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Categorical aspects of Hopf algebras
Robert Wisbauer
Abstract. Hopf algebras allow for useful applications, for example in physics.
Yet they also are mathematical objects of considerable theoretical interest and
it is this aspect which we want to focus on in this survey. Our intention is to
present techniques and results from module and category theory which lead
to a deeper understanding of these structures. We begin with recalling parts
from module theory which do serve our purpose but which may also find other
applications. Eventually the notion of Hopf algebras (in module categories)
will be extended to Hopf monads on arbitrary categories.
Contents
1. Introduction 1
2. Algebras 2
3. Category of A-modules 3
4. Coalgebras and comodules 6
5. Bialgebras and Hopf algebras 9
6. General categories 12
References 18
1. Introduction
The author’s interest in coalgebraic structures and Hopf algebras arose from the
observation that the categories considered in those situations are similar to those in
module theory over associative (and nonassociative) rings. At the beginning in the
1960’s, the study of coalgebras was to a far extent motivated by the classical theory
of algebras over fields; in particular, the finiteness theorem for comodules brought
the investigations close to the theory of finite dimensional algebras. Moreover,
comodules for coalgebras Cover fields can be essentially handled as modules over
the dual algebra C∗.
Bringing in knowledge from module theory, coalgebras over commutative rings
could be handled and from this it was a short step to extend the theory to corings
over non-commutative rings (e.g. [BrWi]). This allows, for example, to consider
for bialgebras Bover a commuatative ring R, the tensorproduct B⊗RBas co-
ring over Band the Hopf bimodules over Bas B⊗RB-comodules. Clearly this
1
2 ROBERT WISBAUER
was a conceptual simplification of the related theory and the basic idea could be
transferred to other situations. Some of these aspects are outlined in this talk.
Since Lawvere’s categorification of general algebra, algebras and coalgebras are
used as basic notions in universal algebra, logic, and theoretical computer science,
for example (e.g. [AdPo], [Gu], [TuPl]).
The categories of interest there are far from being additive. The transfer of Hopf
algebras in module categories to Hopf monads in arbitrary categories provides the
chance to understand and study this notion in this wider context.
Generalisations of Hopf theory to monoidal categories were also suggested in
papers by Moerdijk [Moer], Loday [Lod] and others. Handling these notions in
arbitrary categories may also help to a better understanding of their concepts.
Not surprisingly, there is some overlap with the survey talks [Wi.H] and
[Wi.G]. Here a broader point of view is taken and more recent progress is recorded.
2. Algebras
Let Rbe an associative and commutative ring with unit. Denote by MRthe
category of (right) R-modules.
2.1.Algebras. An R-algebra (A, m, e) is an R-module Awith R-linear maps,
product and unit,
m:A⊗RA→A, e :R→A,
satisfying associativity and unitality conditions expressed by commutativity of the
diagrams
A⊗RA⊗RAm⊗I//
I⊗m
A⊗RA
m
A⊗RAm//A,
AI⊗e//
=
##
G
G
G
G
G
G
G
G
GA⊗RA
m
A
e⊗I
oo
=
{{v
v
v
v
v
v
v
v
v
A.
2.2.Tensorproduct of algebras. Given two R-algebras (A, mA, eA) and
(B, mB, eB), the tensor product A⊗RBcan be made an algebra with product
mA⊗B:A⊗RB⊗RA⊗RBI⊗τ⊗I//A⊗RA⊗RB⊗RBmA⊗mB
//A⊗RB,
and unit eA⊗eB:R→A⊗RB, for some R-linear map
τ:B⊗RA→A⊗RB.
inducing commutative diagrams
B⊗RB⊗RA
I⊗τ
mB⊗I//B⊗RA
τ
B⊗RA⊗RBτ⊗I//A⊗RB⊗BI⊗mB//A⊗RB,
AeB⊗I//
I⊗eB##
G
G
G
G
G
G
G
G
GB⊗RA
τ
A⊗RB,
and similar diagrams derived from the product mAand unit eAof A.
It is easy to see that the canonical twist map
tw : A⊗RB→B⊗R, A, a ⊗b→b⊗a,
satisfies the conditions on τand this is widely used to define a product on A⊗RB.
However, there are many other such maps of interest.
CATEGORICAL ASPECTS OF HOPF ALGEBRAS 3
These kind of conditions can be readily transferred to functors on arbitrary cat-
geories and in this context they are known as distributive laws (e.g. [Be], [Wi.A]).
3. Category of A-modules
Let Abe an associative R-algebra with unit.
3.1.A-modules. A left A-module Mis an R-module with an R-linear map
ρM:A⊗RM→Mwith commutative diagrams
A⊗RA⊗RMI⊗ρM//
m⊗I
A⊗RM
ρM
A⊗RMρM//A,
Me⊗I//
=$$
I
I
I
I
I
I
I
I
IA⊗RM
ρM
M.
The category AMof (unital) left A-modules is a Grothendieck category with A
a finitely generated projective generator.
Properties of (the ring, module) Aare reflected by properties of the category
AM. These interdependencies were studied under the title homological classification
of rings.
To use such techniques for the investigation of the structure of a right A-
module M, one may consider the smallest Grothendieck (full) subcategory of AM
which contains M. For this purpose recall that an A-module Nis called
M-generated if there is an epimorphism M(Λ) →N, Λ an index set, and
M-subgenerated if Nis a submodule of an M-generated module.
3.2.The category σ[M].For any A-module M, denote by σ[M] the full
subcategory of AMwhose objects are all M-subgenerated modules. This is the
smallest Grothendieck category containing M. Thus it shares many properties
with AM, however it need not contain neither a projective nor a finitely generated
generator. For example, one may think of the category of abelian torsion groups
which is just the subcategory σ[Q/Z] of ZM(without non-zero projective objects).
In general, Mneed not be a generator in σ[M]. A module N∈σ[M] with
σ[N] = σ[M] is said to be a subgenerator in σ[M]. Of course, Mis a subgenerator
in σ[M] (by definition). The notion of a subgenerator also plays a prominent role
in the categories considered for coalgebraic structures (e.g. 4.2, 5.3).
An A-module Nis a subgenerator in AMif and only if Aembeds in a finite
direct sum of copies of N, i.e. A →Nk, for some k∈N. Such modules are also
called cofaithful.
The notion of singularity in AMcan be transferred to σ[M]: A module N∈
σ[M] is called singular in σ[M] or M-singular if N'L/K for L∈σ[A] and K⊂L
an essential submodule.
3.3.Trace functor. The inclusion functor σ[P]→AMhas a right adjoint
TM:AM→σ[M], sending X∈AMto
TM(X):=X{f(N)|N∈σ[M], f ∈HomA(N , X)}.
3.4.Functors determined by P∈AM.Given any A-module Pwith S=
EndA(P), there is an adjoint pair of functors
P⊗S−:SM→AM,HomA(P, −) : AM→SM,
4 ROBERT WISBAUER
with (co)restriction
P⊗S−:SM→σ[P],HomA(P, −) : σ[P]→SM.
and functorial isomorphism
HomA(P⊗SX, Y )→HomS(X, HomA(P, Y )),
unit ηX:X→HomA(P, P ⊗SX), x7→ [p7→ p⊗x];
counit εY:P⊗SHomA(P, Y )→Y,p⊗f7→ f(p).
These functors determine an equivalence of categories if and only if ηand εare
natural isomorphisms.
In any category A, an object G∈Ais said to be a generator provided the func-
tor MorA(G, −) : A→Ens is faithful. It is a property of Grothendieck categories
that these functors are even fully faithful ([Nast, III, Teoremˇa 9.1]).
Let P∈AM,S= EndA(P). Then Pis a right S-module and there is a
canonical ring morphism
φ:A→B= EndS(M), a 7→ [m7→ am].
Pis called balanced provided φis an isomorphism.
3.5.Pas generator in AM.The following are equivalent:
(a) Pis a generator in AM;
(b) HomA(P, −) : AM→SMis (fully) faithful;
(c) ε:P⊗SHomA(P, N )→Nis surjective (bijective), N∈AM;
(d) Pis balanced and PSis finitely generated and projective.
Note that the equivalence of (a) and (d) goes back to Morita [Mor]. It need
not hold in more general situations. In [Wi.G, 2.6] it is shown:
3.6.Pas generator in σ[P].The following are equivalent:
(a) Pis a generator in σ[P];
(b) HomA(P, −) : σ[P]→SMis (fully) faithful;
(c) εN:P⊗SHomA(P, N )→Nis sur-(bi-)jective, N∈σ[P];
(d) φ:A→Bis dense, PSis flat and
εVis an isomorphism for all injectives V∈σ[P].
The elementary notions sketched above lead to interesting characterisations of
Azumaya R-algebras (Ra commutative ring) when applied to Aconsidered as an
(A, A)-bimodule, or - equivalently - as a module over A⊗RAo.
In this situation we have for any A⊗RAo-module M,
HomA⊗RAo(A, M ) = Z(M) = {m∈M|am =ma for all a∈A},
and EndA⊗RAo(A)'Z(A), the center of A.
3.7.Azumaya algebras. Let Abe a central R-algebra, that is Z(A) = R.
Then the following are equivalent:
(a) Ais a (projective) generator in A⊗RAoM;
(b) A⊗RAo'EndR(A)and ARis finitely generated and projective;
(c) HomA⊗RAo(A, −) : A⊗RAoM→MRis (fully) faithful;
(d) A⊗R−:MR→A⊗RAoMis an equivalence;
CATEGORICAL ASPECTS OF HOPF ALGEBRAS 5
(e) µ:A⊗RAo→Asplits in A⊗RAoM(Ais R-separable).
The preceding result can also be formulated for not necessarily associative
algebras by referring to the
3.8.Multiplication algebra. Let Abe a (non-associative) R-algebra with
unit. Then any a∈Ainduces R-linear maps
La:A→A, x 7→ ax;Ra:A→A, x 7→ xa.
The multiplication algebra of Ais the (associative) subalgebra
M(A)⊂EndR(A) generated by {La, Ra|a∈A}.
Then Ais a left module over M(A) generated by 1A(in general not projective) and
EndM(A)(A) is isomorphic to the center of A. By σ[M(A)A], or σ[A] for short, we
denote the full subcategory of M(A)Msubgenerated by A. (For algebras Awithout
unit these notions are slightly modified, e.g. [Wi.B]).
This setting allows to define Azumaya also for non-associative algebras (e.g.
[Wi.B, 24.8]).
3.9.Azumaya algebras. Let Abe a central R-algebra with unit. Then the
following are equivalent:
(a) Ais a (projective) generator in M(A)M;
(b) M(A)'EndR(A)and ARis finitely generated and projective;
(c) HomM(A)(A, −) : M(A)M→MRis (fully) faithful;
(d) A⊗R−:MR→M(A)Mis an equivalence.
The fact that the generator property of Aas A⊗RAo-module implies projec-
tivity is a consequence of the commutativity of the corresponding endomorphism
ring (=Z(A)).
Restricting to the subcategory σ[A] we obtain
3.10.Azumaya rings. Let Abe a central R-algebra with unit. Then the
following are equivalent:
(a) Ais a (projective) generator in σ[M(A)A];
(b) M(A)is dense in EndR(A)and ARis faithfully flat;
(c) HomM(A)(A, −) : σ[M(A)A]→MRis (fully) faithful;
(d) A⊗R−:MR→σ[M(A)A]is an equivalence.
For any algebra A, central localisation is possible with respect to the maximal
(or prime) ideals of the center Z(A) and also with respect to central idempotents
of A.
3.11.Pierce stalks. Let Abe a (non-associative) algebra and denote by B(A)
the set of central idempotents of Awhich form a Boolean ring. Denote by Xthe set
of all maximal ideals of B(A). For any x∈ X , the set B(A)\xis a multiplicatively
closed subset of (the center) of Aand we can form the ring of fractions Ax=AS−1.
These are called the Pierce stalks of A(e.g. [Wi.B, Section 18]). They may be
applied for local-global characterisations of algebraic structures, for example (see
[Wi.B, 26.8], [Wi.M]):
3.12.Pierce stalks of Azumaya rings. Let Abe a central (non-associative)
R-algebra with unit. Then the following are equivalent:
6 ROBERT WISBAUER
(a) Ais an Azumaya algebra;
(b) Ais finitely presented in σ[A]and
for every x∈ X ,Axis an Azumaya ring;
(c) for every x∈ X ,Axis an Azumaya ring with center Rx.
Considering the (A, A)-bimodules for an associative ring Amay be regarded
as an extension of the module theory over commutative rings to non-commutative
rings. Using the multiplication algebra M(A) we can even handle non-associative
algebras A. In particular, we can describe a kind of central localisation of semiprime
algebras A. This may help to handle notions in non-commutative geometry.
One problem in transferring localisation techniques from semiprime commu-
tative rings to semiprime non-commutative rings is that the latter need not be
non-singular as one-sided modules. To guarentee this, additional assumptions on
the ring are required (e.g. Goldie’s theorem). This is not the case if we consider A
in the category σ[A].
A module N∈σ[A] is called A-singular if N'L/K for L∈σ[A] and K⊂L
an essential M(A)-submodule (see 3.2). The following is shown in [Wi.B, Section
32].
3.13.Central closure of semiprime algebras. Let Abe a semiprime R-
algebra and b
Athe injective hull of Ain σ[M(A)A]. Then
(i) Ais non-singular in σ[M(A)A].
(ii) EndM(A)(b
A)is a regular, selfinjective, commutative ring, called the extended
centroid.
(iii) b
A=AHomM(A)(A, b
A) = AEndM(A)(b
A)and allows for a ring structure (for
a, b ∈A,α, β ∈EndM(A)(b
A)),
(aα)·(bβ):=ab αβ .
This is the (Martindale) central closure of A.
(iv) b
Ais a simple ring if and only if Ais strongly prime (as an M(A)-module).
Not surprisingly - the above results applied to A=Zyield the rationals Qas
the (self-)injective hull of the integers Z.
A semiprime ring Ais said to be strongly prime (as M(A)-module) if its central
closure is a simple ring, and an ideal I⊂Ais called strongly prime provided the
factor ring A/I is strongly prime.
Using this notion, an associative ring Ais defined to be a Hilbert ring if any
strongly prime ideal of Ais the intersection of maximal ideals. This is the case if
and only if for all n∈N, every maximal ideal J ⊂ A[X1, . . . , Xn] contracts to a
maximal ideal of Aor - equivalently - A[X1, . . . , Xn]/Jis finitely generated as an
A/J ∩A-module (liberal extension). This yields a natural noncommutative version
of Hilbert’s Nullstellensatz (see [KaWi]).
The techniques considered in 3.13 were extended in Lomp [Lomp] to study the
action of Hopf algebras on algebras.
4. Coalgebras and comodules
The module theory sketched in the preceding section provides useful techniques
for the investigation of coalgebras and comodules. In this section Rwill denote a
commutative ring.
CATEGORICAL ASPECTS OF HOPF ALGEBRAS 7
4.1.Coalgebras. An R-coalgebra is a triple (C, ∆, ε) where Cis an R-module
with R-linear maps
∆ : C→C⊗RC, ε :C→R,
satisfying coassociativity and counitality conditions.
The tensor product C⊗RDof two R-coalgebras Cand Dcan be made to
a coalgebra with a similar procedure as for algebras. For this a suitable linear
map τ0:C⊗RD→D⊗RCis needed leading to the corresponding commutative
diagrams (compare 2.2).
The dual R-module C∗= HomR(C, R) has an associative ring structure given
by the convolution product
f∗g= (g⊗f)◦∆ for f, g ∈C∗,
with unit ε.
Replacing g⊗fby f⊗g(as done in the literature) yields a multiplication
opposite to the one given before. This does not do any harm but has some effect
on the formalism considered later on.
4.2.Comodules. Aleft comodule over a coalgebra Cis a pair (M, M) where
Mis an R-module with an R-linear map (coaction)
M:M→C⊗RM
satisfying compatibility and counitality conditions.
A morphism between C-comodules M, N is an R-linear map f:M→N
with N◦f= (I⊗f)◦M. The set (group) of these morphisms is denoted by
HomC(M, N ).
The category CMof left C-comodules is additive, with coproduct and cokernels
- but not necessarily with kernels.
The functor C⊗R−:MR→CMis right adjoint to the forgetful functor
CM→MR, that is, there is an isomorphism
HomC(M, C ⊗RX)→HomR(M, X), f 7→ (ε⊗I)◦f,
and from this it follows that
EndC(C)'HomR(C, R) = C∗,
which is a ring morphism - or antimorphism depending on the choice for the con-
volution product (see 4.1).
Cis a subgenerator in CM, since any C-comodule leads to a diagram
R(Λ)
h
C⊗RR(Λ) '//
I⊗h
C(Λ)
0//M%M
//C⊗RM,
where his an epimorphism for some index set Λ.
Monomorphisms in CMneed not be injective maps and - as a consequence -
generators Gin CMneed not be flat modules over their endomorphism rings and
the functor HomC(G, −) : CM→Ab need not be full.
All monomorphisms in CMare injective maps if and only if Cis flat as an
R-module. In this case CMhas kernels.
There is a close relationship between comodules and modules.
8 ROBERT WISBAUER
4.3.C-comodules and C∗-modules. Any C-comodule M:M→C⊗RM
is a C∗-module by the action
˜M:C∗⊗MI⊗%M
//C∗⊗C⊗Mev⊗I//M.
For any M, N ∈CM, HomC(M, N )⊂HomC∗(M, N ) and hence there is a
faithful functor
Φ : CM→C∗M,(M, M)7→ (M, ˜M)
To make Φ a full functor, the morphism (natural in Y∈MR)
αY:C⊗RY→HomR(C∗, Y ), c ⊗y7→ [f7→ f(c)y],
has to be injective for all Y∈MR(α-condition, see [BrWi, 4.3]):
4.4.CMa full module subcategory. The following are equivalent:
(a) Φ : CM→C∗Mis a full functor;
(b) Φ : CM→σ[C∗C] (⊂C∗M)is an equivalence;
(c) αYis injective for all Y∈MR;
(d) CRis locally projective.
This observation shows that under the given conditions the investigation of the
category of comodule reduces to the study of C∗-modules, more precisely, the study
of the category σ[C∗C] (see [BrWi], [Wi.F]).
As a special case we have (see [BrWi, 4.7]):
4.5.CMa full module category. The following are equivalent:
(a) Φ : CM→C∗Mis an equivalence;
(b) αis an isomorphism;
(c) CRis finitely generated and projective.
4.6.Natural morphism. Applying HomR(X, −) to the morphism αYleads
to the morphism, natural in X, Y ∈MR,
˜αX,Y : HomR(X, C ⊗RY)→HomR(X, HomR(C∗, Y )) '
→HomR(C∗⊗RX, Y ).
If αYis a monomorphism, then αX,Y is a monomorphism,
if αYis an isomorphism, then αX,Y is an isomorphism, X, Y ∈MR.
The latter means that the monad C∗⊗R−and the comonad C⊗R−form an
adjoint pair of endofunctors on MR, while the former condition means a weakened
form of adjunction.
It is known (from category theory) that, for the monad C∗⊗R−, the right
adjoint HomR(C∗,−) is a comonad and the category C∗Mis equivalent to the
category MHomR(C∗,−)of HomR(C∗,−)-comodules (e.g. [B¨oBrWi, 3.5]).
Thus α:C⊗ − → HomR(C∗,−) may be considered as a comonad morphism
yielding a functor
˜
Φ : CM−→ MHomR(C∗,−),
M→C⊗RM7−→ M→C⊗RMαM
−→ HomR(C∗, M).
As noticed in 4.4 and 4.5, this functor is fully faithful if and only if αis injective;
it is an equivalence provided αis a natural isomorphism.
CATEGORICAL ASPECTS OF HOPF ALGEBRAS 9
5. Bialgebras and Hopf algebras
Combining algebras and coalgebras leads to the notion of
5.1.Bialgebras. An R-bialgebra is an R-module Bcarrying an algebra struc-
ture (B, m, e) and a coalgebra structure (B , ∆, ε) with compatibility conditions
which can be expressed in two (equivalent) ways
(a) m:B⊗RB→Band e:R→Bare coalgebra morphisms;
(b) ∆ : B→B⊗RBand ε:B→Rare algebra morphisms.
To formulate this, an algebra and a coalgebra structure is needed on the tensor-
product B⊗RBas defined in 2.2 and 4.1 (with the twist tw map taken for τ). The
twist map (or a braiding) can be avoided at this stage by referring to an entwining
map
ψ:B⊗RB→B⊗RB,
which allows to express compatibility between algebra and coalgebra structure by
commutativity of the diagram (e.g. [B¨oBrWi, 8.1])
B⊗RBm//
∆⊗IB
B∆//B⊗RB
B⊗RB⊗RBIB⊗ψ//B⊗RB⊗RB.
m⊗IB
OO
In the standard situation this entwining is derived from the twist map as
ψ= (m⊗I)◦(I⊗tw) ◦(δ⊗I) : B⊗RB→B⊗RB, a ⊗b7→ a1⊗ba2.
This is a special case of 6.12 (see also [B¨oBrWi, 8.1]).
5.2.Hopf modules. Hopf modules for a bialgebra Bare R-modules Mwith
aB-module and a B-comodule structure
ρM:B⊗RM→M, ρM:M→B⊗RM,
satisfying the compatibility condition
ρM(bm) = ∆(b)·ρM(m),for b∈B, m ∈M.
Here we use that - due to the algebra map ∆ - the tensor product N⊗RMof
two B-modules can be considered as a left B-module via the diagonal action
b·(m⊗n) = ∆(b)(m⊗n) = Xb1n⊗b2m.
This makes the category BMmonoidal.
If the compatibility between mand ∆ is expressed by an entwining map
ψ:B⊗RB→B⊗RB(see 5.1), then the Hopf modules are characterised by
commutativity of the diagram
B⊗RMρM//
I⊗ρM
MρM
//B⊗RM
B⊗RB⊗RMψ⊗I//B⊗RB⊗RM.
I⊗ρM
OO
10 ROBERT WISBAUER
5.3.Category of Hopf modules. Morphisms between two B-Hopf modules
Mand Nare R-linear maps f:M→Nwhich are B-module as well as B-comodule
morphisms. With these morphisms, the Hopf modules form an additive category,
we denote it by B
BM. Certainly Bis an object in B
BM, but in general it is neither a
generator nor a subgenerator.
As mentioned above, B⊗RBhas a (further) left B-module structure induced
by ∆, we denote the resulting module by B⊗bB. It is not difficult to see that
B⊗bBis an object in B
BMand is a subgenerator in this category (e.g. [BrWi,
14.5]).
Similarly, one may keep the trivial B-module structure on B⊗RBbut introduce
a new comodule structure on it. This is again a Hopf module, denoted by B⊗cB,
and is also a subgenerator in B
BM(e.g. [BrWi, 14.5]).
As for comodules, monomorphisms in B
BMneed not be injective maps unless B
is flat as an R-module.
If Bis locally projective as an R-module, the comodule structure of the Hopf
modules may be considered as B∗-module structure and their module and comodule
structures yield a structure as module over the smash product B#B∗. In this case,
B
BMis isomorphic to σ[B#B∗B⊗bB], the full subcategory of B#B∗Msubgenerated
by B⊗bB(or B⊗cB) (e.g. [BrWi, 14.15]).
5.4.Comparison functor. For any R-bialgebra B, there is a comparison
functor
φB
B:RM→B
BM, X 7→ (B⊗RX, m ⊗IX,∆⊗IX),
which is full and faithful since, by module and comodule properties, for any X, Y ∈
MR,
HomB
B(B⊗RX, B ⊗RY)'HomB
R(X, B ⊗RY)'HomR(X, Y ),
with the trivial B-comodule structure on X. In particular, EndB
B(B)'R.
5.5.The bimonad HomR(B, −).As mentioned in 4.6, for a monad (comonad)
B⊗R−, the right adjoint functor HomR(B, −), we denote it by [B, −], is a comonad
(monad).
An entwining ψ:B⊗RB→B⊗RBmay be seen as an entwining between the
monad B⊗R−and the comonad B⊗R−,
˜
ψ:B⊗RB⊗R− → B⊗RB⊗R−
and this induces an entwining between the Hom-functors (see [B¨oBrWi, 8.2])
b
ψ: [B, [B , −]] →[B, [B, −]].
This allows to define [B, −]-Hopf modules (similar to 5.2), the category M[B,−]
[B,−], and
a comparison functor (with obvious notation)
φ[B,−]
[B,−]:RM→M[B,−]
[B,−], X 7→ ([B, X],∆∗
X, m∗
X).
5.6.Antipode. For any bialgebra B, a convolution product can be defined on
the R-module EndR(B) by putting, for f, g ∈EndR(B), (compare 4.1)
f∗g=m◦(f⊗g)◦∆.
This makes (EndR(B),∗) an R-algebra with identity e◦ε.
CATEGORICAL ASPECTS OF HOPF ALGEBRAS 11
An antipode is an S∈EndR(B) which is inverse to the identity map IBof B
with respect to ∗, that is S∗IB=e◦ε=IB∗Sor - explicitely -
m◦(S⊗IB)◦∆ = e◦ε=m◦(IB⊗S)◦∆.
If Bhas an antipode it is called a Hopf algebra.
The existence of an antipode is equivalent to the canonical map
γ:B⊗RBδ⊗I//B⊗RBI⊗m//B⊗RB
being an isomorphism (e.g. [BrWi, 15.2]).
The importance of the antipode is clear by the (see [B¨oBrWi, 8.11])
5.7.Fundamental Theorem. For any R-bialgebra B, the following are equiv-
alent:
(a) Bis a Hopf algebra (i.e. has an antipode);
(b) φB
B:RM→B
BMis an equivalence;
(c) φ[B,−]
[B,−]:RM→M[B,−]
[B,−]is an equivalence;
(d) HomB
B(B, −) : B
BM→RMis full and faithful.
If BRis flat then (a)-(d) are equivalent to:
(e) Bis a generator in B
BM.
Recall that for BRlocally projective, B
BMis equivalent to σ[B#B∗B⊗bB] and
thus we have:
5.8.Corollary. Let Bbe an R-bialgebra with BRlocally projective. Then the
following are equivalent:
(a) Bis a Hopf algebra;
(b) Bis a subgenerator in B
BMand B#B∗is dense in EndR(B);
(c) Bis a generator in B
BM.
These characterisations are very similar to those of Azumaya rings (see 3.10).
This indicates, for example, that Pierce stalks may also be applied to characterise
(properties of) Hopf algebras.
The notion of bialgebras addresses one functor with algebra and coalgebra struc-
tures. More general, one may consider relationships between distinct algebras and
coalgebras:
5.9.Entwined algebras and coalgebras. Given an R-algebra (A, m, e) and
an R-coalgebra (C, ∆, ε), an entwining (between monad A⊗R−and comonad
C⊗R−) is an R-linear map
ψ:A⊗RC→C⊗RA,
inducing certain commutative diagrams. This notion was introduced in Brzezi´nski
and Majid [BrMa] and is a special case of a mixed distributive law (see 6.5).
Entwined modules are defined as R-modules Mwhich are modules (M, M) and
comodules (M, M), inducing commutativity of the diagram (e.g. [BrWi, 32.4])
A⊗M%M//
IA⊗%M
M%M
//C⊗M
A⊗C⊗Mψ⊗I//C⊗A⊗M.
I⊗%M
OO
12 ROBERT WISBAUER
With morphisms which are A-module as well as C-comodule maps, the entwined
modules form a category denoted by C
AM.
C⊗RAis naturally a right A-module and ψcan be applied to define a left
A-module structure on it,
a·(c⊗b) = ψ(a, c)b, for a, b ∈A,c∈C.
Moreover, a coproduct can be defined on C⊗RA, making C⊗RAan A-coring, a
notion which extends the notion of R-coalgebras to non-commutative base rings A.
The category C
AMof entwining modules can be considered as C⊗AM, the category
of left comodules over the coring C⊗RA(e.g. [BrWi, 32.6]).
To get a comparison functor as in 5.4, we have to require that Ais an object
in C
AM; this is equivalent to the existence of a grouplike element in the A-coring
C⊗RA(e.g. [BrWi, 28.1 and 23.16]).
5.10.Galois corings. Let (A, C ) be an entwined pair of an algebra Aand a
coalgebra C. Assume that Ais an entwined module by A:A→C⊗AA. Then
there is a comparison functor
φC
A:MR→C
AM:X7→ (A⊗RX, m ⊗I, A⊗I),
which is left adjoint to the (coinvariant) functor HomC
A(A, −) : C
AM→MR.
Moreover, B= HomC
A(A, A) is a subring of A, HomC
A(A, C ⊗RA)'A, and
evaluation yields a (canonical) map
γ:A⊗BA→C⊗RA.
Now C⊗RAis said to be a Galois A-coring provided γis an isomorphism (e.g.
[BrWi, 28.18]). This describes coalgebra-Galois extensions or non-commutative
principal bundles. If - in this case - ABis a faithfully flat module, then the functor
MB→C
AM:Y7→ (A⊗BY, m ⊗I , A⊗I)
is an equivalence of categories.
This extends the fundamental theorem for Hopf algebras to entwined structures:
If A=C=His a Hopf algebra, then (H, H ) is an entwining, B=R, and the
resulting γis an isomorphsism if and only if Hhas an antipode (see 5.6).
6. General categories
As seen in the preceding sections, the notions of algebras, coalgebras, and Hopf
algebras are all buit up on the tensor product. Hence a first step to generalisation is
to consider monoidal categories (V,⊗,I). For example, entwining structures in such
categories are considered in Mesablishvili [Me]. Furthermore, opmonoidal monads
Ton Vwere considered by Brugui`eres and Virelizier (in [BruVir, 2.3]) which may
be considered as an entwining of the monad Twith the comonad − ⊗ T(I). The
generalised bialgebras in Loday [Lod], defined as Schur functors (on vector spaces)
with a monad structure (operads) and a specified coalgebra structure, may also be
seen as a generalisation of entwining structures [Lod, 2.2.1].
However, algebras and coalgebras also show up in more general categories
as considered in universal algebra, theoretical computer science, logic, etc. (e.g.
Gumm [Gu], Turi and Plotkin [TuPl], Ad´amek and Porst [AdPo]). It is of some
interest to understand how the notion of Hopf algebras can be transferred to these
settings. In what follows we consider an arbitrary category A.
CATEGORICAL ASPECTS OF HOPF ALGEBRAS 13
6.1.Monads on A.Amonad on Ais a triple (F, m, e) with a functor F:A→
Aand natural transformations
m:F F →F,e:IA→F,
inducing commutativity of certain diagrams (as for algebras, see 2.1).
F-modules are defined as X∈Obj(A) with morphisms X:F(X)→Xand
certain commutative diagrams (as for the usual modules, see 3.1).
The catgegory of F-modules is denoted by AF. The free functor
φF:A→AF, X 7→ (F(X), mX)
is left adjoint to the forgetful functor UF:AF→Aby the isomorphism, for X∈A,
Y∈AF,
MorAF(F(X), Y )→MorA(X, UF(Y)), f 7→ f◦eX.
6.2.Comonads on A.Acomonad on Ais a triple (G, δ, ε) with a functor
G:A→Aand natural transformations
δ:G→GG,ε:G→IA,
satisfying certain commuting diagrams (reversed to the module case).
G-comodules are objects X∈Obj(A) with morphisms X:X→G(X) in A
and certain commutative diagrams.
The category of G-comodules is denoted by AG. The free functor
φG:A→AG, X 7→ (G(X), δX)
is right adjoint to the forgetful functor UG:AG→Aby the isomorphism, for
X∈AG,Y∈A,
MorAG(X, G(Y)) →MorA(UG(X), Y ), f 7→ εY◦f.
Monads and comonads are closely related with
6.3.Adjoint functors. A pair of functors L:A→B,R:B→Ais said to be
adjoint if there is an isomorphism, natural in X∈A,Y∈B,
MorB(L(X), Y )'
−→ MorA(X, R(Y)),
also described by natural transformations η:IA→RL,ε:LR →IB. This implies
a monad (RL, RεL, η) on A, a comonad (LR, LηR, ε) on B.
Lis full and faithful if and only if ε:GF →IAis an isomorphism.
Lis an equivalence (with inverse R) if and only if εand ηare natural isomor-
phisms.
6.4.Lifting properties. Compatibility between endofunctors F, G :A→A
can be described by lifting properties. For this, let F:A→Abe a monad and
G:A→Aany functor on Aand consider the diagram
AF
G//
UF
AF
UF
AG//A.
If a Gexists making the diagram commutative it is called a lifting of G. The
questions arising are:
(i) does a lifting Gexist ?
14 ROBERT WISBAUER
(ii) if Gis a monad - is Gagain a monad (monad lifting)?
(iii) if Gis a comonad - is Galso a comonad (comonad lifting)?
For R-algebras Aand B, (i) together with (ii) may be compared with the
definition of an algebra structure on A⊗RBand leads to diagrams similar to those
in 2.1.
For an R-algebras Aand an R-coalgebra C, (i) together with (iii) corresponds
to the entwinings considered in 5.9.
We formulate this in the general case (e.g. [Wi.A, 5.3]).
6.5.Mixed distributive law (entwining). Let (F, m, e)be a monad and
(G, δ, ε)a comonad. Then a comonad lifting G:AF→AFexists if and only if
there is a natural transformations
λ:F G →GF
inducing commutativity of the diagrams
FFG mG//
F λ
F G
λ
F GF λF//GF F Gm //GF,
F G F δ //
λ
F GG λG//GF G
Gλ
GF δF//GGF,
GeG//
Ge !!
C
C
C
C
C
C
C
CF G
λ
GF,
F G F ε //
λ
F
GF.
εF
==
{
{
{
{
{
{
{
{
Entwining is also used to express compatibility for an endofunctor which is a
monad as well as a comonad. Notice that the diagrams in 6.5 either contain the
product mor the coproduct δ, the unit eor the counit ε. Additional conditions
are needed for adequate compatibility.
6.6.(Mixed) bimonad. An endofunctor B:A→Ais said to be a (mixed)
bimonad if it is
(i) a monad (B, m, e) with e:I→Ba comonad morphism,
(ii) a comonad (B, δ, ε) with ε:B→Ia monad morphism,
(iii) with an entwining functorial morphism ψ:BB →BB,
(iv) with a commutative diagram
BB m//
Bδ
Bδ//BB
BBB ψB//BBB.
Bm
OO
6.7.(Mixed) B-bimodules. For a bimonad Bon A,(mixed) bimodules are
defined as B-modules and B-comodules Xsatisfying the pentagonal law
B(X)%X//
B(%X)
X%X
//B(X)
BB(X)ψX//BB(X).
B(%X)
OO
CATEGORICAL ASPECTS OF HOPF ALGEBRAS 15
B-bimodule morphisms are B-module as well as B-comodule morphisms. We denote
the category of B-bimodules by AB
B.
There is a comparison functor (compare 5.4)
φB
B:A→AB
B, A 7−→ [BB(A)µA
→B(A)δA
→BB(A)],
which is full and faithful by the isomorphisms, functorial in X, X0∈A,
MorB
B(B(X), B(X0)) 'MorB(B(X), X 0)'MorA(X, X0).
In particular, this implies EndB
B(B(X)) 'EndA(X), for any X∈A.
Following the pattern in 5.6 we define an
6.8.Antipode. Let Bbe a bimonad. An antipode of Bis a natural transfor-
mation S:B→Bleading to commutativity of the diagram
Bε//
δ
Ie//B
BB
SB//
BS //BB
m
OO
We call BaHopf bimonad provided it has an antipode.
As for Hopf algebras (see 5.6) we observe that the canonical natural transfor-
mation
γ:BB δB//BBB B m //BB
is an isomorphism if and only if Bhas an antipode (e.g. [BrWi, 15.1]).
The Fundamental Theorem for Hopf algebras states that the existence of an
antipode is equivalent to the comparison functor being an equivalence (see 5.7). To
get a corresponding result in our general setting we have to impose slight conditions
on the base category and on the functor (see [MeWi, 5.6]):
6.9.Fundamental Theorem for bimonads. Let Bbe a bimonad on the
category Aand assume that Aadmits colimits or limits and Bpreserves them.
Then the following are equivalent:
(a) Bis a Hopf bimonad (see 6.8);
(b) γ=Bm ·δB :BB →BB is a natural isomorphism;
(c) γ0=mB ·BδB :BB →BB is a natural isomorphism;
(d) the comparison functor φB
B:A→B
BAis an equivalence.
Recall that for an R-module B, the tensor functor B⊗R−has a right adjoint
and we have observed in 5.5 that a bialgebra structure on Bcan be transferred to
the adjoint HomR(B, −).
As shown in [MeWi, 7.5], this applies for general bimonads provided they have
a right adjoint:
6.10.Adjoints of bimonads. Let Bbe an endofunctor of Awith right adjoint
R:A→A. Then Bis a bimonad (with antipode) if and only if Ris a bimonad
(with antipode).
16 ROBERT WISBAUER
As a special case we have that for any R-Hopf algebra H, the functor HomR(H, −)
is a Hopf monad on MR. This is not a tensor functor unless HRis finitely generated
and projective.
As pointed out in 5.1, no twist map (or braiding) is needed on the base category
to formulate the compatibility conditions for bialgebras (and bimonads). There may
exist a kind of braiding relations for bimonads based on distributive laws.
6.11.Double entwinings. Let Bbe an endofunctor on the category Awith
a monad structure B= (B, m, e) and a comonad structure B= (B , δ, ε).
A natural transformation τ:BB →BB is said to be a double entwining
provided
(i) τis a mixed distributive law from the monad Bto the comonad B;
(ii) τis a mixed distributive law from the comonad Bto the monad B.
6.12.Induced bimonad. Let τ:BB →BB be a double entwining with
commutative diagrams
BB B ε //
m
B
ε
Bε//1,
1e//
e
B
δ
BeB //BB,
1e//
=
>
>
>
>
>
>
>B
ε
1,
BB
δδ
m//Bδ//BB
BBBB B τ B //BBBB.
mm
OO
Then the composite
τ:BB δB //BBB Bτ //BBB mB //BB
is a mixed distributive law from the monad Bto the comonad Bmaking (B, m, e, δ, ε, τ )
a bimonad (see 6.6).
It is obvious that for any bimonad B, the product BB is again a monad as well
as a comonad.
BB is also a bimonad provided τsatisfies the Yang-Baxter equation, that is,
commutativity of the diagram
BBB τ B //
Bτ
BBB Bτ //BBB
τB
BBB τ B //BBB B τ //BBB .
If this holds, then BB is a bimonad with
product m:BBBB Bτ B //BBBB mm //BB ,
coproduct δ:BB δδ //BBBB Bτ B //BBBB,
entwining =
τ:BBBB B τ B //BBBB τ τ //BBBB Bτ B //BBBB .
Finally, if τis a double entwining satisfying the Yang-Baxter equation and
τ2= 1, then an opposite bimonad Bop can be defined for Bwith
CATEGORICAL ASPECTS OF HOPF ALGEBRAS 17
product m·τ:BB τ
−→ BB m
−→ B,
coproduct τ·δ:Bδ
−→ BB τ
−→ BB.
If Bhas an antipode S, then S:Bop →Bis a bimonad morphism provided
that
τ·BS =SB and τ·BS =SB.
In the classical theory of Hopf algebras, the category MRof R-modules over a
commutative ring R(or vector spaces) is taken as category Aand tensor functors
B⊗R−are considered (which have right adjoints HomR(B, −)). Here the Fun-
damental theorem for bimonads 6.9 implies that for Hopf algebras 5.7. The twist
map provides a braiding on MRand this induces a double entwining on the tensor
functor B⊗R−.
We conclude with a non-additive example of our notions.
6.13.Endofunctors on Set. On the category Set of sets, any set Ginduces
an endofunctor
G× − :Set →Set, X 7→ G×X,
which has a right adjoint
Map(G, −) : Set →Set, X 7→ Map(G, X).
Recall (e.g. from [Wi.A, 5.19]) that
(1) G× − is a monad if and only if Gis a monoid;
(2) G× − is comonad with coproduct δ:G→G×G,g7→ (g, g );
(3) there is an entwining morphism
ψ:G×G→G×G, (g, h)7→ (gh, g ).
Thus for any monoid G,G× − is a bimonad and
Hopf monads on Set. For a bimonad G× −, the following are equivalent:
(a) G× − is a Hopf monad;
(b) Mor(G, −)is a Hopf monad;
(c) Gis a group.
Here we also have a double entwining given by the twist map
τ:G×G× − 7→ G×G× −,(a, b, −)7→ (b, a, −).
6.14.Remarks. After reporting about bialgebras and the compatibilty of their
algebra and coalgebra part, we considered the entwining of distinct algebras and
coalgebras (see 5.9). Similarly, one may try to extend results for bimonads to the
entwining of a monad Fand a distinct comonad Gon a category Aand to head for
a kind of Fundamental Theorem, that is, an equivalence between the category AG
F
and, say, a module category over some coinvariants. For this one has to extend the
notion of (co)modules over rings to (co)actions of (co)monads on functors and to
introduce the notion of Galois functors. Comparing with 5.10, a crucial question is
when Fallows for a G-coaction. For this a grouplike natural transformation I→G
is needed. In cooperation with B. Mesablishvili the work on these problems is still
in progress.
18 ROBERT WISBAUER
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Mathematical Institute, Heinrich Heine University, 40225 D¨
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E-mail address:wisbauer@math.uni-duesseldorf.de
