’s theorem about the equivalence ofnonstrict monoidal categories to strict ones may be interpreted as implying that therepresentation theory of a quasi-object is quite similar to that of a nonquasi one.10

Tensor equivalence in the realm of monoidal categories translates into thelanguage of equivalence of quasi-bialgebras and quasi-Hopf algebras in terms oftwists in the algebraic realm. In many cases (when a certain cohomology vanishes),the twist will involve a trivial isomorphism on the level of the algebra. In otherwords, the quasi-structure will be “twistable” into a nonquasi one on the sameunderlying algebra. In other cases we will need to modify the underlying set. Onceagain the reader is referred to Etingof and Schiffman (1998) for a discussion of theuse of twists of quasi-bialgebras and quasi-Hopf algebras.

Overall, quasi-Hopf algebras are interesting structures, with many researchersstill investigating their various applications. Their representation theory is also very

9In this context, some authors prefer to use terms like “gauge transformation” or “skrooching”in place of the term “twist.”

10Note that the answer to the following question is not automatically clear and still unknownto the author: Given a monoidal category � which is the representation category of a quasi-Hopfalgebra, how nice is the strict monoidal category tensor equivalent to �? For instance, it is knownthat �′ will not always be the representation category of a Hopf algebra. (This comment is due toG.B., P.E., and the anonymous referee).

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appealing. Moreover, as we mentioned earlier, they can be used to describe theelliptic quantum groups of Felder. Nonetheless, we continue our search for otheralternatives.

This is mainly due to the general fact that algebraists are not much delightedby a nonassociative or a noncoassociative structure. The relaxation of theseproperties makes description of actions and coactions a lot more cumbersome, andthis is typically not very desirable. In short, we do not have a natural notion ofa comodule or a Hopf module over a quasi-object and any other mathematicalconstruction that relies on these cannot be generalized easily to the quasi-setup.(However, also see Hausser and Nill, Preprint; Schauenburg, 2002) Another issue isthe notion of duality for quasi-objects. Since the definition of a quasi-bialgebra isnot self-dual (the underlying set is an associative algebra but the coalgebra structureis not coassociative), the natural object that should be the dual of a quasi-bialgebrais not a quasi-bialgebra; and the natural object which should be the dual of aquasi-Hopf algebra is not a quasi-Hopf algebra. One needs to define separatelya dual quasi-bialgebra and hence a dual quasi-Hopf algebra; see for instance thepreliminaries of Majid (1992).

5. CANDIDATE 2: WEAK HOPF ALGEBRAS

Following the algebraists’ intuition to avoid noncoassociativity and unnaturalways to define duals, we move on to our second candidate: Weak Hopfalgebras. This time our objects are both algebras and coalgebras, associative andcoassociative, respectively, but the relationship between the two types of structuresis weakened. In particular, we no longer force the coalgebra structure to respectthe unit of the algebra structure. In other words, we drop the requirement that thecomultiplication be unit preserving. We write instead

��1� = 1�1� ⊗ 1�2� �= 1⊗ 1�

This, in turn, forces the counit to be at most weakly multiplicative:

��xy� = ��x1�1����1�2�y��

Here is our precise definition11

Definition 5.1. A (finite) weak Hopf algebra over � is a finite-dimensional12

�-vector space H with the structures of an associative algebra �H�m� 1�, and acoassociative coalgebra �H��� �� such that:

11We should point out that there is an alternative use in the literature for the term weak Hopfalgebra. Some authors use the term to refer to a bialgebra B with a weak antipode S in Hom�B� B�,i.e., id ∗ S ∗ id = id and S ∗ id ∗ S = S, where ∗ is the convolution product; see for instance Li (1998)and Li and Duplij (2002). Wisbauer (2001) briefly discusses how the two concepts are related.

12We define only the finite-dimensional version here. In this context, it seems to be standard torestrict to finite dimensions because it is much easier to dualize the notion in that case. For a moredetailed discussion of why finite-dimensionality is in general preferred, we refer the reader to Böhmet al. (1999).

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(1) The comultiplication � is a (not necessarily unit-preserving) homomorphism ofalgebras such that:

��⊗ id���1� = ���1�⊗ 1��1⊗ ��1�� = �1⊗ ��1�����1�⊗ 1�

(2) The counit � is a �-linear map satisfying the identity:

��fgh� = ��fg�1����g�2�h� = ��fg�2����g�1�h�

for all f� g� h ∈ H ;(3) There is a linear map S � H → H , called an antipode, such that for all h ∈ H ,

m�id⊗ S���h� = ��⊗ id����1��h⊗ 1��

m�S ⊗ id���h� = �id⊗ ����1⊗ h���1��

m�m⊗ id��S ⊗ id⊗ S���⊗ id���h� = S�h��

We can easily see that the condition on the counit implies weakmultiplicativity. We also note that the defining equations for the antipode givenabove would coincide with Eq. (2.1) if � were to preserve the unit. This would alsoimply that the counit would be a homomorphism of algebras, and we would thusend up with a Hopf algebra.

Weak Hopf algebras behave much better with respect to duality. In otherwords, given a weak Hopf algebra A over a field �, the space Hom��A��� canbe given a natural weak Hopf algebra structure, using the canonical pairing �� �Hom��A���× A → �.

Weak Hopf algebras in the above sense were introduced in Böhm et al. (1999),Böhm and Szlachányi (1996), Nill (Preprint), and Szlachányi (1997). Later on it wasobserved that the paragroups of Ocneanu (1988), and the face algebras of Hayashi(1993, 1994, 1998) fit nicely into this description. Many of the original constructionsof weak Hopf algebras were motivated by applications to operator algebras, butmany saw from early on their relationship to “dynamical deformations of quantumgroups.” We refer the reader to Nikshych and Vainerman (2002) for a more detailedexposition of weak Hopf algebras and their relationship to various generalizationsof the idea of quantum groups. This reference also contains discussion of variousexamples of weak Hopf algebras, one of which we present here.

Recall that group algebras were basic examples of Hopf algebras. We look ata nice generalization of these to find some basic examples of weak Hopf algebras.For this purpose, we first need to generalize the notion of a group via the followingdefinition.

Definition 5.2. A groupoid over a set X is a set G together with the followingstructure maps:

(1) A pair of maps s� t � G → X, respectively called the source and the target;(2) A product m, i.e., a partial function m � G×G → G satisfying the following two

properties;

(a) t�m�g� h�� = t�g�, s�m�g� h�� = s�h� whenever m�g� h� is defined;

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(b) m is associative: m�m�g� h�� k� = m�g�m�h� k�� whenever the relevant termsare defined;13

(3) An embedding � � X → G called the identity section such that m���t�g��� g� =g = m�g� ��s�g��� for all g ∈ G;

(4) An inversion map ı � G → G such that m�ı�g�� g� = ��s�g�� and m�g� ı�g�� =��t�g�� for all g ∈ G.

Note that a groupoid over a singleton X = �x� is a group.We should point out here that there are other ways to define a groupoid.

One way is to view it as a particular type of category �X�G� with X makingup the set of objects, such that the morphisms (elements of G) are all invertible.We choose not to consider this categorical description. However, simply by drawingsome diagrams to represent the structure maps defined above, one can clearly seehow the categorical notion may be derived easily.

Groupoids were first introduced in 1926, and since then found applications indifferential topology and geometry, algebraic topology and geometry, and analysis.An accessible introduction to groupoids can be found in Weinstein (1996). For morerigorous accounts see the bibliography there.

Nowwe start with a finite groupoidG and consider the algebra�G. Here we areconsidering the product m�g� h� of two elements ofG to be defined as in the groupoiditself (and in cases when it is not defined, we set the relevant product equal to zero).We define the comultiplication �, the counit � and the antipode S on G by:

��g� = g ⊗ g ��g� = 1 S�g� = ı�g� for all g ∈ G�

(When G is a group, �G is its group algebra, a Hopf algebra). The dual of thegroupoid algebra is also a weak Hopf algebra and can be viewed as a functionalgebra on G. See Nikshych and Vainerman (2002) about this and other examples.

Due to the inherent symmetry in their construction, weak Hopf algebrasappeal to many researchers with strong algebraic preferences. Thus manyalgebraically natural constructions have been generalized to their context, see forinstance Böhm et al. (1999), Böhm and Szlachányi (2000), and Vecsernyés (2003).Even the curious fact that the trivial representation of a weak Hopf algebra mayor may not be indecomposable (Böhm et al., 1999, Prop. 2.15) has interestingimplications; see Böhm and Szlachányi (2000).

Weak Hopf algebras have nice representation categories (Böhm and Szlachányi,2000; Nikshych et al., 2003). In particular, representation categories of semisimplefinite weak Hopf algebras are (multi)fusion categories. A fusion category is asemisimple rigid monoidal category with finitely many simple objects and finite-dimensional homomorphism spaces, such that the endomorphism space of the unitobject is one-dimensional. Relaxing the condition that the endomorphism space of theunit object be one-dimensional gives us a multifusion category. Fusion categories havebeen of much interest in the past few years. We refer the reader to Etingof et al. (2005)

13Defining the multiplication as a partial function allows us to avoid giving a precise domainfor it. However, when it is relevant, the domain of m is called the set of composable pairs. Clearly, itis a subset of G×G.

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ON HOPF ALGEBRAS AND THEIR GENERALIZATIONS 4355

for a presentation of recent results about them, and Calaque and Etingof (Preprint)for another accessible exposition.

A multifusion category can be viewed as the representation category of a (non-unique) semisimple weak Hopf algebra (Hayashi, Preprint; Szlachányi, 2001). In factmany natural category-theoretical constructions can be restated in the language ofweak Hopf algebras. See, for instance, Etingof et al. (2005), where many resultson (multi)fusion categories stated and proved in terms of weak Hopf algebras. Theauthors of Etingof et al. (2005) also show that a certain class of fusion categoriescan also be realized as the representation categories of finite-dimensional semisimplequasi-Hopf algebras. Thus the notion of a fusion category allows us to build aframework in which both quasi-Hopf algebras and weak Hopf algebras can beunderstood.

At this point, we move on to our third candidate, the Hopf algebroid, eventhough our second one is still quite promising. We provide the reader with only onereason, and that is generality. Böhm and Szlachányi (2004) showed that weak Hopfalgebras with bijective antipodes are in fact Hopf algebroids. Xu (1999) constructedparticular Hopf algebroids which fully encoded the information of certain quasi-Hopf algebras associated with solutions of the quantum dynamical Yang–Baxterequation. In other words, our third candidate will in fact provide us with a finestructure which incorporates most of the interesting weak Hopf algebras and all ofthe dynamical quantum groups realized as quasi-Hopf algebras, and then offers ussome more.

6. CANDIDATE 3: HOPF ALGEBROIDS

A natural question for an algebraist would be: What if we do not restrictourselves to commutative rings �? Can one develop the theory of bialgebras inthis more general setting? Takeuchi (1977) described and studied a new algebraicstructure generalizing bialgebras to the noncommutative setting. His structures wereoriginally called ×A-bialgebras. Then in 1996, when developing a geometricallymotivated generalization of the theory of quantum groups, Lu defined similarstructures (Lu, 1996) and called them bialgebroids, to emphasize that the way thesestructures generalized bialgebras resembled the way Poisson groupoids generalizedPoisson groups. A bit later, Xu (2001) gave a similar definition and also impliedthe equivalence of these three notions. Brzezinski and Militaru (2002) providesa complete algebraic proof of this result. Böhm (2005a), Böhm and Szlachányi(2004), and Brzezinski and Militaru (2002) all include a historical account of thesedevelopments. In this note we will mainly be following Böhm (2005a) and Böhmand Szlachányi (2004) for our definitions.

In short, a bialgebroid should be the natural extension of the notion of abialgebra to the world of groupoids. This then implies that a bialgebroid is no longeran algebra, but a bimodule over a noncommutative ring. Here we will describe twosymmetrical notions, that of a left bialgebroid and a right bialgebroid, first developedin Kadison and Szlachányi (2003). We begin first with a left bialgebroid �L, givenby the following data:

(1) Two associative unital rings: the total ring B and the base ring L;

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(2) Two ring homomorphisms: the source sL � L → B and the target tL � Lop → Bsuch that the images of L in B commute, making B an L−L bimodule;

(3) Two maps �L � B → BL ⊗ LB and �L � B → L, which make �B� �L� �L� acomonoid in the category of L−L bimodules.

We also need the following to be satisfied:

b�1�tL�l�⊗ b�2� = b�1� ⊗ b�2�sL�l��

�L�1B� = 1B ⊗ 1B and �L�bb′� = �L�b��L�b

′��

�L�1B� = 1L�

�L�bsL � �L�b′�� = �L�bb

′� = �L�btL � �L�b′���

Above, we used a modification of the Sweedler notation for �L:

�L�b� = b�1� ⊗ b�2��

Note that the first axiom above is essential in dealing with the noncommutativebase ring case; without it, even talking about the multiplicativity of �L (�L�bb

′� =�L�b��L�b

′�) does not make sense. Note also that the source and target maps sL andtL may be used to define four commuting actions of L on B; these then give us, inan obvious way, three other bimodule structures on B.

Similarly, we define a right bialgebroid �R using the following data:

(1) Two associative unital rings: the total ring B and the base ring R;(2) Two ring homomorphisms: the source sR � R → B and the target tR � Rop → B

such that the images of R in B commute, making B an R−R bimodule;(3) Two maps �R � B → BR ⊗ RB and �R � B → R, which make �B� �R� �R� a

comonoid in the category of R−R bimodules.

As in the case of left bialgebroids, we need some conditions on sR� tR� �R� �R

analogous to the conditions on sL� tL� �L� �L:

sR�r�b�1� ⊗ b�2� = b�1� ⊗ tR�r�b

�2��

�R�1B� = 1B ⊗ 1B and �R�bb′� = �R�b��R�b

′��

�R�1B� = 1R�

�R�sR � �R�b�b′� = �R�bb

′� = �R�tR � �R�b�b′��

Here, we used another version of the Sweedler notation for �R:

�R�b� = b�1� ⊗ b�2��

Once again, we can define three other bimodule structures on B using the sourceand the target.

Now all these bimodule structures and more generally the two notions ofbialgebroids are very much related to one another, as expected. For instance if

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�L = �B� L� sL� tL� �L� �L� is a left bialgebroid, then its co-opposite is again a leftbialgebroid: ��L�cop = �B� Lop� tL� sL� �

opL � �L�. Here �

opL is defined as T � �L, where,

as before, T is an analogue of the usual twist map, sending a⊗ b to b ⊗ a.14

The opposite ��L�op defined by the data �Bop� L� tL� sL� �L� �L� is a right bialgebroid.

We refer the reader to Brzezinski and Militaru (2002) for several concreteexamples of bialgebroids. In this reference, Brzezinski and Militaru show a way toassociate a bialgebroid to a braided commutative algebra in the category of Yetter–Drinfeld modules. Moreover, they show that the smash product of a Hopf algebrawith an algebra in the Yetter–Drinfeld category is a bialgebroid if and only if thealgebra is braided commutative. Another interesting construction from Brzezinskiand Militaru (2002) gives a generic method of obtaining bialgebroids from solutionsof the quantum Yang–Baxter equation.

As mentioned above, Takeuchi’s ×A-bialgebras (Takeuchi, 1977), Lu’sbialgebroids (Lu, 1996) and Xu’s bialgebroids (Xu, 2001) were all shown to beequivalent (Brzezinski and Militaru, 2002), and all these are compatible with thedefinition above which we took from Böhm (2005a) and Böhm and Szlachányi(2004). Currently, therefore, there is a universal consensus on what should beaccepted as the correct structure generalizing bialgebras to the noncommutative basering case. How, then, does one develop the theory of Hopf algebras in this setting?In the following paragraphs, we briefly look at how several algebraists approachedthis problem. But first we ask a different question: Why were algebraists interestedin generalizing to the noncommutative base ring case in the first place?

In the mid 1990s, certain finite index depth 2 ring extensions from the theoryof Von Neumann algebras were shown to be related to Hopf algebras (Szymanski,1994). A search for connections with more general extensions followed. In 2003,connections with the bialgebroids of Takeuchi were discovered (Kadison andSzlachányi, 2003). Thus the case corresponding to a bialgebroid with a generalizedantipode was naturally of some interest.

However, this problem proved to be somewhat complicated. The originalantipode suggested in Lu (1996) was not universally accepted, and various otherformulations followed. We refer the reader to Böhm (2005a) for a comparative studyof these various antipodes. Another relevant discussion may be found in Day andStreet (2004). The definition used here is from Böhm and Szlachányi (2004).

In particular, to define a Hopf algebroid, we need two associative unitalrings H and L, and set R = Lop. We consider a left bialgebroid structure �L =�H� L� sL� tL� �L� �L� and a right bialgebroid structure �R = �H�R� sR� tR� �R� �R�

associated to this pair of rings. We require that sL�L� = tR�R� and tL�L� = sR�R� assubrings of H , and

��L ⊗ idH� � �R = �idH ⊗ �R� � �L���R ⊗ idH� � �L = �idH ⊗ �L� � �R�

14Even though the category of L-bimodules is not necessarily symmetric, it is still possible todefine an isomorphism T � B ⊗L B → B ⊗Lop B given by the flip.

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The last ingredient is the antipode, a bijection S � H → H , satisfying

S�tL�l�htL�l′�� = sL�l

′�S�h�sL�l��

S�tR�r′�htR�r�� = sR�r�S�h�sR�r

′�

for all l� l′ ∈ L, r� r ′ ∈ R, and h ∈ H .15

Our final constraint on the antipode S is as follows:

S�h�1��h�2� = sR � �R�h��

h�1�S�h�2�� = sL � �L�h�

for any h ∈ H . Once again, the subscripts and the superscripts on h come from ageneralized version of the Sweedler notation which we use to define the two maps�L and �R:

�L�h� = h�1� ⊗ h�2� ∈ HL ⊗ LH

�R�h� = h�1� ⊗ h�2� ∈ HR ⊗ RH�

Then, we say that the triple � = ��L��R� S� is a Hopf algebroid.16

It is easy to see how this symmetric definition, in terms of two bialgebroidsand a bijective map called an antipode, is analogous to the definition of a Hopfalgebra from a bialgebra and a bijective map called an antipode. However, it maynot be nearly as easy to see why the particular conditions above describe the “rightantipode.” We accept the definition given above without further analysis; a sufficientdiscussion of this issue is provided already in the introduction to Böhm (2005a).

There are many examples of Hopf algebroids in recent literature. Here we willdescribe an interesting example from Böhm and Szlachányi (2004). Let � be a fieldof characteristic different from 2. Consider the group bialgebra �2 presented as aleft bialgebroid with the relevant operations on the single generator t as follows:

t2 = 1 ��t� = t ⊗ t ��t� = 1�

The source and the target maps are the natural ones: s� t � � → �2 definedas s��� = �1 = t���. Now if we define an antipode S � �2 → �2 by settingS�t�= − t, we obtain a Hopf algebroid. (Note that the given antipode is not a Hopfalgebra antipode.)

The notions of integrals and duals for a Hopf algebroid have already beenstudied (Böhm and Szlachányi, 2004; Böhm, 2005b). The natural self-duality ofweak Hopf algebras is not available in this theory. In order to define duals one

15We require that the antipode S be a bijection for the sake of simplicity. For a study of Hopfalgebroids with antipodes which are not necessarily bijective, see Böhm (2005b).

16We should note here that some of the above information in our definition is redundant. Infact one can start with a left bialgebroid �L = �H� L� sL� tL� �L� �L� and an anti-isomorphism S of thetotal ring H satisfying certain conditions, and from here can reconstruct a right bialgebroid �R usingthe same total ring such that the triple ��L��R� S� is a Hopf algebroid. See Böhm (2005a) for moredetails. An even more streamlined set of defining axioms is proposed in Böhm and Brzezinski (2006).

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needs to study the theory of integrals (Böhm and Szlachányi, 2004).17 However, therepresentation theory is still quite nice. In particular, the category of (co)modulesof a bialgebroid is monoidal. Furthermore, left L-bialgebroid structures on anL⊗ Lop ring B are in bijection with monoidal structures on the category ofleft B-modules such that the forgetful functor B� → L�L is strict monoidal(Schauenburg, 1998). Moreover, the category of (left-left) Hopf modules of a Hopfalgebroid � = ��L��R� S� is equivalent to the category of (right) modules over theleft base ring L (Böhm, 2005b).

Hopf algebroids can be used to describe the constructions discussed earlier.As we briefly mentioned at the end of the previous section, both quasi-Hopfobjects and weak Hopf objects fit into the Hopf algebroid picture with somemodifications. In particular, weak Hopf algebras with bijective antipodes are in factHopf algebroids (Böhm and Szlachányi, 2004). Moreover, we already know thatusing the framework of fusion categories, we can describe certain quasi-Hopf objectsin terms of weak Hopf algebras. We can see a connection between quasi-Hopfobjects and Hopf algebroids even more directly, if we look at Xu (1999). There,Xu constructs Hopf algebroids which fully encode the information of certain quasi-Hopf algebras associated with solutions of the quantum dynamical Yang–Baxterequation. Hence, it is clear that there are some very interesting connections betweenthese three generalizations of the notion of Hopf algebras.

7. SHORT INTERLUDE: IS THIS ALL THAT THERE IS?

In Section 3, we motivated our interest in the search for the correctgeneralization of Hopf algebras by emphasizing the need for a Hopf-like object thatcan be used to develop sufficiently the theory of dynamical quantum groups.

Indeed, several researchers already investigated each of these structures witha view toward the theory of dynamical quantum groups. We saw, already, howthese three fit into the theory of dynamical quantum groups and also how theyrelate to one another. In particular, the theory of (multi)fusion categories providesa fresh point of view and may be the right framework for understanding all of thesestructures.

Nonetheless, the search for other generalizations of Hopf algebras still goeson. In the rest of this note, we introduce two more recently developed structuresgeneralizing Hopf algebras. These came up in contexts that are not directlyconnected to the theory of dynamical quantum groups. However they both raisemany new questions, and so they may be of interest to readers who are lookingfor general themes or for other new structures that may be useful for their ownpurposes.

8. A CATEGORY-THEORETICAL APPROACH: HOPF MONADS

As mentioned in Section 2, any finitely generated commutative Hopf algebraover a field � with characteristic 0 is the function algebra of an affine group G over

17It may be interesting to note here that duality for bialgebroids (over a noncommutative base)is much simpler, see Kadison and Szlachányi (2003).

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�. Similarly, a cocommutative Hopf algebra generated by its primitive elements isthe universal enveloping algebra of a Lie algebra �. Readers may find material onthese well-known results in classics like Hochschild (1981) and Milnor and Moore(1965) and in more modern texts like Etingof and Schiffman (1998) and FerrerSantos and Rittatore (2005).

Both these results are in fact particular instances of Tannaka–Krein duality.Traditionally, the origins of Tannaka–Krein theory are attributed to Groethendieck.We refer the reader to Hewitt and Ross (1970) for a comprehensive accountrelating the original works of Tannaka and Krein with more modern treatments.The first modern references in the subject that make extensive use of category theoryare Saavedra Rivano (1972) and Deligne and Milne (1982). Expositions of someTannaka–Krein type theorems presented in the flavor closest to the perspectiveof this note may be found in Etingof and Schiffman (1998). For more details onthe modern approach, with an emphasis on Hopf algebras, quantum groups andmonoidal categories, the reader may find Joyal and Street (1991a) and Majid (1995)useful.

In the language of categories, the philosophy underlying the various resultsthat can be gathered under the heading of Tannaka–Krein theory can be statedas follows: it should be possible to view any nice category as the representationcategory of some algebraic structure. This vague assertion becomes accuratemathematics when one chooses appropriate descriptions for the italicized phrases.

For instance, one precise formulation of the statement of Section 2 aboutcommutative Hopf algebras is the following theorem.

Theorem 8.1 (Deligne and Milne, 1982; Etingof and Schiffman, 1998). Let be asymmetric monoidal additive category defined over an algebraically closed field � ofcharacteristic zero. Suppose that is abelian and has finitely many indecomposableobjects. Suppose, in addition, that there is an exact faithful tensor functor F � → Vec�.Then the groupG of tensor automorphisms of F is a finite group and � Rep�G�.

In literature, one can find similar results for more general Hopf algebras(Schauenburg, 1992a; Ulbrich, 1990) and quasi-Hopf algebras (Majid, 1992). In fact,the search for ever more general Hopf-like objects always includes category-theoretic investigations which look at the representation categories of these objectsand attempt to describe them as some nice types of categories.

Our fourth candidate for the correct generalization of Hopf algebras fitsperfectly into the spirit of this categorical approach to Hopf algebras. This is theso-called Hopf monad structure.18 Very briefly, a Hopf monad is a Hopf-like object ina general category.

More precisely, we begin with a monoidal category , and assume that itis rigid; in other words, we assume that every object in has a left dual and aright dual. We consider first a monad in the sense of MacLane (1998). In short, a

18We note here that there is at least one other category-theoretic Hopf-like object studied inthe recent years. We do not focus on the relevant constructions here; we simply refer the reader toDay et al. (2003) and Day and Street (2004), where the ideas are developed in great detail. Someconnections to the picture from Section 6 may be found in Böhm and Szlachányi (2004) and Böhm(To appear).

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monad on is an algebra in the monoidal category End��. More specifically amonad is an endofuctor T of equipped with two functorial morphisms � � T2 → Tand � � 1 → T which satisfy certain conditions analogous to those describing theproduct and the unit of an algebra.

The appropriate definition in this context for the notion generalizingbialgebras was introduced by Moerdijk (2002) and asserts that a bimonad is a monadT which is also comonoidal. In the precise statement of the definition, there arefunctorial morphisms playing the analogous roles of the coproduct and the counit.The notion of a bimonad is not self-dual, but defining duals is not too complicated.

To make a bimonad T into a Hopf monad, we only require the introductionof two new functorial morphisms, the so-called left antipode and the right antipode,which make use of the left and right duals of the category to encode the left andright duals in the category of T modules. Thus a Hopf monad is a bimonad with twofunctorial morphisms, a left antipode and a right antipode.

We need to remark on our specific terminology here. The term “Hopfmonad” appeared earlier, in Moerdijk (2002). In particular, the “Hopf monads” ofMoerdijk (2002) are precisely the structures that we here called “bimonads.” We arefollowing Bruguières and Virelizier (2007) and Szlachányi (2003) with this choiceof terminology, because it seems quite natural linguistically to reserve the Hopfterm for bimonads with antipode. However, for readers familiar with operad theory,the terminology choice of Moerdijk (2002) may seem more reasonable; see alsoMoerdijk (2001).

Hopf monads in the sense we are using here were introduced and studiedin detail in Bruguières and Virelizier (2007). The authors’ motivation there stemsfrom the various topological invariants constructed via methods of what is nowcalled quantum topology (Kauffman, 1993). The original construction by Reshetikhinand Turaev (1991) of invariants for 3-manifolds using quantum sl2��� and related(ribbon) Hopf algebras has been generalized in recent years to constructions in moreand more general settings. The authors of Bruguières and Virelizier (2007) studyHopf monads as a generalization of Hopf algebras to categories with no braidingso as to provide the ultimate framework to understand all these newer invariants ina uniform manner.

Any finite-dimensional Hopf algebra H over � can be used to construct aHopf monad on the category of finite-dimensional �-vector spaces. More generally,Hopf monads can be constructed using any rigid monoidal category with anunderlying algebraic structure. In other words, rigid monoidal categories whicharise as module categories of most algebraic objects can be used to construct Hopfmonads. In particular, the Hopf monad concept brings together all of the earlierstructures we studied in this note: quasi-Hopf algebras, weak Hopf algebras, andHopf algebroids all can be used to construct Hopf monads in some category of�-vector spaces. For a clarification of the relationship between bimonads andbialgebroids, see Szlachányi (2003). The quasi- and weak Hopf algebra cases areexplicitly discussed in Bruguières and Virelizier (2007).

9. A (POISSON-)GEOMETRIC APPROACH: HOPFISH ALGEBRAS

Finally, we briefly describe a fifth Hopf-like object which generalizes thenotion of a Hopf algebra.

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This object, playfully called a hopfish algebra by its creators, was introducedfirst in Tang et al. (2007) and applied to the problem of describing modules ofirrational rotation algebras in Blohmann et al. (2008). The ideas in its developmentfit in the context of (Poisson) geometry; the reader may find interesting backgroundand motivational discussions in the recent survey Blohmann and Weinstein (2008)of algebraic structures in Poisson geometry.

Hopfish algebras live in a category � whose objects are �-unital algebras andwhose morphisms are bimodules. Then, as the authors of Tang et al. (2007) show,the notions of comultiplication, counit, and the antipode can all be developed in thisframework as particular types of bimodules (i.e., morphisms in the new category).

More precisely, we start with the notion of a sesquiunital sesquialgebra,which is the relevant generalization of bialgebras in this context: A sesquiunitalsesquialgebra over a commutative ring � is a unital �-algebra B equipped with a�B ⊗ B�B�-bimodule � (the coproduct), and a ��� B�-bimodule (ie a right B-module)� (the counit), with the following properties:

(1) (coassociativity axiom) The two �B ⊗ B ⊗ B�B� bimodules �B ⊗ ��⊗B⊗B � and��⊗ B�⊗B⊗B � are isomorphic; and

(2) (counit axiom) The two �B� B� bimodules ��⊗ B�⊗B⊗B � and �B ⊗ ��⊗B⊗B � areboth isomorphic to B.

To move on to the Hopf-like objects one needs an antipode appropriate forthe context. In order to define the right structure that should correspond to theantipode, we begin with a preantipode. In particular, a preantipode for a sesquiunitalsesquialgebra B over � is a left �B ⊗ B�-module S together with an isomorphism ofits �-dual with the right �B ⊗ B�-module HomB�����. One can view S as a �B� Bop�-bimodule, and hence an �-morphism in Hom�B� Bop�. If a preantipode S on B isa free left B-module of rank 1 when considered as a �B� Bop�-bimodule, then it iscalled an antipode for B. A sesquiunital sesquialgebra H equipped with an antipodeS is called a hopfish algebra.

To see the standard examples, we need to begin with a well-known functorof Morita theory. This functor, called modulation in Tang et al. (2007), goes fromthe category of �-unital algebras with algebra homomorphisms as morphisms, tothe category �. Using this functor, one can see that the modulation of biunitalbialgebras will be basic examples of sesquiunital sesquialgebras and the modulationof Hopf algebras will be basic examples of hopfish algebras. In Tang et al. (2007)the authors also show that quasi-Hopf and weak Hopf algebras algebras can bemodulated to yield hopfish algebras.

For more details on hopfish algebras we refer the reader to Tang et al. (2007),Blohmann et al. (2008), and Blohmann and Weinstein (2008). Readers who preferthe more general setting of higher category theory may find that Day et al. (2003)and Day and Street (2004) can perhaps complement the above references on hopfishalgebras.

10. CONCLUSION: WHERE DO WE GO NEXT?

In this note, we focused on five structures that have been introduced in therecent years as generalizations of Hopf algebras. One thing is clear: The notion of

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Hopf algebras proved useful in so many diverse ways that mathematicians of manydifferent persuasions decided that the search for the correct generalization was quitean important task. More significantly, a lot of new and interesting mathematicscame up during these investigations.

Some readers may also have noticed along the way a few open problems thatare looking for solutions. For instance, the question whether the Hopf algebroidsfrom Lu (1996) are a proper subclass of the Hopf algebroids of Section 6 or not isstill not settled. The example we gave at the end of Section 6 would not be a Hopfalgebroid in the sense of Lu (1996). So far, we do not have an example which wouldbe a Hopf algebroid for Lu (1996) and would not be a Hopf algebroid in our sense,but it is not certain that we will never find one. This is an interesting question, asthe Hopf algebroids in Lu (1996) are quite natural structures themselves.

There are also many open questions about (multi)fusion categories and henceabout (weak) Hopf algebras. One of the most fascinating questions involves theexistence (or nonexistence) of finite dimensional semisimple Hopf algebras withnongroup theoretical representation categories. For details on this and many otheropen questions about (multi)fusion categories, we refer the reader once again toEtingof et al. (2005).

The relationship between representation categories of all the variousgeneralizations of Hopf algebras also needs to be studied further. Indeed, a lot ofthe pieces are out there, but there is still more work to be done.

In this note, we pointed out some such connections. We also provided somedetails on the motivations of the mathematicians who developed the relevantstructures. Along the way, we gave our own reasons for being interested ingeneralizations of Hopf algebras. We hope that at this point, the readers havealready decided if any of the five structures fits their own particular needs ormathematical inclinations. If not, they at least have an idea of what leads to followto decide on this matter for themselves.

ACKNOWLEDGMENTS

The author would like to thank Marcelo Aguiar, Gabriella Böhm, PavelEtingof, Ignacio Lopez Franco, Ludmil Hadjiivanov, Mitja Mastnak, SusanMontgomery, Frederic Patras, Fernando Souza, Ivan Todorov, Alexis Virelizier,Alan Weinstein, Milen Yakimov, and the anonymous referee, for helpful commentsand useful questions during various stages of the work that led to this article.

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