Table Of ContentA NOTE ON SPLIT EXTENSIONS OF BIALGEBRAS
7
1 XABIERGARCÍA-MARTÍNEZANDTIMVANDERLINDEN
0
2
Abstract. We prove a universal characterization of Hopf algebras among
r cocommutative bialgebrasoveranalgebraicallyclosedfield: acocommutative
a
bialgebraisaHopfalgebrapreciselywheneverysplitextensionoveritadmits
M
a joindecomposition. We alsoexplainwhythis resultcannot be extended to
anon-cocommutative setting.
3
1
]
A 1. Introduction
R An elementary result in the theory of modules says that in any short exact
. sequence
h
t
ma 0 ,2K k ,2X lr f ,2 Y ,20 f ˝s“1Y
s
[ wherethecokernelf admitsasections,themiddleobjectX decomposesasadirect
sum X –K‘Y. If, however,the given sequence is a short exact sequence of, say,
2
v groups or Lie algebras, then this is of course no longer true: then we can at most
5 deduce that X is a semidirect product K¸Y of K and Y. In a fundamental way,
6 this interpretation depends on, or even amounts to, the fact that X is generated
6
by its subobjects kpKq and spYq. One may argue that, in a non-additive setting,
0
the join decomposition X “kpKq_spYq in the lattice of subobjects of X is what
0
. replaces the direct sum decomposition, valid for split extensions of modules.
1
When the given split extension is a sequence of cocommutative bialgebras (over
0
7 a commutative ring with unit K), we may ask ourselves the question whether such
1 a join decomposition of the middle object in the sequence always exists. Although
: kernels are not as nice as one could expect [2, 3], it is not difficult to see that if Y
v
i is a Hopf algebra then the answer is yes.
X The main point of this note is that this happens only then, at least when K is
r an algebraically closed field. We shall prove, in other words, the following new
a
universal characterization of cocommutative Hopf algebras among cocommutative
bialgebras over K:
All split extensions over a bialgebra Y admit a join decomposition
if and only if Y is a Hopf algebra.
Thisresultisalongthelinesof,andisactuallyavariationon,asimilarcharacteriza-
tionofgroupsamongmonoids,recentlyobtainedin[12,7]. Theretheauthorsshow
2010 Mathematics Subject Classification. 16T05,18A40, 18E99,18G15,20J15.
ThisworkwaspartiallysupportedbyMinisteriodeEconomíayCompetitividad(Spain),grant
MTM2016-79661-P (AEI/FEDER, UE, support included). The first author was also supported
by Xunta de Galicia, grant GRC2013-045 (European FEDER support included), by an FPU
scholarshipoftheMinisteriodeEducación,CulturayDeporte(Spain)andbyaFundaciónBarrié
scholarship. ThesecondauthorisaResearchAssociateoftheFondsdelaRechercheScientifique–
FNRS.
1
2 XABIERGARCÍA-MARTÍNEZANDTIMVANDERLINDEN
that all split extensions (of monoids) over a monoid Y admit a join decomposition
if and only if Y is a group.
In fact something stronger than the existence of a join decomposition may be
proved in a more general context; this will be the subject of Section 2, where we
explore some basic aspects of split extensions of cocommutative bialgebras. In
particular,weshowthatoveraHopf algebra,allsplitextensionsofcocommutative
bialgebras admit a join decomposition (Corollary 2.5). In Section 3 we focus on
the other implication and prove that among cocommutative bialgebras over an
algebraically closed field, only Hopf algebras admit join decompositions of their
splitextensions(Theorem3.5). InthefinalSection4weexplainwhytheconstraint
that the bialgebras in this characterization are cocommutative is essential. As it
turns out, in a non-cocommutative setting, even the very weakest universal join
decomposition condition is too strong.
2. Split extensions over Hopf algebras
A split extension in a pointed category with finite limits C is a diagram
f
K k ,2X lr ,2Y
s
where k is a kernel and s is a section of f. So f ˝s “ 1 , but a priori we are
Y
not asking that f is a cokernel of k, so that pk,fq is a short exact sequence, and
this is not automatically the case. We do always have that K and Y, considered
as subobjects of X,have a trivialintersection. Indeed, using that k is the pullback
of 0ÑY along f, it is easy to check that the pullback of k and s is zero.
In this general context, a join of two subobjects may not always exist, but the
concept introduced in the next definition expresses what we want, and agrees with
the condition that X “kpKq_spYq whenever that expression makes sense—as it
does in any regular category with binary coproducts, for instance [4].
Definition 2.1. A pair of arrows pk,sq with the same codomain X is jointly
extremally epimorphicwhenthearrowsk andscannotbothfactorthroughone
and the same proper subobject of X: whenever we have a diagram
M
⑧⑧:D (cid:12)(cid:18) Zd❄❄❄
⑧ ❄
⑧⑧ m ❄❄
⑧ ❄
⑧ ❄
⑧ (cid:12)(cid:18)
K ,2 X lr Y
k s
wherem isamonomorphism,necessarilymis anisomorphism. We saythatasplit
extension as above is strong when pk,sq is a jointly extremally epimorphic pair;
thecouplepf,sqisthencalledastrong point. Whenwesaythatasplitextension
admits a join decomposition,we mean that it is strong.
The given split extension is said to be stably strong (the couple pf,sq is a
stably strong point) when all of its pullbacks (along any morphism g: W ÑY)
are strong. Following [12], we say that an object Y is protomodular when all
split extensions over Y are stably strong.
Remark 2.2. It is easily seen [12] that the split epimorphism f in a strong point
pf,sq is alwaysthe cokernelof its kernelk. This means, in particular,that all split
A NOTE ON SPLIT EXTENSIONS OF BIALGEBRAS 3
extensionsoveraprotomodularobjectY,aswellasalloftheirpullbacks,are(split)
short exact sequences which admit a join decomposition.
Remark 2.3. When allobjects in C areprotomodular,C is a protomodular cat-
egory in the sense of[5]. Next to Barrexactness,this is one of the keyingredients
in the definition of a semi-abelian category [9], and crucial for results such as the
3ˆ3 Lemma,the Snake Lemma, the Short Five Lemma [6,4], orthe existenceofa
Quillen model category structure for homotopy of simplicial objects [15]. Typical
examplesarethecategoriesofgroups,Liealgebras,crossedmodules,loops,associa-
tive algebras,etc. As recently shownin[8,10], alsothe categoryofcocommutative
Hopf algebras over a field of characteristic zero is semi-abelian.
Given a category with finite products C, we write MonpCq for the category of
internal monoids, and GppCq for the category of internal groups in C. For a com-
mutativering with unitK, we let CoAlgK,coc denote the categoryofcocommutative
coalgebras over K. It is well known [14] that there is an equivalence between the
categoryBiAlgK,coc ofcocommutativebialgebrasoverKandMonpCoAlgK,cocq,which
restricts to an equivalence between the category HopfK,coc of cocommutative Hopf
algebrasoverKandGppCoAlgK,cocq. Thisiseasilyseenusingthatin CoAlgK,coc the
product X ˆY is XbY and 1 is K.
Theorem 2.4. Let C be a category with finite limits. If Y P GppCq then all split
extensions in MonpCq over Y are stably strong. In other words, any internal group
in C is a protomodular object in MonpCq.
Proof. Consider in MonpCq the commutative diagram
Kerpπ1q Kerpfq
l k
M(cid:1)(cid:9) ,2 m ,2W ˆ(cid:12)(cid:18) X π2 ,2 X(cid:12)(cid:18)
U] LR Y LR
x1W,s˝gy π1 s f
(cid:12)(cid:18) (cid:12)(cid:18)
W ,2 Y
g
where the bottom right square is a pullback, m is a monomorphism, and Y is
an internal group. We shall see that m is an isomorphism. Since only limits are
considered, the whole commutative diagram is sent into a category of presheaves
of sets by the Yoneda embedding, in such a way that the internal groups and
internal monoids in it are mapped to ordinary groups and monoids, respectively.
Since the Yoneda embedding reflects isomorphisms, it now suffices to give a proof
inSet. There,itiseasytoseethatmisanisomorphism,sinceeveryelementpw,xq
of W ˆ X can be written as p1,x¨spgpwq´1qq¨pw,sgpwqq, where clearly the first
Y
element belongs to the kernel of π1 and the second one comes from W. (cid:3)
Corollary 2.5. Cocommutative Hopf algebras are protomodular in BiAlg . (cid:3)
K,coc
It follows that, over a Hopf algebra, split extensions of bialgebras are well-
behaved; not only are they short exact sequences, but it is also not hard to see
that the Split Short Five Lemma holds for them, so that equivalences classes of
split extensions may be considered as in ordinary group cohomology.
4 XABIERGARCÍA-MARTÍNEZANDTIMVANDERLINDEN
3. A universal characterization of cocommutative Hopf algebras
Theconverseislessstraightforward. Inthecaseofgroupsandmonoids(C “Set
in Theorem 2.4), it was shown in [12] (resp. in [7]) that all points in Mon over Y
are stably strong (resp. strong) if and only if Y is a group. However, those proofs
involvecoproducts,andsoaYonedaembeddingargumentasinTheorem2.4would
not work.
We now let K be an algebraically closed field. We consider the adjoint pair
G
,2
BiAlgK,coc lr J Mon
Kr´s
where the left adjoint Kr´s is the monoid algebra functor and the right adjoint G
sends a bialgebra B (with comultiplication ∆ and counit ε ) to its monoid of
B B
grouplike elements GpBq“txPB |∆ pxq“xbx and ε pxq“1u.
B B
Lemma 3.1. Kr´s preserves monomorphisms.
Proof. ThefunctorKr´ssendsanymonoidmonomorphismtoabialgebramorphism
of which the underlying vector space map is an injection. (cid:3)
Our aim is to prove that G preservesprotomodular objects: then for any proto-
modularbialgebraB, the monoidofgrouplikeelements GpBq is agroup,sothatB
is a Hopf algebra by [14, 8.0.1.c and 9.2.5].
Proposition 3.2. For any monoid M we have GpKrMsq – M. For any bialge-
braB,thecounitǫ : KrGpBqsÑB oftheadjunctionatB isasplitmonomorphism
B
with retraction π : B ÑKrGpBqs, determined in a way which is functorial in B.
B
Proof. The first statement follows immediately from the definition of KrMs, while
the second depends on [14, 8.0.1.c and 8.1.2]. (cid:3)
Since protomodular objects are closed under retracts [12], it follows that if B is
a protomodular bialgebra, then so is KrGpBqs.
Proposition 3.3. The functor G preserves jointly extremally epimorphic pairs.
Proof. Let pk,sq be a jointly extremally epimorphic pair in BiAlgK,coc. Then the
commutativity of the diagram
KrGpKqsKrGpkqs,2 KrGpXqsKlr rGpsqsKrGpYqs
LR LR LR
πK πX πY
K ,2X lr Y
k s
obtainedviaProposition3.2andthefactthattheupwardpointingarrowsaresplit
epimorphisms imply that the pair pKrGpkqs,KrGpsqsq is jointly extremally epimor-
phic. Now suppose that m is a monomorphism making the diagram on the left
M KrMs
✉✉✉✉✉✉✉✉✉6? (cid:12)(cid:18)(cid:12)(cid:18)m^h❍❍❍❍❍❍❍❍❍ rrrrrrrrrr5= (cid:12)(cid:18)(cid:12)(cid:18)Krmai▲▲s▲▲▲▲▲▲▲▲
GpKq ,2GpXqlr GpYq KrGpKqs ,2 KrGpXqslr KrGpYqs
Gpkq Gpsq KrGpkqs KrGpsqs
A NOTE ON SPLIT EXTENSIONS OF BIALGEBRAS 5
commute. Applying Kr´s we obtain the diagram on the right, in which Krms is
a monomorphism by Lemma 3.1. Since, by the above, the bottom pair is jointly
extremally epimorphic, we see that Krms is an isomorphism. But then also m “
GpKrmsq is an isomorphism, which proves our claim that pGpkq,Gpsqq is a jointly
extremally epimorphic pair. (cid:3)
Proposition3.4. Ifall split extensions overabialgebra Y arestrong, thenall split
extensions over GpYq are strong. In particular, G preserves protomodular objects.
Proof. Consider a split extension
f
K k ,2 X lr ,2GpYq
s
over GpYq. We apply the functor Kr´s, then take the kernel of Krfs to obtain the
split extension of bialgebras
Krfs
L l ,2KrXslr ,2KrGpYqs.
Krss
From Proposition 3.2 it follows that all split extensions over KrGpYqs are strong.
Hence pl,Krssqis a jointly extremally epimorphic pair. Applying the functor G, we
regainthe originalsplitextension,sinceGisarightadjoint,thuspreserveskernels;
but G also preserves jointly extremally epimorphic pairs by Proposition 3.3, so
that the pair pk,sq is jointly extremally epimorphic. As a consequence, all split
extensions over the monoid GpYq are strong, and GpYq is protomodular [7]. (cid:3)
Theorem 3.5. If K is an algebraically closed field and Y is a cocommutative bial-
gebra over K, then the following conditions are equivalent:
(i) Y is a Hopf algebra;
(ii) in BiAlgK,coc, all split extensions over Y admit a join decomposition;
(iii) Y is a protomodular object in BiAlgK,coc.
Proof. (i) implies (iii) is Theorem 2.4, and (ii) is obviously weaker than (iii). For
the proof that (ii) implies (i), suppose that all split extensions over Y admit a join
decomposition. Then Proposition 3.4 implies that in Mon all split extensions over
GpYqarestrong. HenceGpYqisagroupbytheresultin[7],whichmakesY aHopf
algebra by [14, 8.0.1.c and 9.2.5]. (cid:3)
Remark 3.6. This implies that the category BiAlgK,coc cannot be protomodular:
otherwiseallbialgebraswouldbe Hopf algebras. Inparticular,the Split Short Five
Lemma is not generally valid for bialgebras.
4. On cocommutativity
In this final section we study what happens beyond the cocommutative setting.
Here K is a field.
Allobjectsinthe categoryofcocommutativeK-bialgebrassatisfyacertainweak
join decomposition property: being a category of internal monoids (in CoAlgK,coc),
the category BiAlg is unital in the sense of [4]. Given an object Y, it is said
K,coc
to be a unital object [12] when every split extension of the type
πX πY
X lr ,2XˆY lr ,2Y
x1X,0y x0,1Yy
6 XABIERGARCÍA-MARTÍNEZANDTIMVANDERLINDEN
is strong. Notice how this condition is symmetric in X and Y. So protomodular
objects are always unital of course,but in fact this condition is weak enough to be
satisfied by all cocommutative bialgebras over K.
Letus nowleavethe cocommutativesetting andaskourselveswhatitmeansfor
an object Y in BiAlgK to be unital—a very weak thing to ask, compared with the
condition that all split extensions over Y are (stably) strong.
Proposition 4.1. If Y is a unital object of BiAlg , then for every object X we
K
have an isomorphism X ˆY –X bY.
Proof. Given any bialgebra X we may consider the diagram
X lr ρX XbKlr1XbεY ,2X bY lrεXb1Y ,2KbY λY ,2 Y.
– 1XbηY ηXb1Y –
We are first going to prove that the comparison morphism
m“xρ ˝p1 bε q,λ ˝pε b1 qy: X bY ÑXˆY
X X Y Y X Y
is a monomorphism.
Notethatitisalmostneveraninjection;forinstance,takingX “Y tobeaten-
soralgebraTpVq(withcounitε pvq“0forv PV)yieldseasycounterexamples.
TpVq
However,in the category BiAlgK, monomorphisms need not be injective [13, 1].
Let h: Z ÑX bY be a morphism of bialgebras. We write
f “ρ ˝p1 bε q˝h: Z ÑX and g “λ ˝pε b1 q˝h: Z ÑY.
X X Y Y X Y
It suffices to prove that h “ pf bgq˝∆ as vector space maps for our claim to
Z
hold. Indeed,ifhandh1 induce thesamef andg,thenthe givenequalityofvector
space maps proves that h“h1.
Since h is a coalgebra map, we have that ∆ ˝h “ phbhq˝∆ . Writing
XbY Z
τ : X bY ÑY bX for the twist map, we calculate:
X,Y
pf bgq˝∆
Z
“pρ bλ q˝p1 bε bε b1 q˝phbhq˝∆
X Y X Y X Y Z
“pρ bλ q˝p1 bε bε b1 q˝∆ ˝h
X Y X Y X Y XbY
“pρ bλ q˝p1 bε bε b1 q˝p1 bτ b1 q˝p∆ b∆ q˝h
X Y X Y X Y X X,Y Y X Y
“pρ bλ q˝p1 bε bε b1 q˝p∆ b∆ q˝h
X Y X X Y Y X Y
“pρ bλ q˝pρ´1bλ´1q˝h“h.
X Y X Y
It follows that m is a monomorphism. Moreover,m makes the diagram
X bY
p1XbηtYtqt˝tρt´Xtt1tttt5? (cid:12)(cid:18)(cid:12)(cid:18)m_i❏❏❏❏p❏η❏X❏❏b❏1❏Y❏q˝λ´Y1
X ,2 X ˆY lr Y
x1X,0y x0,1Yy
commute. The assumption that Y is unital tells us that m is an isomorphism. (cid:3)
ThisimmediatelyimpliesthatanyunitalobjectY inBiAlgK hastobecocommu-
tative,since ∆ : Y ÑY bY is the morphismofbialgebrasx1 ,1 y: Y ÑY ˆY.
Y Y Y
In particular,the categoryBiAlgK is not unital, so it cannot be protomodular, and
not even Mal’tsev [4].
A NOTE ON SPLIT EXTENSIONS OF BIALGEBRAS 7
However, the situation is actually much worse, since it almost never happens
that XbY is the product of X in Y in the category of all K-bialgebras—noteven
whenbothX andY arecocommutative. Infact,Kitselfcannotbeaprotomodular
object in BiAlgK, since this would imply that all objects of BiAlgK are unital [12].
As we have just seen, this is manifestly false.
ThesameholdsforthecategoryHopf ofHopfalgebrasoverK. Atfirstthismay
K
seem to contradict results in [11] on split extensions of Hopf algebras. We must
keep in mind, though, that for a Hopf algebra H, the map x1 ,0y in the diagram
H
π1 π2
H lr ,2H ˆH lr ,2 H
x1H,0y x0,1Hy
is the kernel of π2, but π2 need not be its cokernel, unless H is cocommutative.
Hence this diagramdoes not representa shortexact sequence,and so neither The-
orem 4.1 nor Theorem 4.2 in [11] saying that H ˆH –H bH applies.
We conclude that it makes no sense to study protomodular objects in BiAlgK or
in HopfK, and we thus restrict our attention to the cocommutative case.
Acknowledgements
ThankstoJoséManuelFernándezVilaboa,IsarGoyvaerts,MarinoGran,James
R. A. Gray, Gabriel Kadjo and Joost Vercruysse for fruitful discussions and useful
comments. We would also like to thank the University of Cape Town and Stellen-
bosch University for their kind hospitality during our stay in South Africa.
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8 XABIERGARCÍA-MARTÍNEZANDTIMVANDERLINDEN
(Xabier García-Martínez) Department of Algebra, University of Santiago de Com-
postela, 15782 Santiago de Compostela, Spain.
E-mail address: [email protected]
(TimVanderLinden)Institut de Recherche en Mathématique et Physique,Univer-
sité catholique de Louvain,chemin du cyclotron 2 bte L7.01.02, B–1348 Louvain-la-
Neuve, Belgium
E-mail address: [email protected]
