Hopf algebras [Lecture notes]

116 Pages·2014·0.699 MB·English

by  Stefaan Caenepeel and J. Vercruysse

Hopf algebras
S. Caenepeel and J. Vercruysse
Syllabus106bijWE-DWIS-12762“Hopfalgebrasenquantumgroepen-Hopfalgebrasandquantumgroups”
MasterWiskundeVrijeUniversiteitBrusselenUniversiteitAntwerpen. 2014
Contents
1 Basicnotionsfromcategorytheory 3
1.1 Categoriesandfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Naturaltransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Adjointfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.5 Equivalencesandisomorphismsofcategories . . . . . . . . . . . . . . . . 7
1.2 Abeliancategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Equalizersandcoequalizers . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Kernelsandcokernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 Limitsandcolimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.4 AbeliancategoriesandGrothendieckcategories . . . . . . . . . . . . . . . 11
1.2.5 Exactfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.6 Grothendieckcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Tensorproductsofmodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Universalproperty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Existenceoftensorproduct . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.3 Iteratedtensorproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.4 Tensorproductsoverfields . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.5 Tensorproductsoverarbitraryalgebras . . . . . . . . . . . . . . . . . . . 15
1.4 Monoidalcategoriesandalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 Monoidalcategoriesandcoherence . . . . . . . . . . . . . . . . . . . . . 16
1.4.2 Monoidalfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.3 Symmetricandbraidedmonoidalcategories . . . . . . . . . . . . . . . . . 18
2 Hopfalgebras 20
2.1 Monoidalcategoriesandbialgebras . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Hopfalgebrasandduality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Theconvolutionproduct,theantipodeandHopfalgebras . . . . . . . . . . 23
2.2.2 Projectivemodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Propertiesofcoalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Examplesofcoalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Subcoalgebrasandcoideals . . . . . . . . . . . . . . . . . . . . . . . . . 30
1
2.4 Comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 ExamplesofHopfalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Hopfmodulesandintegraltheory 44
3.1 Integralsandseparability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Hopfmodulesandthefundamentaltheorem . . . . . . . . . . . . . . . . . . . . . 47
4 GaloisTheory 58
4.1 Algebrasandcoalgebrasinmonoidalcategories . . . . . . . . . . . . . . . . . . . 58
4.2 Corings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Faithfullyflatdescent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 Galoiscorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 MoritaTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.6 GaloiscoringsandMoritatheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.7 Hopf-Galoisextensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.8 Stronglygradedrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.9 ClassicalGaloistheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5 Examplesfrom(non-commutative)geometry 102
5.1 Thegeneralphilosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 Hopfalgebrasinalgebraicgeometry . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2.1 Coordinatesasmonoidalfunctor . . . . . . . . . . . . . . . . . . . . . . . 104
5.3 Aglimpseonnon-commutativegeometry . . . . . . . . . . . . . . . . . . . . . . 111
5.3.1 Non-commutativegeometrybyHopf(Galois)theory . . . . . . . . . . . . 111
5.3.2 Deformationsofalgebraicgroups: algebraicquantumgroups . . . . . . . . 112
5.3.3 Morequantumgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2
Chapter 1
Basic notions from category theory
1.1 Categories and functors
1.1.1 Categories
AcategoryC consistsofthefollowingdata:
• aclass|C| = C = C ofobjects,denotedbyX,Y,Z,...;
0
• foranytwoobjectsX,Y,asetHom (X,Y) = Hom(X,Y) = C(X,Y)ofmorphisms;
C
• foranythreeobjectsX,Y,Z acompositionlawforthemorphisms:
◦ : Hom(X,Y)×Hom(Y,Z) → Hom(X,Z), (f,g) (cid:55)→ g ◦f;
• foranyobjectX aunitmorphismonX,denotedby1 orX forshort.
X
Thesedataaresubjectedtothefollowingcompatibilityconditions:
• for all objects X,Y,Z,U, and all morphisms f ∈ Hom(X,Y), g ∈ Hom(Y,Z) and h ∈
Hom(Z,U),wehave
h◦(g ◦f) = (h◦g)◦f;
• forallobjectsX,Y,Z,andallmorphismsf ∈ Hom(X,Y)andg ∈ Hom(Y,Z),wehave
Y ◦f = f g ◦Y = g.
Remark1.1.1 Ingeneral,theobjectsofacategoryconstituteaclass,notaset. Thereasonbehind
this is the well-known set-theoretical problem that there exists no “set of all sets”. Without going
into detail, let us remind that a class is a set if and only if it belongs to some (other) class. For
similar reasons, there does not exist a “category of all categories”, unless this new category is “of
a larger type”. (A bit more precise: the categories defined above are Hom-Set categories, i.e. for
any two objects X,Y, we have that Hom(X,Y) is a set. It is possible to build a category out of
this type of categories that will no longer be a Hom-Set category, but a Hom-Class category: in a
Hom-ClasscategoryHom(X,Y)isaclassforanytwoobjectsX andY. Ontheotherhand,ifwe
restrict to so called small categories, i.e. categories with only a set of objects, then these form a
Hom-Setcategory.)
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Examples1.1.2 1. The category Set whose objects are sets, and where the set of morphisms
betweentwosetsisgivenbyallmappingsbetweenthosesets.
2. Let k be a commutative ring, then M denotes the category with as objects all (right) k-
k
modules,andwithasmorphismsbetweentwok-modulesallk-linearmappings.
3. If A is a non-commutative ring, we can consider also the category of right A-modules M ,
A
the category of left A-modules M, and the category of A-bimodules M . If B is an-
A A A
other ring, we can also consider the category of A-B bimodules M . Remark that if A is
A B
commutative,thenM and Mcoincide,buttheyaredifferentfrom M !
A A A A
4. Top is the category of topological spaces with continuous mappings between them. Top is
0
the category of pointed topological spaces, i.e. topological spaces with a fixed base point,
andcontinuousmappingsbetweenthemthatpreservethebasepoint.
5. Grpisthecategoryofgroupswithgrouphomomorphismsbetweenthem.
6. Ab is the category of Abelian groups with group homomorphisms between them. Remark
thatAb = M .
Z
7. Rng isthecategoryofringswithringhomomorphismsbetweenthem.
8. Alg is the category of k-algebras with k-algebra homomorphisms between them. We have
k
Alg = Rng.
Z
9. All previous examples are concrete categories: their objects are sets (with additional struc-
ture), i.e. they allow for a faithful forgetful functor to Set (see below). An example of a
non-concrete category is as follows. Let M be a monoid, then we can consider this as a
categorywithoneobject∗,whereHom(∗,∗) = M.
10. The trivial category ∗ has only one object ∗, and one morphism, the identity morphism of ∗
(thisisthepreviousexamplewithM thetrivialmonoid).
11. Another example of a non-concrete category can be obtained by taking a category whose
classofobjectsisN ,andwhereHom(n,m) = M (k): alln×mmatriceswithentriesin
0 n,m
k (wherek ise.g.acommutativering).
12. If C is a category, then Cop is the category obtained by taking the same class of objects as in
C,butbyreversingthearrows,i.e.Hom (X,Y) = Hom (Y,X). Wecallthistheopposite
Cop C
categoryofC.
13. If C and D are categories, then we can construct the product category C ×D, whose objects
are pairs (C,D), with C ∈ C and D ∈ D, and morphisms (f,g) : (C,D) → (C(cid:48),D(cid:48)) are
pairsofmorphismsf : C → C(cid:48) inC andg : D → D(cid:48) inD.
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1.1.2 Functors
LetC andD betwocategories. A(covariant)functorF : C → D consistsofthefollowingdata:
• foranyobjectX ∈ C,wehaveanobjectFX = F(X) ∈ D;
• foranymorphismf : X → Y inC,thereisamorphismFf = F(f) : FX → FY inD;
satisfyingthefollowingconditions,
• forallf ∈ Hom(X,Y)andg ∈ Hom(Y,Z),wehaveF(g ◦f) = F(g)◦F(f);
• forallobjectsX,wehaveF(1 ) = 1 .
X FX
A contravariant functor F : C → D is a covariant functor F : Cop → D. Most or all functors that
we will encounter will be covariant, therefore if we say functor we will mean covariant functor,
unless we say differently. For any functor F : C → D, one can consider two functors Fop : Cop →
D andFcop : C → Dop. ThenF iscontravariantifandonlyifFop andFcop arecovariant(andvisa
versa). ThefunctorFop,cop : Cop → Dop isagaincontravariant.
Examples1.1.3 1. The identity functor 1 : C → C, where 1 (C) = C for all objects C ∈ C
C C
and1 (f) = f forallmorphismsf : C → C(cid:48) inC.
C
2. The constant functor C → D at X, assigns to every object C ∈ C the same fixed object
X ∈ D, and assigns to every morphism f in C the identity morphism on X. Remark that
definingtheconstantfunctor,isequivalenttochoosinganobjectD ∈ D.
3. The tensor functor −⊗− : M ×M → M , associates two k-modules X and Y to their
k k k
tensorproductX ⊗Y (seeSection1.3).
4. All“concrete”categoriesfromExample1.1.2(1)–(8)allowforaforgetfulfunctortoSet,that
sends the objects of the concrete category to the underlying set, and the homomorphisms to
theunderlyingmappingbetweentheunderlyingsets.
5. π : Top → Grp is the functor that sends a pointed topological space (X,x ) to its funda-
1 0 0
mentalgroupπ (X,x ).
1 0
6. Anexample of a contravariant functor isthe following (−)∗ : M → M , which assigns to
k k
everyk-moduleX thedualmoduleX∗ = Hom(X,k).
All functors between two categories C and D are gathered in Fun(C,D). In general, Fun(C,D) is
aclass,butnotnecessarilyaset. Hence,ifonewantstodefine‘acategoryofallcategories’where
themorphismsarefunctors,somecareisneeded(seeabove).
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1.1.3 Natural transformations
Let F,G : C → D be two functors. A natural transformation α : F → G (sometimes denoted by
α : F ⇒ G),assignstoeveryobjectC ∈ C amorphismα : FC → GC inD renderingforevery
C
f : C → C(cid:48) inC thefollowingdiagraminD commutative,
Ff (cid:47)(cid:47)
FC FC(cid:48)
αC αC(cid:48)
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47)
GC GC(cid:48)
Gf
Remark1.1.4 If α : F → G is a natural transformation, we also say that α : FC → GC is a
C
morphismthatisnaturalinC. Insuchanexpression,thefunctorsF andGareoftennotexplicitly
predescribed. E.g. the morphism X ⊗ Y∗ → Hom(Y,X), x ⊗ f (cid:55)→ (y (cid:55)→ xf(y)), is natural
both in X and Y. Here X and Y are k-modules, x ∈ X, y ∈ Y, f ∈ Y∗ = Hom(Y,k), the dual
k-moduleofY.
Ifα isanisomorphisminD forallC ∈ C,wesaythatα : F → Gisanaturalisomorphism.
C
Examples1.1.5 1. If F : C → D is a functor, then 1 : F → F, defined by (1 ) = 1 :
F F X F(X)
F(X) → F(X)istheidentitynaturaltransformationonF.
2. The canonical injection ι : X → X∗∗,ι (x)(f) = f(x), for all x ∈ X and f ∈ X∗, from
X X
a k-module X to the dual of its dual, defines a natural transformation ι : 1 → (−)∗∗. If
M
k
we restrict to the category of finitely generated and projective k-modules, then ι is a natural
isomorphism.
1.1.4 Adjoint functors
Let C and D be two categories, and L : C → D, R : D → C be two functors. We say that
(equivalently)
• LisaleftadjointforR;
• R isarightadjointforL;
• (L,R)isanadjointpair;
• thepair(L,R)isanadjunction,
ifandonlyifanyoffollowingequivalentconditionshold:
(i) thereisanaturalisomorphism
θ : Hom (LC,D) → Hom (C,RD), (1.1)
C,D D C
withC ∈ C andD ∈ D;
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(ii) there are natural transformations η : 1 → RL, called the unit, and ε : LR → 1 , called the
C D
counit,whichrenderthefollowingdiagramscommutativeforallC ∈ C andD ∈ D,
LC (cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)L(cid:81)(cid:81)η(cid:81)(cid:81)C(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:47)(cid:47)L(cid:81)(cid:81)RL(cid:15)(cid:15)εLCC RD(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)η(cid:81)(cid:81)R(cid:81)(cid:81)D(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:81)(cid:47)(cid:47)R(cid:81)(cid:81)LR(cid:15)(cid:15)RDεD
LC RD
This means that we have the following identities between natural transformations: 1 =
L
εL◦Lη and1 = Rε◦ηR.
R
A proof of the equivalence between condition (i) and (ii) can be found in any standard book on
category theory. We also give the following list of examples as illustration, without proof. Some
oftheexampleswillbeusedorprovedlaterinthecourse.
Examples1.1.6 1. Let U : Grp → Set be the forgetful functor that sends every group to the
underlyingset. ThenU hasaleftadjointgivenbythefunctorF : Set → Grpthatassociates
to every set the free group generated by the elements of this set. Remark that equation (1.1)
expresses that a group homomorphism (from a free group) is determined completely by its
actionongenerators.
2. Let X be a k-module. The functor − ⊗ X : M → M is a left adjoint for the functor
k k
Hom(X,−) : M → M .
k k
3. Let X be an A-B bimodule. The functor − ⊗ X : M → M is a left adjoint for the
A A B
functorHom (X,−) : M → M .
B B A
4. Let ι : B → A be a morphism of k-algebras. Then the restriction of scalars functor R :
M → M is a right adjoint to the induction functor −⊗ A : M → M . Recall that
A B B B A
forarightA-moduleX,R(X) = X ask-module,andtheB-actiononR(X)isgivenbythe
formula
x·b = xι(b),
forallx ∈ X andb ∈ B.
5. Let k− : Grp → Alg be the functor that associates to any group G the group algebra kG
k
over k. Let U : Alg → Grp be the functor that associates to any k-algebra A its unit group
k
U(A). Thenk−isaleftadjointforU.
1.1.5 Equivalences and isomorphisms of categories
LetF : C → D beafunctor. ThenF inducesthefollowingmorphismthatisnaturalinC,C(cid:48) ∈ C,
F : Hom (C,C(cid:48)) → Hom (FC,FC(cid:48)).
C,C(cid:48) C D
Thefunctorissaidtobe
• faithfulifF isinjective,
C,C(cid:48)
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• fullifF issurjective,
C,C(cid:48)
• fullyfaithfulifF isbijective
C,C(cid:48)
forallC,C(cid:48) ∈ C.
If (L,R) is an adjoint pair of functors with unit η and counit ε, then L is fully faithful if and only
ifη isanaturalisomorphismandR isfullyfaithfulifandonlyifεisanisomorphism.
Examples1.1.7 1. All forgetful functors from a concrete category (as in Example 1.1.2 (1)–
(8))toSetarefaithful(infact,admittingafaithfulfunctortoSet,isthedefinitionofbeinga
concretecategory,andthisfaithfulfunctortoSetisthencalledtheforgetfulfunctor).
2. Let ι : B → A be a surjective ring homomorphism, then the restriction of scalars functor
R : M → M isfull.
A B
3. TheforgetfulfunctorAb → Grpisfullyfaithful.
A functor F : C → D is called an equivalence of categories if and only if one of the following
equivalentconditionsholds:
1. F isfullyfaithfulandhasafullyfaithfulrightadjoint;
2. F isfullyfaithfulandhasafullyfaithfulleftadjoint;
3. F is fully faithful and essentially surjective, i.e. each object D ∈ D is isomorphic to an
objectoftheformFC,forC ∈ C;
4. F hasaleftadjointandtheunitandcounitoftheadjunctionarenaturalisomorphisms;
5. F hasarightadjointandtheunitandcounitoftheadjunctionarenaturalisomorphisms;
6. thereisafunctorG : D → C andnaturalisomorphismsGF → 1 andFG → 1 .
C D
Thereisasubtledifferencebetweenanequivalenceofcategoriesandthefollowingstrongernotion:
A functor F : C → D is called an isomorphism of categories if and only if there is a functor
G : D → C suchthatGF = 1 andFG = 1 .
C D
Examples1.1.8 1. Letk beacommutativeringandR = M (k)then×nmatrixringoverk.
n
ThenthecategoriesM andM areequivalent,(butnotnecessarilyisomorphicifn (cid:54)= 1).
k R
2. ThecategoriesAbandM areisomorphic.
Z
∼
3. ForanycategoryC,wehaveanisomorphismC ×∗ = C.
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1.2 Abelian categories
1.2.1 Equalizers and coequalizers
Let A be any category and consider two (parallel) morphisms f,g : X → Y in A. The equalizer
of the pair (f,g), is a couple (E,e) consisting of an object E and a morphism e : E → X, such
thatf ◦e = g ◦e,andthatsatisfiesthefollowinguniversalproperty. Forallpairs(T,t : T → X),
suchthatf ◦t = g ◦t,theremustexistauniquemorphismu : T → E suchthatt = e◦u.
∃!uE(cid:79)(cid:79) (cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:112)et(cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:112)(cid:47)(cid:47)(cid:56)(cid:56)X fg (cid:47)(cid:47)(cid:47)(cid:47)Y
T
The dual notionof an equalizer isthat of acoequalizer. Explicitly: the coequalizerof a pair(f,g)
is a couple (C,c), consisting of an object C and a morphism c : Y → C, such that c◦f = c◦g.
Moreover, (C,c) is required to satisfy the following universal property. For all pairs (T,t : Y →
C),suchthatt◦f = t◦g,theremustexistauniquemorphismu : C → T suchthatt = u◦c.
X fg (cid:47)(cid:47)(cid:47)(cid:47)Y (cid:77)(cid:77)(cid:77)(cid:77)(cid:77)(cid:77)t(cid:77)c(cid:77)(cid:77)(cid:77)(cid:77)(cid:77)(cid:77)(cid:47)(cid:47)(cid:38)(cid:38)C(cid:15)(cid:15)∃!u
T
Bytheuniversalproperty,itcanbeprovedthatequalizersandcoequalizers,iftheyexist,areunique
uptoisomorphism. Explicitly,thispropertytellsthatifforagivenpair(f,g),onefindstocouples
(E,e)and(E(cid:48),e(cid:48))suchthatf ◦e = g◦eandf ◦e(cid:48) = g◦e(cid:48) andbothcouplessatisfytheuniversal
property,thenthereexistsanisomorphismφ : E → E(cid:48) inA,suchthate = e(cid:48) ◦φ.
Let (E,e) be the equalizer of a pair (f,g). An elementary but useful property of equalizers tells
thateisalwaysamonomorphism. Similarly,foracoequalizer(C,c),cisanepimorphism.
1.2.2 Kernels and cokernels
AzeroobjectforacategoryA,isanobject0inA,suchthatforanyotherobjectAinA,Hom(A,0)
andHom(0,A)consistsofexactlyoneelement. Ifitexits,thezeroobjectofAisunique. Suppose
that A has a zero object and let A and B be two objects of A. A morphism f : B → A is called
the zero morphism, if f factors trough 0, i.e. f = f ◦f where f and f are the unique elements
1 2 1 2
in Hom(0,A) and Hom(B,0), respectively. From now on, we denote any morphism from, to, or
factorizingtrough0alsoby0.
The kernel of a morphism f : B → A, is the equalizer of the pair (f,0). The cokernel of f
is the coequalizer of the pair (f,0). Remark that in contrast with the classical definition, in the
categoricaldefinitionofakernel,akernelconsistsofapair(K,κ),whereK isanobjectofA,and
κ : K → B is a morphism in A. The monomorphism κ corresponds in the classical examples to
thecanonicalembeddingofthekernel.
Theimageofamorphismf : B → A,isthecokernelofthekernelκ : K → B off. Thecoimage
isthekernelofthecokernel.
9