The pro-unipotent completion [expository notes]

11 Pages·2012·0.277 MB·English

by  Alberto Vezzani

THE PRO-UNIPOTENT COMPLETION
ALBERTOVEZZANI
The aim of these notes is to give an overview of Quillen’s construction of the pro-unipotent completion of an
abstractgroup(oraLiealgebra). Someconsequencesoftheformulasandspecialcasesareexplainedinmoredetail.
1. INTRODUCTION
WestartbypresentingtheworkofQuillen[5,AppendixA],andtranslatingitintothesettingofalgebraicgroups,
followingtheapproachof[2]. WealsofollowCartier[1]forspecificfactsonHopfalgebras. Fromnowon,wework
overthebasefieldQ.
Definition1. GivenanabstractgroupΓ[resp. aLiealgebrag],thepro-unipotentcompletionΓun [resp. gun]isthe
universalpro-unipotentalgebraicgroupGendowedwithamapΓ→G(Q)[resp. g→Lie(G)].
Let us focus on the case of groups. The meaning of the definition is that there is a map u: Γ → Γun(Q) such
that for any map f: Γ → G(Q) to the Q-points of a pro-unipotent algebraic group G, there exists a unique map
φ: Γun → G such that f = φ(Q)◦u. In other words, we are looking for a left adjoint to the functor G (cid:55)→ G(Q)
defined from pro-unipotent algebraic groups to abstract groups. Sadly enough, we anticipate that we will need to
restricttoasubcategoryofabstractgroupsinordertofindsuchafunctor.
Thecategoryofpro-unipotentalgebraicgroupsisafullsubcategoryofthecategoryofpro-affinealgebraicalgebraic
groups (the category of formal filtered limits of affine algebraic groups over quotients). This category is clearly
equivalent to the opposite category of Hopf algebras (not necessarily finitely presented). What we need to do is
thereforetoassociatetoanabstractgroupaparticularcommutativeHopfalgebraoverQ. Therearesomewell-known
examplesofadjointpairswhichareclosetoreachingthisaim.
Proposition2. Thereisanadjointpairoffunctors
Q[·]:Gps(cid:29)Alg:(·)×
betweenthecategoryofabstractgroupsGpsand(notnecessarilycommutative)Q-algebrasAlg. ThealgebraQ[Γ]
is called the group algebra associated to Γ, it is isomorphic to (cid:76) Qg as a vector space and the product is the
g∈Γ
Q-bilinearextensionofg·h=ghforallg,h∈Γ.
AnygroupalgebraQ[Γ]canbeendowedwiththestructureofaHopfalgebrawithrespecttothemaps
∆: g (cid:55)→g⊗g S: g (cid:55)→g−1 (cid:15): g (cid:55)→1
forallg ∈Γ.ThereforethefunctorQ[·]factorsoverHA,thecategoryofHopfalgebrasoverQ.Alsothisnewfunctor
hasanadjoint:
Proposition3. Thereisanadjointpairoffunctors
Q[·]:Gps(cid:29)HA:G
betweenthecategoryofabstractgroupsandthecategoryof(notnecessarilycommutative)Q-HopfalgebraswhereG
associatestoaHopfalgebraRthesetofgroup-likeelements:
GR:={x∈R: ∆x=x⊗x,(cid:15)(x)=1}
endowedwiththeproductinheritedfromR.
1
Proof. This comes from the previous proposition. Indeed, given a map Q[Γ] → R induced by f: Γ → R×, the
diagram
Q[Γ] (cid:47)(cid:47)Q[Γ]⊗Q[Γ]
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47)
R R⊗R
commutesifandonlyifallimagesoftheelementsofGsatisfy∆x=x⊗x. Moreover,sinceanygroup-likeelement
xsatisfies(cid:15)(x)=1,alsotheaugmentationispreserved. Wearelefttoprovethatallgroup-likeelementsareinvertible
andthatGRisagroup. Thisfollowsfromthefollowinglemma. (cid:3)
Lemma4. Thesetofgroup-likeelementscanequivalentlybedefinedinthefollowingway:
GR:={x∈R×: ∆x=x⊗x}.
Moreover,ifx∈GRthenS(x)∈GRandS(x)x=xS(x)=1,whereS istheantipodemap.
Proof. Ifx(cid:54)=0and∆x=x⊗x,thenx=∇((cid:15)⊗id)(∆x)=(cid:15)(x)x,hence(cid:15)(x)=1. FromtheHopfalgebraaxioms
∇◦(S⊗id)◦∆=1=∇◦(id⊗S)◦∆weconcludeS(x)x=1=xS(x). Also,from(S⊗S)◦∆=∆op◦S,we
conclude∆(S(x))=S(x)⊗S(x). (cid:3)
ThesituationforLiealgebrasissimilar: if(g,[·,·])isaLiealgebraoverQ,thentheuniversalenvelopingalgebra
UgisthequotientofthetensoralgebraT(g)=(cid:76) g⊗nbytherelationsx⊗y−y⊗x=[x,y]forallx,y ∈g. It
n∈N
satisfiesauniversalproperty:
Proposition5. (i) Thereisanadjointpairoffunctors
U:LA(cid:29)Alg:for
between the category LA of Lie algebras over Q and (not necessarily commutative) Q-algebras. The functor
forsendsaQ-algebraRtotheLiealgebrastructureoverRinducedbycommutators.
(ii) TheleftadjointfactorsoverthecategoryofHopfalgebrasbyendowingtheuniversalenvelopingalgebraUgof
aLiealgebragwiththestructureofaHopfalgebrawithrespecttothemaps
∆: x(cid:55)→x⊗1+1⊗x S: x(cid:55)→−x (cid:15): x(cid:55)→0
forallx∈g.
(iii) Thereisanadjointpairoffunctors
U:LA(cid:29)HA:P
betweenthecategoryofLiealgebrasand(notnecessarilycommutative)Q-HopfalgebraswhereP associatesto
aHopfalgebraRthesetofprimitiveelements:
PR:={x∈R: ∆x=x⊗1+1⊗x}
endowedwiththeLiebracketinducedbycommutators.
Proof. Theproofisanalogoustothecaseofgroups. Weremarkthatanyprimitiveelementxsatisfies(cid:15)(x) = 0from
theequalityx=∇((cid:15)⊗1)(∆x)=(cid:15)(x)+x. (cid:3)
Note that Q[Γ] and Ug are cocommutative, but not necessarily commutative (this happens iff Γ or g is abelian).
Sinceourinitialaimwastoassociatetoagroup(ortoaLiealgebra)acommutative,butnotnecessarilycocommutative
Hopfalgebra,thenaturalideaisnowto“takeduals”. Takingdualsofvectorspacesisadelicateoperationwhenever
thedimensionisnotfinite. Hence,wewillneedtorestricttoaparticularcasewherethesituationisself-reflexiveas
inthefinite-dimensionalcase.
Definition 6. A topological vector space V is linearly compact if it is homeomorphic to limV/V , where V/V are
←− i i
discreteandfinitedimensional,andthemapsinthediagramarequotients.
Wewilldenoteby(·)∨thedualspaceandby(·)∗thetopologicaldual.
Example 7. If V is a discrete vector space, we will always enodow its dual V∨ with the linearly compact topology
limW∨bylettingW varyamongthesubvectorspacesofV whicharefinitedimensional.
←− i i
Proposition8. (1) IfV isdiscrete[resp. linearlycompact],then(V∨)∗ ∼=V [resp. (V∗)∨ ∼=V].
2
(2) IfV isdiscrete[resp. linearlycompact],then(V ⊗V)∨ ∼=V∨⊗ˆV∨[resp. (V⊗ˆV)∗ ∼=V∗⊗V∗].
Inparticular,dualitydefinesanequivalenceofcategoriesbetweencommutativeHopfalgebraandthecategoryof
linearlycompactHopfalgebras.
Our attempt is now to use these dualities in order to obtain a commutative and cocommutative Hopf algebra out
of Q[Γ] or Ug. By what just stated, we need to get a complete topological Hopf algebra. Any Hopf algebra R is
augmented by the counit (cid:15). Let I denote the augmentation ideal. We can endow R with the I-adic topology, and
completeitwithrespecttoit.
Definition9. AcompleteHopfalgebraisacompletetopologicalaugmentedalgebra(cid:15): R → R/I ∼= Q,homeomor-
phictolimR/Ik andendowedwithamap∆: R→R⊗ˆRthatfitsintheusualdiagramsofHopfalgebras.Wedenote
←−
thecategoryofcompleteHopfalgebrasbyCHA.
We remark that our definition differs slightly from the one of [5] since Quillen introduces also the choice of a
filtration.
Example10. IfRisaHopfalgebra,thenitsI-adiccompletionRˆisacompleteHopfalgebra. Inparticular,Γ(cid:55)→Q(cid:100)[Γ]
andg(cid:55)→U(cid:99)gdefinefunctorstothecategoryCHA.
Thefollowingpropositionisaformalconsequenceofthepreviousones.
Proposition11. Thereareadjointpairsoffunctors
Qˆ[·]:Gps(cid:29)CHA:G
Uˆ:LA(cid:29)CHA:P
whereG andP aredefinedlikebefore.
WeremarkthatifoneconsiderstheadjunctionQ[·]: Gps (cid:29) HA :G,thenΓ ∼= GQ[Γ]forallabstractgroupsΓ.
Thisisnolongertruefortheaboveadjunctionsince, aswewillsee, GQˆ[Γ]definesthenilpotent, uniquelydivisible
closureofΓ. ThesameholdsalsointhecaseofLiealgebras: g∼=P(Ug)butP(Uˆg)isthenilpotentclosureofg.
WecannowisolateinCHAthefullsubcategoryCofthosealgebrasRwhicharealsolinearlycompact.SinceR∼=
limR/Ik,thisconditionisequivalenttoaskingthatR/Ik isfinitedimensionalforallk. Sincethisisobviouslytrue
←−
fork =1,byinductionweconcludethatthisisequivalenttothefinitedimensionalityofallIk/Ik+1. Multiplication
definesasurjection(I/I2)⊗k →Ik/Ik+1,andthereforethisisequivalenttoimposingI/I2finitedimensional.
Example12. (1) LetΓbeanabstractgroup. Then
IQˆ[Γ]/IQˆ2[Γ] ∼=IQ[Γ]/IQ2[Γ] ∼=Γab⊗Z Q
whereΓabistheabelianizationofΓ,andwherethelastisomorphismisinducedby(g−e)(cid:55)→g. Inparticular,
ifΓissuchthatΓab⊗ZQhasfiniterank,thenQˆ[Γ]islinearlycompact.WedenotebyG(cid:103)psthefullsubcategory
ofGpsofobjectssatisfyingthisproperty.
(2) LetgbeaLiealgebra. Then
I /I2 ∼=I /I2 ∼=g/[g,g]
Uˆg Uˆg Ug Ug
wherethelastisomorphismisinducedbyx (cid:55)→ x. Inparticular,ifgissuchthatg/[g,g]hasfiniterank,then
Uˆgislinearlycompact. WedenotebyL(cid:103)AthefullsubcategoryofLAofobjectssatisfyingthisproperty.
Byduality,thecategoryCisequivalenttoafullsubcategoryofHAop,andhencetoasubcategoryofpro-affine
algebraicgroups.
Proposition13. LetG=SpecRbeapro-affinealgebraicgroup.ThenR∨ ∈CifandonlyifPRisfinitedimensional
andthe“conilpotencyfiltration”
(1) 0⊂C :=Ann I ⊂...⊂C :=Ann Ik+1 ⊂...
0 R k R
isexhaustive,i.e. ifR=(cid:83)C ,whereI istheaugmentationidealofR∨.
i
3
Proof. WeknowthatR∨ liesinCifI/I2 isfinitedimensional,andifR∨ ∼= limR∨/Ik. Thedualspace(R∨/Ik)∗
←−
coincideswithC . Inparticular,C =QandC =Q⊕PR. Indeed,anelementxofRliesinAnnI2 ifandonlyif
k 0 1
∆x=y⊗1+1⊗z,andbyusingtheaxiomsofHopfalgebra,thisturnsouttobeequivalentto∆x=x⊗1+1⊗x
if(cid:15)(x)=0.
WethenconcludethatI/I2 isfinitedimensionalifandonlyif(I/I2)∗ =(ker(R/I2 →R/I))∗ =C /C =PR
1 0
isfinitedimensional,andthatR∨ ∼=limR∨/Ik ifandonlyifR=lim(R∨/Ik)∗ =limC . (cid:3)
←− −→ −→ k
Weremarkthattheconilpotencyfiltration{C }justdefinedcoincideswiththeoneofCartier[1,3.8(A)].Thisis
i
partofthefollowingproposition,whoseproofcomesbyinductionfromthepreviousone.
Proposition14. LetG = SpecRbeapro-affinealgebraicgroupandletR¯ beitsaugmentationideal. Theelements
ofthefiltration(1)canbedefinedequivalentlyinthefollowingways:
(1) C = Q⊕ker∆¯ ,where∆¯: R¯ → R¯⊗R¯ mapsxto∆x−x⊗1−1⊗xand∆¯ : R¯ → R¯⊗n mapsxto
i n n
(∆¯ ⊗id⊗...⊗id)(∆¯ x).
n
(2) C /C isthetrivialsubrepresentationofGinsideR/C .
i+1 i i
Definition 15. A pro-unipotent algebraic group is a pro-affine algebraic group SpecR such that the conilpotency
filtration(1)isexhaustive.Aunipotentalgebraicgroupisapro-unipotentalgebraicgroupSpecRsuchthatRisfinitely
presented. Thecategorydefinedby[pro-]unipotentalgebraicgroupswillbedenotedwithUAG[resp. pUAG].
Inparticular,aunipotentalgebraicgroupGsuchthattheLiealgebraPO(G)isfinitedimensionaldefinesanobject
O(G)∨ofC.
Ourdefinitionisdifferentfromthe“standard”one. Wenowprovetheequivalenceofthetwonotions. Recallthat
UT isthesubgroupofGL definedbyupper-triangularmatrices,whichhave1’sonthemaindiagonal.
n n
Proposition16. LetGbeapro-affinealgebraicgroup. Thefollowingareequivalent:
(i) Gispro-unipotent.
(ii) Foreverynon-zerorepresentationV ofG,thereexistsanon-zerovectorv ∈V suchthatG·v =v.
IncaseGisanalgebraicgroup,thepreviousconditionsareequivalentto:
(iii) GisisomorphictoasubgroupofUT forsomen.
n
Proof. Let G = SpecR. Suppose (i) is satisfied. Then any representation ρ: V → V ⊗R admits an exhaustive
filtration {V } where V := {v ∈ V ⊗C }. In particular, V is a trivial subrepresentation since if v ∈ V , then
k k k 0 0
ρv =v⊗1. Wenowprove(ii)byshowingthatV =0impliesV =0.
k k+1
(cid:80)
It can be explicitly seen that ∆C ⊂ C ⊗ C . Therefore if x ∈ V , then (1 ⊗ ∆)(ρx) lies in
i a+b=i a b k+1
(cid:80)
V ⊗C ⊗C . Since a and b can’t be both bigger than k, if follows that V is mapped to 0 via the
a+b=k+1 a b k+1
compositemap
1⊗∆ π
V →V ⊗R → V ⊗R⊗R→V ⊗R/C ⊗R/C
k k
Ontheotherhand,thepreviousmapcoincides(bytheaxiomsofcomodules)with
ρ⊗1 π
V →V ⊗R → V ⊗R⊗R→V ⊗R/C ⊗R/C
k k
which is an injection since V = 0. Viceversa, if any representation V has a non-zero trivial subrepresentation, by
k
inductiononecandefineanascendingfiltration{V }suchthatV /V isthetrivialsubrepresentationofV/V . Since
i i+1 i i
any element of V generates a finite dimensional subrepresentation, it follows that this filtration is exhaustive. The
conilpotency filtration corresponds to the filtration associated to the representation defined on R itself. This proves
(i)⇔(ii).
If V is finite dimensional, then (by induction on its dimension, since V (cid:54)= 0) it is an extension of trivial repre-
0
sentations. Itfollowsinparticularthat, withrespecttoasuitablebasis, ρ: G → GL factorsoverUT . IfGisan
V n
algebraicgroup,onecanapplythisfacttoafaithfulfinitedimensionalrepresentationtoprove(iii). (cid:3)
Being pro-unipotent is closed under quotients (using the condition (ii) for example). Hence, if SpecR is a pro-
unipotentalgebraicgroup,thenanysub-HopfalgebraR(cid:48) ofRdefinesapro-unipotentalgebraicgroup. Itfollowsthat
thecategoryofpUAGcoincideswiththepro-objectsofUAGandCisasubcategoryofit.
WeremarkthatourdefinitionisslightlydifferentfromtheoneofCartier[1,endofp. 53],sincewedonotimpose
thatRhascountabledimension.
4
Example17. (1) LetΓbeanabstractgroupsuchthatΓab⊗ZQhasfiniterank. ThenQˆ[Γ]endowedwiththeI-
adictopologyislinearlycompact,sinceitishomeomorphictolimQˆ[Γ]/IkandallI havefinitecodimension.
←− k
Inparticular,Spec(Qˆ[Γ]∗)ispro-unipotent.
(2) Similarly,ifgisaLiealgebrasuchthatg/[g,g]hasfiniterank,thenSpec(Uˆg∗)ispro-unipotent.
(3) ConsiderG = G . TheHopfalgebraisR = Q[t], itsdualvectorspaceis(cid:81)Q(cid:15) where(cid:15) (ti) = δ . By
a k k k,i
duality,theaugmentationidealisI = ker((Q → R)∨) = {φ: R → Q: φ(1) = 0} = (cid:104)(cid:15) (cid:105) . Theproduct
k k>0
isdefinedviadualityfromthecoproductofRwhichsendsttot⊗1+1⊗t,sothat
(cid:88)
((cid:15) ·(cid:15) )(ti)=((cid:15) ⊗(cid:15) )(∆ti)=((cid:15) ⊗(cid:15) )( tα⊗tβ)=δ
h k h k h k h+k,i
α+β=i
andhence(cid:15) ·(cid:15) =(cid:15) .
h k h+k
Therefore,I isgeneratedasanidealby(cid:15) := (cid:15) . Inparticular,R∨ ∼= Q[[(cid:15)]],whichisI-adicallycomplete.
1
WeconcludethatG isunipotent.
a
(4) ConsiderG=G . Inthiscase,thecoproductonR=Q[t,t−1]sendsttot⊗t. Therefore
m
((cid:15) ·(cid:15) )(ti)=((cid:15) ⊗(cid:15) )(∆ti)=((cid:15) ⊗(cid:15) )(ti⊗ti)=δ
h k h k h k h,k,i
WeconcludeinparticularthatI2 =I,andhenceG isnotunipotent.
m
Let’snowconsiderthefunctorswehaveobtainedfrompUAGtoGpsandtoLA.
Proposition18. LetG=SpecRbeapro-affinealgebraicgroup. ThenG(R∨)∼=G(Q)andP(R∨)∼=LieG.
Proof. The unit of R∨ is the counit (cid:15) of R and the counit ε of R∨ is φ (cid:55)→ φ(1). Also, for any φ ∈ R∨ and any
x,y ∈R,(∆φ)(x⊗y)=φ(xy). Therefore
∆φ=φ⊗φ⇔φ(xy)=φ(x)φ(y) ε(φ)=1⇔φ(1)=1
and
∆φ=φ⊗1+1⊗φ⇔φ(xy)=φ(x)(cid:15)(y)+(cid:15)(x)φ(y)⇔φ(I2)=φ(R/I)=0
sothatGR∨ ∼=G(Q)andPR∨ ∼=(I/I2)∨ ∼=LieG. (cid:3)
Thepreviouspropositioncanbeslightlygeneralized.
Proposition 19. Let G = SpecR be a pro-affine algebraic group. For any Q-algebra S, we consider the natural
structureofS-coalgebrawithcounitonR∨⊗S anddenotewithG(R∨⊗S)thegroup-likeelementswithrespectto
it(i.e. elementsxsuchthat∆x=x⊗x,(cid:15)(φ)=1). ThenG(R∨⊗S)∼=G(S).
Proof. WehaveR∨ ⊗S ∼= HomQ(R,S)and(R∨ ⊗S)⊗S (R∨ ⊗S) ∼= HomQ(R⊗R,S). Withrespecttothese
identifications,thecomultiplicationsendsaQ-linearmapφto∆φ: x⊗y (cid:55)→φ(xy)andthecounit(cid:15)sendsφtoφ(1).
Therefore,φisgroup-likeifandonlyifitisaringhomomorphism,asclaimed. (cid:3)
Proposition20. IfGisaunipotentalgebraicgroup,thenG(Q)ab⊗ZQandLieGhavefiniterank.
Proof. ThisistrueforUT ,henceforanyunipotentalgebraicgroup. (cid:3)
n
RecallthatwehavedenotedbyG(cid:103)ps[resp. byL(cid:103)A]thesubcategoryofGps[resp. ofLA]constitutedbygroups
Γ such that Γab ⊗Z Q has finite rank [resp. by Lie algebras g such that g/[g,g] has finite rank]. Let pG(cid:103)ps [resp.
pL(cid:103)A] denote the associated category of pro-objects. It is equivalent to the category of topological groups Γ which
are homeomorphic to l←im−Γ/Γi, with Γ/Γi discrete and lying in G(cid:103)ps [resp. topological Lie algebras g which are
homeomorphictol←im−g/gi,withg/gidiscreteandlyinginL(cid:103)A].
Byourconstruction,wehavethereforeobtainedadjunctionpairs
Spec((Qˆ[·])∗):pG(cid:103)ps(cid:29)pUAG:(Q)
(2)
Spec(Uˆ(·)∗):pL(cid:103)A(cid:29)pUAG:Lie
whichareactuallywhatwewerelookingforfromtheverybeginning!
Corollary21(Quillen’sformula). (1) LetΓbeanobjectofG(cid:103)ps(e.g.ifΓisfinitelygenerated).ThenSpec((Qˆ[Γ])∗)∼=
Γun.
5
(2) LetgbeanobjectofL(cid:103)A(e.g. ifgisfinitedimensional). ThenSpec((Uˆg)∗)∼=gun.
Proof. This follows formally from the previous adjunctions. We focus on the case of groups. Suppose that G =
Spec(O(G))isinpUAG.ThenGisafilteredlimitofunipotentalgebraicgroupsG withPO(G )finite-dimensional.
i i
Inparticular,O(G )∨aswellasQˆ[Γ]lieinCandtherefore:
i
Hom(Γ,G(Q))=limHom(Γ,G (Q))=limHom(Γ,GO(G )∨)=limHom(Qˆ[Γ],O(G )∨)=
←− i ←− i ←− i
i i i
=limHom(Spec((Qˆ[Γ])∗),G )=Hom(Spec((Qˆ[Γ])∗),G).
←− i
i
(cid:3)
Weconcludeourpanoramaonadjunctionsbythefollowingremark. TherearewellknownadjunctionsfromSet
toGpsandfromSettoLA. Wewonderwhethertheyarecompatiblewiththerestofthediagram. Inwhatfollows
wearecruciallyusingthefactthatweareworkingincharacteristic0.
Let R be in CHA. Suppose that x lies in the augmentation ideal. Then the series (cid:80)xk has a limit which we
k!
denotebyexpx.
Proposition22. Theadjunctiondiagram
(cid:110)(cid:118)(cid:118)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:110)(cid:47)(cid:47)(cid:110)(cid:54)(cid:54) Set(cid:104)(cid:104)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:47)(cid:47)(cid:40)(cid:40)
Gps(cid:111)(cid:111) CHA(cid:111)(cid:111) LA
commutesuptoanequivalenceoffunctorsinducedbytheexponentialmap.
Proof. It suffices to prove that the two right adjoints are equivalent. The claim then follows from the following
lemma. (cid:3)
Lemma23. LetRbeanobjectofCHA. Thenx∈PR⇔expx∈GR.
Proof. Thisfollowsfromtheequalities
∆x=x⊗1+1⊗x⇔∆expx=exp(∆x)=exp(x⊗1+1⊗x)=exp(x)⊗exp(x)
whichcomefromthedefinitionoftheexponential. (cid:3)
2. QUILLEN’STHEOREMANDCOROLLARIES
Theorem24(Quillen). LetMGps[resp. MLA]bethesubcategoryofG(cid:103)ps[resp. L(cid:103)A]constitutedbynilpotent,
uniquelydivisiblegroups[resp. nilpotentalgebras]. Thentheadjunctions(2)induceequivalenceofcategories:
(cid:47)(cid:47) (cid:47)(cid:47)
pMGps(cid:111)(cid:111) pUAG(cid:111)(cid:111) pMLA
∼ ∼
Wedevotetherestofthesectiontosktchingtheproofofthistheorem.
We begin with a useful fact from category theory. It is a generalization of well-known cases (Galois correspon-
dences,closuresofsubsets,algebraicsetsetc.) whichusuallyinvolveorderedsetsratherthangeneralcategories.
Proposition25. LetF:C(cid:29)D:U beanadjunction. Thefollowingareequivalent
(1) FUF →F isanisomorphismoffunctors.
(2) U →UFU isanisomorphismoffunctors.
Moreover,ifthepreviousconditionsaresatisfied,thentheadjointpairdecomposesintothreeadjointpairs
UF (cid:47)(cid:47) F (cid:47)(cid:47) (cid:47)(cid:47)
C(cid:111)(cid:111) CUF (cid:111)(cid:111) ∼ DFU (cid:111)(cid:111) D
U FU
whereCUF [resp. DFU]isthefullsubcategoryofC[resp. D]constitutedbytheobjectsX suchthatX → UFX
is an isomorphism [resp. FUX → X is an isomorphism], and where the pair in the center is an equivalence of
categories.
Proof. Thefirstpartisstandardcategorytheory(e.g. [3,Lemma4.3]),thesecondisastraightforwardexercise. (cid:3)
6
Inparticular,inordertoprovethetheoremwearelefttoprovethefollowingfacts:
(i) FU ∼=id.
UF
(ii) Γ∈pMGps⇔Γ∈pG(cid:103)ps .
UF
(iii) g∈pMLA⇔g∈pL(cid:103)A .
whereU iseither(Q)orLieandF isitsrespectiveleftadjoint.
Corollary26. Inordertoprovethetheorem,itsufficestoprove
(I) IfG∈pUAG,thenG(Q)∈pMGpsandLieG∈pMLA.
UF
(II) Γ∈pMGps⇒Γ∈pG(cid:103)ps .
UF
(III) g∈pMLA⇒g∈pL(cid:103)A .
(IV) (Q)andLiereflectisomorphisms.
Proof. The only non-trivial fact is the proof of condition (i). Let X be in pUAG. Then by (I) and (II), UX →
UFUX isanisomorphism. BecausethecompostionUX → UFUX → UX istheidentity, wealsoconcludethat
UFUX →UX isanisomorphism. By(IV),weconcludeFUX ∼=X,aswanted. (cid:3)
SketchoftheproofofQuillen’stheorem. Conditions(II),(III),(IV)areprovedbyQuillenatthelevelofGps (cid:28)
CHA (cid:28) LA (see [5, Theorem A.3.3]). Condition (I) comes from the fact that if G is unipotent then LieG and
G(Q)arenilpotent(itsufficestocheckthisforUTn), LieGisfinitedimensionalandG(Q)ab ⊗Z Qhasfiniterank
(alreadyremarked),andG(Q)isuniquelydivisible(itisisomorphictoexpLieGbyProposition22). (cid:3)
Corollary27. (1) LetΓbeinG(cid:103)ps. ThenΓunischaracterizedbythefactthatitispro-unipotentandΓun(Q)is
theuniversalpro-nilpotentuniquelydivisiblegroupassociatedtoΓ.
(2) Let g be in L(cid:103)A. Then gun is characterized by the fact that it is pro-unipotent and Liegun is the universal
pro-nilpotentLiealgebraassociatedtoΓ.
Proof. ThiscomesfromQuillen’stheoremandthelateraladjunctionsofProposition25. (cid:3)
Corollary28. Thecategoryofunipotentalgebraicgroupisequivalenttothecategoryoffinitedimensionalnilpotent
Liealgebras.
Proof. ThefunctorfromUAGtonilpotentLiealgebrasisfullyfaithfulbyQuillen’stheorem. Itisessentiallysurjec-
tiveby[4,Theorem3.27]. (cid:3)
FromthepreviouscorollaryandProposition22,wecandeduceasimilarequivalencebetweenunipotentalgebraic
groupsandabstractgroupswhichareexponentialsofnilpotent,finitedimensionalLiealgebras.
3. THEFREECASE
Wenowconsiderthefreepro-unipotentgroupG associatedtoafinitesetS ={e ,...,e }.Bythecommutativity
S 1 n
oftheadjunctionofProposition22,itisisomorphictothepro-unipotentcompletionofthefreegroupΓ overS,and
S
ofthefreeLiealgebraLS overS. Foritsconstruction,wechoosethissecondoption.
By what we already proved, G ∼= Spec(((ULS)∧)∗), where L is the free Lie algebra functor. The functor UL
S
from Set to Alg is left adjoint to the forgetful functor, and therefore ULS ∼= Q(cid:104)e ,...,e (cid:105), the algebra of non-
1 n
commutativepolynomialsinnvariables. Itisstraightforwardtocheckthat(ULS)∧ ∼= Q(cid:104)(cid:104)e ,...,e (cid:105)(cid:105),theformal
1 n
non-commutativepowerseriesinnvariables. Itscoproductisdefinedviatherelationse (cid:55)→ e ⊗1+1⊗e . IfI =
i i i
(cid:80)
(i ,...,i )isamulti-index,weindicatewithe theproducte ·...·e . Byinduction,itfollows∆e = e ⊗e ,
1 k I 1 k I σ J K
whereI,J varyamongmulti-indicesandσvariesamongthepermutationsSym(|J|,|K|)suchthatσ(J,K)=I. We
clarifythiswithanexample.
Example29.
∆(e2e )=e2e ⊗1+e2⊗e +2e e ⊗e +2e ⊗e e +e ⊗e2+1⊗e2e .
1 2 1 2 1 2 1 2 1 1 1 2 2 1 1 2
Thisalgebraisgradedwithrespecttothedegree,isomorphicto(cid:81)V⊗k,whereV isthefreevectorspacegenerated
byS. Itfollowsthat((ULS)∧)∗ ∼=(cid:76)(V⊗k)∨ ∼=(cid:76)(V∨)⊗k ∼=T(V∨). WenowinvestigateitsHopfoperations. We
denoteby(cid:15) thedualofe .
I I
7
Fromtheformulas
(cid:32) (cid:33)
(cid:88)
((cid:15) ·(cid:15) )(e )=((cid:15) ⊗(cid:15) )(∆e )=((cid:15) ⊗(cid:15) ) e ⊗e
I J K I J K I J M N
σ
wededucethat((cid:15) ·(cid:15) )(e )=1ifthereisawaytoshuffleI andJ toformK andis0otherwise. Hence,theproduct
I J K
ofT(V∨)istheshuffleproduct.
Fromtheformula
(∆(cid:15) )(e ⊗e )=(cid:15) (e )
I J K I JK
(cid:80)
wededucethat(∆(cid:15) )(e ⊗e )=1ifJK =I andis0otherwise. Therefore,∆(cid:15) = (cid:15) ⊗(cid:15) ,theso-called
I J K I JK=I J K
deconcatenationcoproduct.
Werecaptheresultswehaveprovedaboutthefreecaseinthefollowingproposition.
Proposition 30. Let Γ be the free abstract group generated by the finite set S = {e ,...,e } and let Γun be its
S 1 n S
pro-unipotentcompletion.
(1) TheexponentialmapdefinesanisomorphismO(Γun)∨ = limQ[Γ ]/In ∼= Q(cid:104)(cid:104)e ,...,e (cid:105)(cid:105), andhencean
S ←− S 1 n
isomorphismO(Γun)∼=T(V∨),whereV isthevectorspacegeneratedbyS.
S
(2) Lie(Γun)iscanonicallyisomorphictotheuniversalpro-nilpotentalgebraassociatedtothefreeLiealgebra
S
LS associatedtoS,i.e. tothecompletionofLS bythelowercentralseries.
(3) ForanyQ-algebraT,Γun(T)isisomorphictothegroup-likeelementsofthecoalgebraT(cid:104)(cid:104)e ,...,e (cid:105)(cid:105).
S 1 n
Proof. Asnotedatthebeginningofthissection, theexponentialmapinducesanisomorphismfromΓun toG , the
S S
pro-unipotent completion of LS. The first two points then follow from our previous calculations and Corollary 27.
ThelastpointfollowsfromProposition19. (cid:3)
4. MALCEVORIGINALCONSTRUCTION
Wenowgiveanexplicitdescriptionofanotherspecialcase,originallystudiedbyMalcev. Wereferto[6]forthe
group theory facts we need here. Suppose Γ is nilpotent and finitely generated. In this case, the torsion elements
constituteasubgroupH. Sinceanyuniquelydivisiblegrouphasnotorsion,byCorollary22weconcludethatΓun ∼=
(Γ/H)un. WecanthereforesupposethatΓhasnotorsion.
Let
Γ=Γ ≥Γ ≥...≥Γ =1
1 2 k
be the lower central series (Γ = [Γ ,Γ]). Each factor Γ /Γ is abelian and finitely generated since [Γ ,Γ ] ≤
i i−1 i i+1 i i
[Γ ,Γ] = Γ . Wecanthenrefinethelowercentralseriestoobtainanothercentralserieswithcyclicquotients. A
i i+1
groupwithsuchacentralseriesiscalledpolycyclic. Quotientsandsubgroupofpolycycliconesareagainpolycyclic
(bystudyingtheinducedfiltrations). Inparticular,thequotientsoftheuppercentralseries(Z /Z =Z(G/Z ))
i i+1 i+1
Γ=Z ≥Z ≥...≥Z =1
1 2 k
arepolycyclic. Theyarealsowithouttorsionbythenextlemma.
Lemma31. IfΓisnilpotentandwithouttorsion,thenallquotientsΓ/Z arewithouttorsion.
i
Proof. Since Γ/Z ∼= (G/Z )/(Z /Z ) ∼= (G/Z )/(Z(Γ/Z )), it suffices to prove that if Γ is nilpotent and
i+1 i i+1 i i i
withouttorsion,thenΓ/Z(Γ)iswithouttorsion.
Supposexm iscentral. Weneedtoprovethatalsoxis. Ifxm iscentral,thenforanyywehave(y−1xy)m =xm.
Itsufficestoproveuniquenessofrootsinatorsion-freenilpotentgroup. Wemakeinductiononthenilpotencyclass,
beingthecaseofanabeliangrouptrivial.
Letam = bm inatorsion-freenilpotentgroupΓ. Inordertoprovea = b,itsufficestoprove[a,b] = 1sinceΓis
torsion-free. Sinceb−1ab = a[a,b],bothb−1abandalieinthesubgroupH = (cid:104)[Γ,Γ],a(cid:105),whichhasastriclylower
nilpotencyclass(see[6,Proposition2.5.5]). Byinduction,fromtheequality(b−1ab)m = b−1amb = bm = am,we
conclude[a,b]=1aswanted. (cid:3)
Inconclusion,theuppercentralseriescanbeenrichedintoafiltration
Γ=Γ1 ≥Γ2 ≥...Γs+1 =1
such that Γi/Γi+1 ∼= (cid:104)e (cid:105) ∼= Z. The set {e ,...,e } is called a Malcev basis for Γ. We can associate to any
i 1 s
element g ∈ Γ a unique set of s coordinates t (g) ∈ Z such that g = (cid:81)eti(g), and Γi coincides with the subset
i i
8
{g ∈ Γ: t (g) = 0,j < i}. ThisdefinesabijectionfromΓtoZs. Wenowrecoveralsotheproductintermsofthe
j
coordinates.
Proposition32. LetΓbefinitelygenerated,nilpotentandwithouttorsion. Let{e ,...,e }beaMalcevbasisforΓ
1 s
andt (g)theMalcevcoordinatesofanelementg.
i
(1) The product is polynomial in the coordinates, i.e. there exist polynomials P with rational coefficients such
i
that
t (gh)=t (g)+t (h)+P (t (g),t (h)).
i i i i j j
Moreover,thepolynomialP dependsonlyont ’swithj <i.
i j
(2) Theinverseispolynomialinthecoordinates,i.e. thereexistpolynomialsQ withrationalcoefficientssuch
i,k
that
t (gk)=kt (g)+Q (t (g)).
i i i,k j
Moreover,thepolynomialQ dependsonlyont ’swithj <i.
i,k j
Proof. MakeinductiononthecardianlityoftheMalcevbasis. Detailsin[6,Proprie´te´ 3.1.5]. (cid:3)
SincethepolynomialsP andQ haverationalcoefficients,theydefineanalgebraicgroupG: R(cid:55)→(Rs,·)where
i i,k
·istheproductdefinedusingtheaboveformulas.
Proposition33. ThegroupGisunipotent.
Proof. Since O(G) is a polynomial ring, G(Q) is dense in G. Therefore, if we prove that there is a faithful finite
dimensionalrepresentationofGsuchthatG(Q)→GL (Q)ismadeofunipotentmorphisms(i.e. ,forallg ∈G(Q),
V
(g−id)n =0forn(cid:29)0),weconcludethat,withrespecttoasuitablebasis,G(Q)factorsthroughUT (Q).Bydensity,
n
we can isomorphically embed G in UT , as wanted. Because the regular representation contains all representation,
n
we can equivalently prove that all endomorphisms of G(Q) are unipotent with respect to it (i.e. (g −id)n = 0 for
n(cid:29)0whenrestrictedtoanysubrepresentationoffinitedimension).
ThisrepresentationsendsgtotheendomorphismT (cid:55)→t (g)+T +P (t ,T ). Bytheformulasabove,itfollows
i i i i j j
that(g−id)itsendsamonomialTI = Ti1 ·...Tis toalinearcombinationofmonomialswhicharestriclysmaller
1 s
withrespecttothelexicographicorder. Inparticular,foranymulti-indexI,(g−id)n(TI)=0forn(cid:29)0. (cid:3)
WeremarkthatthemapΓ→G(Q)isinducedbytheinclusionZs →Qs. Itsatisfiesauniversalproperty:
Proposition34. TheabstractgroupG(Q)isthenilpotent,uniquelydivisibleclosureofΓ.
Proof. Using the formulas and induction on i, it is straightforward to see that G(Q) is uniquely divisible and if
x∈G(Q),thenxn ∈Γforn(cid:29)1. (cid:3)
Corollary35. Γun ∼=G.
Proof. ThiscomesfromthepreviouspropositionsandCorollary27. (cid:3)
5. TORSORS
Let Γ be an abstract group. We remark that the functors we used to define Γun make sense more generally for
Γ-sets:
Γ-SetQ→[·]Q[Γ]-Mod→∧ Qˆ[Γ]-ModSpec→((·)∗)Γun-Var
andwedenoteagaintheircompositionwithS (cid:55)→ Sun. Moreover,allthesefunctorsaretensorialwithrespecttothe
tensorsdefinedineachcategory. Thefirstandthelastonearetensorialwithrespecttothecartesianproduct.Itfollows
thatifS ∈Γ-Setisatorsor,i.e. if
Γ×S →S×S (g,s)(cid:55)→(g·s,s)
isanisomorphism,thenalso
Γun×Sun ∼=(Γ×S)un →(S×S)un ∼=Sun×Sun
isanisomorphism. Therefore,S (cid:55)→Sunmapstorsorstotorsors.
WerecallthatoneoftheformsofChen’stheoremstatesthatthemaps
c : Q[π (M;a,b)]/In+1 →∼ Q ⊕H (Mn,Zn )
n 1 a,b n a,b
9
defineanisomorphismofdirectedsystems(seethenotationsandthedetailsin??). Wecanthendeduce,bypassing
tothedualandthedirectlimitovern,thefollowingfact.
Corollary36. Theisomorphismsc induceanisomorphismofvectorspaces
n
c∨: Q ⊕limHn(Mn,Zn )→∼ O(π (M;a,b)un).
a,b −→ a,b 1
n
6. THETANNAKIANAPPROACH
Proposition37. LetΓbeanabstractgroupsuchthatΓun ilwelldefined. ThenΓun isthepro-affinealgebraicgroup
associatedtotheTannakiancategoryofunipotentrepresentationsofΓ.
Proof. ThefunctorΓun-Rep → Γ-RepsendingΓun → GL toΓ → Γun(Q) → GL (Q)factorsoverunipotent
V V
representations of Γ since Γun is pro-unipotent. Viceversa, if Γ → GL (Q) is unipotent then, with respect to a
V
suitablebasis,itfactorsoverUT . ItfollowsthatthesubgroupH ofGL generatedbyΓisisomorphictoasubgroup
n V
ofUT ,henceunipotent. Bytheuniversalproperty,themapΓ → H(Q)theninducesamapΓun → H → GL ,as
n V
wanted. (cid:3)
Sincewehaveprovedtheexistence(andQuillen’sconstruction)ofΓunonlyforgroupsΓwithniceproperties(i.e.
Γab ⊗Z Q has finite dimension), we wonder if this proposition gives a more general construction of Γun, i.e. if the
TannakadualofunipotentrepresentationsofΓsatisfiestheuniversalpropertyofthepro-unipotentcompletion.
7. MOREONTHECONILPOTENCYFILTRATION
WenowpresentalastcorollaryofthefirstsectionandQuillen’spaper. SupposeGisunipotent,andletgbeitsLie
algebra. Wehaveg∼=PO(G)∨,anditinheritstheI-adicfiltrationfromO(G)∨:
PO(G)∨ =PO(G)∨∩I ⊃PO(G)∨∩I2 ⊃...
On the other hand, we can consider the graded Hopf algebra gr•O(G)∨, obtained via the I-adic filtration. Its
primitiveelementsconstituteagradedsubsetPgr•O(G)∨.
Proposition38. [5,PropositionA.2.14]Thenaturalmapgr•PO(G)∨ →Pgr•O(G)∨isanisomorphism.
Inparticular, wededucegr1g ∼= I/I2 ∼= g/[g,g]. ThisabelianLiealgebrahasanaturalmaptogr•PO(G)∨ ∼=
Pgr•O(G)∨. It is a quotient of the free Lie algebra LS generated by a chosen basis S of g/[g,g]. By adjunction,
weobtainamapUˆLS →gr•O(G)∨,andbydualityamap(gr•O(G)∨)∗ →(ˆULS)∗,andamapSpec((UˆLS)∗)→
Spec((gr•O(G)∨)∗).
WerecallthattheconilpotencyfiltrationonO(G)
0⊂C ⊂C ⊂...
1 2
isdualtotheI-adicfiltrationonO(G)∨
O(G)∨ ⊃I ⊃I2 ⊃...
inthesensethat(griO(G)∨)∗ ∼=gr O(G).Henceinparticular,g/[g,g]∗ ∼=(I/I2)∗ ∼=(gr O(G))and(gr•O(G)∨)∗ ∼=
i 1
gr O(G).
•
We remark that Spec((UˆLS)∗) is a free pro-unipotent group generated by S. By what we proved in Section 3,
(ˆULS)∗ ∼=T(g/[g,g]∨)∼=T(gr O(G)).
1
Proposition39. Themapgr O(G)→T(gr O(G))isinjective.
• 1
Proof. We prove that the dual is surjective. It is the map UˆLS → gr•O(G)∨ ∼= gr•Uˆg, where the last equality
followsfromtheunipotencyofG.
The graded algebra gr•Uˆg is generated by its first graded piece gr1Uˆg = Pgr1Uˆ ∼= gr1PUˆ ∼= g/[g,g], hence
UˆLS →gr•Uˆgissurjective,asclaimed. (cid:3)
10