Gast´ on Andr´ es Garc´ıa
Universidad Nacional de C´ ordoba, Argentina XVIII Latin American Algebra Colloquium
S˜ ao Pedro, Brazil
August 3rd to August 8th, 2009
Abstract
These notes correspond to a mini-course for graduate and undergraduate students given in the XVIII Latin American Algebra Colloquium in S˜ao Pedro, Brazil. Because of the lenght of the course, we intend only to give the basic ideas of the subject and to show, by means of examples, how quantum groups enter into the scene of the classification problem of finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero. The reader who is interested in this subject may have a look at the bibliography on quantum groups and Hopf algebras and references therein. We did not intend to give an exhaustive or comprehensive list of the references in these subjects, but just give some of them to serve as a guide.
1 Introduction
Quantum groups, introduced in 1986 by Drinfeld [Dr2], form a certain class of Hopf algebras. Up to date there is no rigorous, universally accepted definition, but it is generally agreed that this term includes certain deformations in one or more parameters of classical objects associated to algebraic groups, such as enveloping algebras of semisimple Lie algebras or algebras of regular functions on the corresponding algebraic groups. As one can relate algebraic groups with commutative Hopf algebras via group schemes, it is also agreed that the category of quantum groups should correspond to the opposite category of the category of Hopf algebras. This is why some authors define quantum groups as non-commutative and non-cocommutative Hopf algebras.
Hopf algebras were introduced in the 50’s, and from the 60’s they have been intensively studied. First in relation with algebraic groups and later as objects of self interest. One of the main open problems in the theory of Hopf algebras is the classification of Hopf algebras H of a fixed dimension over an algebraically closed field of characteristic zero. Up to now, very few general results are known and the classification is solved only if the dimensionN ofH is smaller or equal than 19, ifN can be factorized in a simple way, i.e. N = p, p2,2p2 with pa prime number, or if the Hopf algebra has additional properties such as semisimplicity or pointness. It turns out that there is a deep relation between semisimple Hopf algebras and group theory, and between pointed Hopf algebras and Lie theory. Since we do not assume some knowledge on Lie theory, we shall not talk about this relation. Nevertheless, it can be traced back from the examples coming from quantum groups.
One of the obstructions in solving the classification problem is the lack of enough examples. Hence, it is necessary to find new families of Hopf algebras. From the very beginning, this role was played by quantum groups. They consists of a large family with different structural properties and were used with profit to solve the classification problem for fixed dimensions.
After introducing Hopf algebras, together with some basic examples, we give in Section 3 the definition of the simpliest quantum groups Oq(SL2(k)) and Uq(sl2)(k) over an algebraically closed field k of characteristic zero. Ifq is a primitive`-th root of unity ofk, then one can define thesmall quantum group uq(sl2)(k), which is a finite-dimensional quotient ofUq(sl2)(k) of dimension`3.
Finally in Section 4, we show how these small quantum groups and variation of them enter into the scene of the classification problem of Hopf algebras of dimension p2 and pointed Hopf algebras of dimensionp3.
1
2 Hopf algebras
One of the major threads running through this subject has it roots in a philosophy proposed by Grothendieck, which states that one should study objects by means of the functions on them. This allows to relate algebraic objects to geometric objects andvice versa. In this section we outline how this philosophy leads naturally to the concept of a Hopf algebra.
LetX be a finite set. Then
A=CX={f :X→C|f function}
is a unital algebra overCof finite dimension. Indeed,Ais an algebra over a fieldk if (i) A is a vector space overk,
(ii) A has a multiplication
m:A×A→A, (f, g)7→f g, with (f g)(x) =f(x)g(x), which is associative, i.e. (f g)h=f(gh) for allf, g, h∈A.
(iii) A has a unit
u:k→A, λ7→λ·1A,
where 1A=u(1k) is the unit of the algebra and the image ofuis contained in the centre ofA.
Sincemis a bilinear map (i.e. linear in each component), we may consider it as a linear map m:A⊗A→A, f⊗g7→f g, with (f g)(x) =f(x)g(x).
In this case, the associativity can be described by the commutative diagram A⊗A⊗A m⊗id //
id⊗m
A⊗A
m
A⊗A m //A.
The corresponding diagram for the unit is the following A⊗A
m
k⊗A
u⊗idtttttttt::
t
'JJJJJJJ$$
JJ
J A⊗k
id⊗u
ddJJJJJJJJJ
zztttttt'tttt
A
Note that in this case, A is commutative, that is (f g)(x) = f(x)g(x) = g(x)f(x) = (gf)(x) for all f, g∈A, x∈X. Another way of saying this is the following:
Definition 2.1 LetV andW be twok-vector spaces. Theflip map τis the linear mapτ :V ⊗W → W ⊗V given byτ(v⊗w) =w⊗vfor allv∈V, w∈W.
ThenAis commutative if and only ifm◦τ=min A⊗A.
Letx∈X and defineδx∈Aby
δx(y) =δx,y=
( 1 ifx6=y,
0 ifx=y. (1)
Then the set {δx}x∈X is a linear basis of A. In particular, dimA=|X|. The multiplication ofA is given byδxδy=δxy and the unit by 1 =P
x∈Xδx.
Conversely, given a finite-dimensional commutative algebra overCwithout nilpotent elements, we can associate to it the set
X = SpecA={α:A→C|αis an algebra map}= AlgC(A,C),
where α is an algebra map if α(1) = 1 and α(ab) = α(a)α(b) for all a, b ∈ A. Thus, we have an equivalence
{finite sets}oo /o /o /o /o /o /o /o //{comm. finite-dim. C−alg. without nilp. elem.}
X //
CX
SpecAoo A
Clearly, ifA=CX, thenX⊆SpecCX, since for anyx∈X we may define ¯x(α) =α(x) for allα∈A and it holds that ¯x(αβ) = (αβ)(x) =α(x)β(x) = ¯x(α)¯x(β) for allα, β ∈A; that is, ¯xis an algebra map. The other equality follows from Hilbert’s Nullstellensatz.
Suppose now thatX =Gis a finite group. Then we have maps
m:G×G→G u:{1} →G S :G→G
(g, h)7→gh 17→e g7→g−1,
where mis the product of the group, ugives the unit andS gives the inverse. What are the corre- sponding maps inCG? Using these maps we may define inCG the following linear maps
∆ :CG→CG⊗CG ε:CG→C S :CG→CG
f 7→∆(f) f 7→f(1) f 7→ S(f),
where ∆(f)(g⊗h) := f(gh) and S(f)(g) = f(g−1) for all f ∈ CG, g, h ∈ G; that is, ∆, ε, S are the transpose maps ofm,u andS, respectively. In particular, ∆ is coassociative: since f((gh)k) = f(g(hk)) for allf ∈CG, g, h, k∈G, we have that
(∆⊗id)∆(f)(g⊗h⊗k) =f((gh)k) =f(g(hk)) = (id⊗∆)∆(f)(g⊗h⊗k),
which implies that (∆⊗id)∆(f) = (id⊗∆)∆(f) for allf ∈CG. This motivates the following definition.
From now on,k will denote a field.
Definition 2.2 A k-colgebrais a triple (C,∆, ε), where C is a k-vector space ∆ :C → C⊗C and ε:C→k are linear maps that satisfy the following commutative diagrams
Coassociativity: Counit:
C ∆ //
∆
C⊗C
∆⊗id
C⊗C
id⊗∆ //C⊗C⊗C
C
∆
'
zztttttttttt
'
%%J
JJ JJ JJ JJ J
k⊗C C⊗k
C⊗C
ε⊗id
ddJJJJJJJJJ id⊗ε
::t
tt tt tt tt
The map ∆ is calledcoproduct orcomultiplication and the mapεis calledcounit. Usually we refer to a coalgebra only by C, if no confusion arrives. We say that C iscocommutative if τ◦∆ = ∆ inC.
Note that the commutative diagrams are the same diagrams as in the definition of algebra, but with inverted arrows.
Example 2.3 Let X be a non-empty set and consider the k-vector space kX with basis{ex}x∈X. ThenkX is a coalgebra with the linear maps defined on the basis elements by
∆(ex) =ex⊗ex, ε(ex) = 1 for allx∈X.
Indeed, since ∆ andεare defined on a basis ofkX, it suffices to check the axioms on these elements.
For the coassociativity we have
(∆⊗id)∆(ex) = (∆⊗id)(ex⊗ex) = ∆(ex)⊗ex=ex⊗ex⊗ex=ex⊗∆(ex) = (ex⊗ex)
= (id⊗∆)(ex⊗ex) = (id⊗∆)∆(ex),
for allx ∈X. For the counit we have to check that ex =m(ε⊗id)∆(ex) = m(id⊗ε)∆(ex) for all x∈X. But
m(ε⊗id)∆(ex) =m(ε⊗id)(ex⊗ex) =m(ε(ex)⊗ex) =m(1⊗ex) =ex and m(id⊗ε)∆(ex) =m(id⊗ε)(ex⊗ex) =m(ex⊗ε(ex)) =m(ex⊗1) =ex.
Example 2.4 LetGbe a finite group. ThenkG is a coalgebra with the comultiplication and counit given by
∆(δg) =X
h∈G
δh⊗δh−1g and ε(δg) =δg,e,
for allg∈G, wheree∈Gis the identity of the group. Note that ∆ andεare the linear maps induced by the group operationsmandu. Indeed,
∆(δg)(h⊗t) =δg(ht) =
( 1 ifg=ht 0 ifg6=ht =
( 1 ifh−1g=t, 0 ifh−1g6=t Since{δg}g∈G is a linear basis ofkG, it follows that ∆(δg) =P
h∈Gδh⊗δh−1g. Analogously,ε(δg) = δg(e) =δg,e. Sincemis associative andugives the unit, it follows thatkG is a coalgebra.
Definition 2.5 Let C andD be two colgebras with comultiplication ∆C and∆D and counit εC and εD, respectively.
(i) A linear map f :C→D is called a colgebra map if ∆D◦f = (f⊗f)∆C andεC=εD◦f. (ii) A linear subspace E⊆C is a subcoalgebra if∆(E)⊆E⊗E.
(iii) A linear subspace I⊆C is a coideal if∆(I)⊆I⊗C+C⊗I andεC(I) = 0.
The following theorem says that coideals can be viewed as kernels of coalgebra maps andvice versa.
We leave its proof as exercise.
Theorem 2.6 [Sw, Thm. 1.4.7]LetC be a coalgebra,Ia coideal ofCandπ:C→C/I the canonical linear map onto the quotient vector space. Then
(a) C/I has a unique coalgebra structure such that πis a coalgebra map. This structure is induced by
∆eC/I :C−∆→C⊗C−−−→π⊗π (C/I)⊗(C/I) and εC/I(c+I) =ε(C).
(b) If f :C→D is any coalgebra map thenKerf is a coideal.
(c) If I⊆Kerf then there is a unique coalgebra mapf¯such that the following diagram commutes
C f //
πBBBBB!!
BB
B D
C/I
f¯
=={
{{ {{ {{ {
In particular, from the theorem follows that for all coalgebra C, C+ = Kerε⊆C is a coideal ofC, sinceεC is a coalgebra map.
Remark 2.7 In coalgebra theory, one uses usually the Sweedlersigma notationfor the comultiplica- tion: ifc∈C, we denote ∆(c) =P
iai⊗bi∈C⊗Cby
∆(c) =c(1)⊗c(2).
For example, the coassociativity axiom ofCgiven by the equality (∆⊗id)◦∆ = (id⊗∆)◦∆, can be written as follows
(c(1))(1)⊗(c(1))(2)⊗c(2) =c(1)⊗(c(2))(1)⊗(c(2))(2)=c(1)⊗c(2)⊗c(3),
for allc∈C. We recommend the reader to do the exercises in [Sw, Section 1.2] about sigma notation.
The use of this notation turns to be very fruitful in the theory of Hopf algebras.
We have seen in Example 2.4 that the algebra kG, G a finite group, has a coalgebra structure.
Moreover, these two structures are compatible: with the comultiplication and counit defined above,
∆ andεare algebra maps. Indeed,
∆(δgδh)(s⊗t) =δgδh(st) =δg(st)δh(st) = [∆(δg)(s⊗t)][∆(δh)(s⊗t)],
for alls, t∈G. This implies that ∆(δgδh) = ∆(δg)∆(δh) for allg, h∈G. Moreover, as the unit ofkG is given by 1kG =P
g∈Gδg, we have
∆(1kG) = ∆(X
g∈G
δg) =X
g∈G
∆(δg) = X
g,h∈G
δh⊗δh−1g= X
k,h∈G
δh⊗δk = (X
h∈G
δh)⊗(X
k∈G
δk) = 1kG⊗1kG. That is, ∆ is an algebra map. Analogously, it can be seen thatεis an algebra map and we leave it as exercise for the reader.
Furthermore, this happens to be equivalent to m and u being algebra maps. This motivates the following definition.
Definition 2.8 A bialgebrais a k-vector spaceB endowed with an algebra structure (B, m, u) and a coalgebra structure (B,∆, ε) such that ∆ and ε are algebra maps, or equivalently, m and u are coalgebra maps. That is, ∆ andεmust satisfy
∆(ab) = (ab)(1)⊗(ab)(2)=a(1)b(1)⊗a(2)b(2)= ∆(a)∆(b), ∆(1) = 1⊗1 and ε(ab) =ε(a)ε(b), ε(1) = 1, for alla, b∈B.
The corresponding commutative diagrams are the following B⊗B ∆⊗∆//
m
B⊗B⊗B⊗B
id⊗τ⊗id
B⊗B⊗B⊗B
m⊗m
B ∆ //B⊗B
B⊗B
m
ε⊗ε //k⊗k
'
B ε //k.
As expected, a linear map f :B →B0 between two bialgebras is abialgebra map iff is an algebra map and a coalgebra map. A subspaceI⊆Bis called abi-idealif it is a two-sided ideal and a coideal.
As before,I is a bi-ideal of a bialgebraB if and only if thek-vector spaceB/I is a bialgebra with the structure induced by the quotient.
Example 2.9 LetG be a finite group. We have seen in Example 2.3 that the linear spacekGwith basis{eg}g∈Gis a coalgebra. It also has an algebra structure with unit 1kG =e1and the multiplication defined byegeh=egh for allg, h∈G. The algebrakGis usually called thegroup algebra. Moreover, since
∆(e1) =e1⊗e1 and ∆(egeh) = ∆(egh) =egh⊗egh= (eg⊗eg)(eh⊗eh) = ∆(eg)∆(eh), it follows thatkGis a bialgebra.
Example 2.10 LetGbe a finite group. In Example 2.4 we showed thatkGhas a coalgebra structure.
It is easy to see that this coalgebra structure is compatible with the algebra structure defined above, givingkG a bialgebra structure. We leave the proof as exercise for the reader.
Example 2.11 Consider the k-vector space Mn(k) of n×n matrices with coefficients in k. It has a monoid structure with respect to the multiplication, since not all elements are invertible. Let O(Mn(k)) be the commutative algebra over k generated by the elements {Xij| 1 ≤ i, j ≤ n}. As algebra, it is simply the commutative ring of polynomials inn2variables
O(Mn(k)) =k[Xij|1≤i, j≤n].
Moreover, O(Mn(k)) is a subalgebra of the algebra of functions {f : Mn(k) → k| f function} on Mn(k), whereXij is the function defined by the matrix coefficient
Xij(A) =aij for allA= (aij)1≤i,j≤n ∈Mn(k).
If we denote byEij the matrix with a 1 in the entry (i, j) and zero in all others, the set{Eij}1≤i,j≤n is a linear basis ofMn(k) and the set{Xij}1≤i,j≤n is the correspondig dual basis with
hXij, Ekli=δikδjl.
Therefore,O(Mn(k)) is the algebra of regular functions onMn(k). O(Mn(k)) is a bialgebra with the coalgebra structure determined by
∆(Xij) =
n
X
k=1
Xik⊗Xkj, ε(Xij) =δij for all 1≤i, j≤n.
Indeed, sinceO(Mn(k)) is generated as a free algebra by the elements {Xij| 1≤i, j ≤n}, to define the algebra maps ∆ and ε, it suffices to define them on the generators. Moreover, since both maps are uniquely determined by their values on the generators, it is enought to check the coassociativity and the counit axioms on them. For the coassociativity we have
(∆⊗id)∆(Xij) = (∆⊗id)
n
X
k=1
Xik⊗Xkj
!
=
n
X
k=1
∆(Xik)⊗Xkj=
n
X
k,l=1
Xil⊗Xlk⊗Xkj and
(id⊗∆)∆(Xij) = (id⊗∆)
n
X
l=1
Xil⊗Xlj
!
=
n
X
l=1
Xil⊗∆(Xlj) =
n
X
k=1
Xil⊗Xlk⊗Xkj,
for all 1≤i, j≤n. Thus, ∆ is coassociative. For the counit we have m(ε⊗id)∆(Xij) =m(ε⊗id)
n
X
k=1
Xik⊗Xkj
!
=m
n
X
k=1
ε(Xik)⊗Xkj
!
=m
n
X
k=1
δik⊗Xkj
!
=m(1⊗Xij) =Xij and
m(id⊗ε)∆(Xij) =m(id⊗ε)
n
X
k=1
Xik⊗Xkj
!
=m
n
X
k=1
Xik⊗ε(Xkj)
!
=m
n
X
k=1
Xik⊗δkj
!
=m(Xij⊗1) =Xij,
for all 1≤i, j≤n; which proves thatεis a counit and thusO(Mn(k)) is a bialgebra.
Definition 2.12 LetC be a coalgebra and letc∈C.
(i) We say that c is a group-like element if ∆(c) = c⊗c and ε(c) = 1. We denote the set of group-like elements by G(C). If C has a bialgebra structure, then G(C) is a group under the multiplication.
(ii) For a, b ∈ G(C), c is called a (a, b)-primitive element if ∆(c) = a⊗c+c⊗b. The set of (a, b)-primitive elements is denoted by
Pa,b ={c∈C|∆(c) =a⊗c+c⊗b};
in particular, k(a−b)⊆Pa,b. If C is a bialgebra and we take a= 1 =b, the elements ofP1,1
are called simplyprimitive elements.
Examples 2.13 (i) Let G be a finite group and consider the bialgebra structure on kG. Then G⊆G(kG), since ∆(g) =g⊗g for allg∈G. Moreover, on has that G=G(kG).
(ii) Consider now the bialgebra structure in kG. Then G(kG) = Algk(kG,k) = G, whereb Gb is the character group ofG.
LetCbe a coalgebra andAan algebra. The set Homk(C, A) becomes an algebra under theconvolution productgiven by
(f ∗g)(c) =f(c(1))g(c(2)) for allf, g∈Homk(C, A), c∈C.
The unit element in Homk(C, A) isuεwithuε(c) =ε(c)1Afor allc∈C. Note that whenA=k, then Homk(C,k) =C∗ and the algebra structure is the one defined in Exercise 5.
Definition 2.14 Let (H, m, u,∆, ε) be a bialgebra and consider Endk(H) = Homk(H, H) as algebra with the convolution product.
(i) An endomorphismS ofH is called anantipode for the bialgebra H if
S ∗idH =uε= idH∗S. (2) That is,S is the inverse of the identity in Endk(H).
(ii) We say that (H, m, u,∆, ε,S) is a Hopf algebra ifS is an antipode for H. We shall denote it only byH if no confusion arrives.
Remarks 2.15 (i) A bialgebra does not need necessarily to have an antipode. If it does, it has only one, simply becauseS is the inverse of the identity in Endk(H).
(ii) Using the definition of the convolution product, we can re-write equation (2) to the following equality for allh∈H
S(h(1))h(2)=uε(h) =h(1)S(h(2)). (3) This equation is usually stated as the antipode axiom for the definition of a Hopf algebra. The corresponding commutative diagram is the following
H
∆
{{wwwwwwwww ∆
##G
GG GG GG GG
uε
H⊗H
id⊗S
H⊗H
S⊗id
H⊗H
mGGGGGG##
GG
G H⊗H
{{wwwwwwmwww
H
since by definition we must have that m(S⊗ ∈)∆(h) =m(id⊗S)∆(h), for allh∈H.
Example 2.16 Let G be a finite group. Then the group algebrakGis a Hopf algebra, where the antipode is defined by S(eg) = eg−1 for allg ∈G. Indeed, since the elements {eg}g∈G for a linear basis ofkG,S is defined as a linear map by its values on the basis. Let us check thatS is an antipode by showing equality (3) oneg for allg∈G. Recall that ∆(eg) =eg⊗eg.
S(eg)eg=eg−1eg=eg−1g=e1= 1kG=ε(eg)1kG, egS(eg) =egeg−1 =egg−1 =ε(eg)1kG.
Example 2.17 LetGbe a finite group. Then the algebrakG is a Hopf algebra, where the antipode is defined byS(δg) =δg−1 for allg ∈G. Indeed, since the elements{δg}g∈G for a linear basis of kG, S is defined as a linear map by its values on the basis. Let us check thatS is an antipode by showing equality (3) onδgfor allg∈G. Recall that ∆(δg) =P
h∈Gδh⊗δh−1g. X
h∈G
S(δh)δh−1g=X
h∈G
δh−1δh−1g==X
h∈G
δh−1,h−1gδh−1=X
h∈G
δg,1δh−1 =ε(δg)X
h∈G
δh−1=ε(δg)1kG, X
h∈G
δhS(δh−1g) =X
h∈G
δhδg−1h==X
h∈G
δh,g−1hδh=X
h∈G
δg−1,1δh=ε(δg−1)X
h∈G
δh=ε(δg)1kG.
Proposition 2.18 [Sw, Prop. 4.0.1]Let H be a Hopf algebra with antipodeS. Then
(a) S(hk) = S(k)S(h) and S(1) = 1, for all h, k ∈ H. In particular, S defines an algebra map S :H →Hop.
(b) ∆(S(h)) = S(h(2))⊗ S(h(1)) for all h ∈H. In particular, S : H → Hcop defines a coalgebra map.
Remark 2.19 Suppose that B is a bialgebra over k which is generated as algebra by the elements {bi}i∈I. Then to define an antipode on B it is enough to define S on the generators such that S:B→Bop is an algebra map and equality (3) holds for allbi, i∈I.
As for coalgebras an bialgebras, we have the obvious definitions for Hopf algebra maps and Hopf ideals.
A linear map f :H →K between two Hopf algebras is called aHopf algebra map iff is a bialgebra map and f(SH(h)) = SK(f(h)) for all h ∈ H. Actually, it can be proved using the uniqueness of the antipode that iff : H → K is a bialgebra map between two Hopf algebras, then necessarilyf preserves the antipode,i.e. f is a Hopf algebra map.
A linear subspaceIof a Hopf algebraH is calleda Hopf ideal ifIis a bi-ideal andS(I)⊆I. Clearly, I ⊆H is a Hopf ideal if and only if the quotient vector space H/I is a Hopf algebra. For example, H+= Kerεis a Hopf ideal ofH and it is called theaugmentation ideal ofH.
With this definition, it is straighforward to check that for any finite group G, (kG)∗ 'kG as Hopf algebras. See the exercises.
Example 2.20 Let (H, m, u,∆, ε,S) be a Hopf algebra over k. Then using the flip map τ, one can easily prove that (Hop, mop, u,∆, ε,S−1), (Hcop, m, u,∆cop, ε,S−1) and (Hop,cop, mop, u,∆cop, ε,S) are Hopf algebras, whereHop=H as coalgebra but with the opposite multiplicationmop(h⊗k) = m◦τ(h⊗k) =m(k⊗h) andHcop =H as algebra but with the opposite comultiplication, that is,
∆cop(h) =τ◦∆(h) =h(2)⊗h(1) for allh∈H. We leave the proof of these claims as exercise for the reader.
Example 2.21 Recall from Example 2.11 that forn= 2, the algebra O(M2(k)) =k[X11, X12, X21, X22]
has a bialgebra structure. To make the notation not so heavy we write from now on O(M2) = O(M2(k)) and
a=X11, b=X12, c=X21, and d=X22.
Then, the comultiplication is determined by
∆(a) =a⊗a+b⊗c, ∆(b) =a⊗b+b⊗d,
∆(c) =c⊗a+d⊗c, ∆(d) =c⊗b+d⊗d, and the counit byε(a) = 1 =ε(d),ε(b) =ε(c) = 0.
We define nowO(SL2) by the commutative algebra generated by the elementsa, b, c, d satisfying the relationad−bc= 1. For short we write
O(SL2) =k[a, b, c, d|ad−bc= 1].
To see thatO(SL2) inherits the bialgebra structure ofO(M2), it is enough to prove that the comulti- plication ∆ and the counits ε are well-defined algebra maps on the quotient, which is the same as saying that the idealI=O(M2)(ad−bc−1) generated by the elementad−bc−1 is a bi-ideal. Thus we have to prove thatε(ad−bc) =ε(1) = 1 and ∆(ad−bc) = (ad−bc)⊗(ad−bc) = ∆(1) = 1⊗1 hold in O(Mn). Indeed, letA=O(M2) and denote byπ:A→A/I the canonical quotient. By Theorem 2.6, the comultimplication ∆A/I is induced by the composition (π⊗π)∆:
A ∆ //
π
A⊗A
π⊗π
A/I A/I⊗A/I
Hence, we have to see thatI⊆Ker(π⊗π)∆. But ift=ad−bcand ∆(t) = 1, then
∆(A(t−1)) = ∆(A)∆(t−1) = (A⊗A)(t⊗t−1⊗1) = (A⊗A)((t−1)⊗t+ 1⊗(t−1))
=A(t−1)⊗At+A⊗A(t−1)⊆I⊗A+A⊗I,
which implies that (π⊗π)∆(I) = 0. Analogously, εA/I is induced byε:A→k andI⊆Kerεsince ε(t) = 1. Thus
ε(ad−bc) =ε(a)ε(d)−ε(b)ε(c) = 1 and
∆(ad−bc) = ∆(a)∆(d)−∆(b)∆(b) = (a⊗a+b⊗c)(c⊗b+d⊗d)−(a⊗b+b⊗d)(c⊗a+d⊗c)
=ac⊗ab+bc⊗cb+ad⊗ad+bd⊗cd−bc⊗da−bd⊗dc−ac⊗ba−ad⊗bc
=ad⊗(ad−bc)−bc⊗(da−cb) = (ad−bc)⊗(ad−bc).
ThusO(SL2) is a bialgebra. This implies in particular that thedeterminantt=ad−bcis a group-like element inO(Mn).
Furthermore,O(SL2) is a Hopf algebra with the antipode given by
S(a) =d, S(b) =−b, S(c) =−c, and S(d) =a.
If we write
a b c d
⊗ a b
c d
=
a⊗a+b⊗c a⊗b+b⊗d c⊗a+d⊗c c⊗b+d⊗d
, we can write
S(a) S(b) S(c) S(d)
=
d −b
−c a
Note that the antipode matrix is given by the inverse matrix, since the determinant is equal to 1 in O(SL2). Actually, O(Mn) is not a Hopf algebra with the bialgebra structure defined above, since the determinantt is a group-like element which is not invertible. Notably, adding an inverse t−1 to O(Mn) is enough to give a Hopf algebra structure on the localization O(Mn)[t−1] ofO(Mn) att−1. This Hopf algebra is calledO(GLn) and corresponds to the algebra of regular functions on GLn(k).
Another way to obtain a Hopf algebra is to take the quotient by the relation t = 1, which defines O(SLn).
In matrix notation, the coassociativity follows from the equality a b
c d
⊗ a b
c d
⊗ a b
c d
= a b
c d
⊗
a b c d
⊗ a b
c d
,
and the counit from
a b c d
1 0 0 1
= a b
c d
= 1 0
0 1
a b c d
.
To prove that S defines an antipode for O(SL2), by Remark 2.19 we have to prove first that S : O(SL2)→ O(SL2)opis a well-defined algebra map, and then check equation (3) for the generators.
SinceS(1) = 1 andS(ad−bc) =S(ad)− S(bc) =S(d)S(a)− S(c)S(b) =ad−(−c)(−b) =ad−cb= ad−bc, it follows thatS :O(SL2)→ O(SL2)opis a well-defined algebra map.
To check equation (3) for the generators is equivalent to prove the following matrix equality a b
c d
S(a) S(b) S(c) S(d)
=
S(a) S(b) S(c) S(d)
a b c d
=
ε(a) ε(b) ε(c) ε(d)
1 0 0 1
,
which follows from the equalityad−bc= 1 inO(SL2) and we leave it as exercise.
Remark 2.22 Recall that SL2(k) is the subgroup of matrices ofGL2(k) given by SL2(k) ={A∈k2×2: detA= 1}.
Then,O(SL2(k)) is the commutative algebra of rational functions onSL2(k) generated by the matrix coefficients via
a(A) =a11, b(A) =a12, c(A) =a21 and d(A) =a22, for allA= (aij)1≤i,j≤2. Note that
(ad−bc)(A) =a(A)d(A)−b(A)c(A) =a11a22−a12a21= detA= 1.
Moreover, every matrix A∈ SL2(k) defines an element of the group Algk(O(SL2(k)),k) of algebra maps fromO(SL2(k)) tok, by
A(a) =a11, A(b) =a12, A(c) =a21, A(d) =a22 and A(1) = 1.
It is well-defined sinceA(ad−bc) =A(a)A(d)−A(b)A(c) =a11a22−a12a21= detA= 1. Conversely, every elementαof Algk(O(SL2(k)),k) defines a matrix in SL2(k) by
α(a) α(b) α(c) α(d)
,
and it holds thatα(a)α(d)−α(b)α(c) =α(ad−bc) = 1. Hence we have a group isomorphism SL2(k)'Algk(O(SL2(k)),k).
We have constructed Hopf algebras coming from groups, which are commutative and represent algebras of functions on these groups. We end this section with the following theorem that states that if the field k is algebraically closed of characteristic zero, then all commutative Hopf algebras arise in this way.
Theorem 2.23 [Cartier]Let k be an algebraically closed field of characteristic zero.
(a) Let H be a finite-dimensional commutative Hopf algebra. Then H is isomorphic to kG, where Gis the finite group given byG= Spec(H) = Algk(H,k).
(b) Let H be a commutative Hopf algebra, thenH is isomorphic to the algebra of regular functions O(G)on a (pro) algebraic groupG.
2.1 Exercises
1) Prove that the set{δx}x∈X defined in (1) is a linear basis ofA=kX and it is an algebra with the multiplication given byδxδy =δxy for allx, y∈X and the unit by 1 =P
x∈Xδx.
2) LetGbe a finite group. Prove thatkG is a coalgebra whose dimension is equal to the order of the group.
3) Let C be a k-vector space with basis {cm| m ∈ N∪ {0}}. Prove that C is a coalgebra with comultiplication ∆ and counitεdefined for allm∈N∪ {0}by
∆(cm) =
m
X
i=0
ci⊗cm−i, ε(cm) =δ0,m.
4) Let C be a k-vector space with basis{s, c}. Prove thatC is a coalgebra with comultiplication ∆ and counitεdefined by
∆(s) =s⊗c+c⊗s, ε(s) = 0,
∆(c) =c⊗c−s⊗s, ε(s) = 1.
5) LetC be coalgebra overk.
(a) Prove that the dual spaceC∗ ={f :C →k| f is linear} is an algebra with the multiplication and unit defined by
(f·g)(c) =f(c(1))g(c(2)) and 1(c) =ε(c) for allf, g∈C∗, c∈C, where ∆(c) =c(1)⊗c(2) is the comultiplication ofc∈C.
(b) Prove thatD is a subcoalgebra ofC if and only ifD⊥={f :C→k|f(D) = 0} is a two-sided ideal ofC∗.
(c) Prove thatI is a coideal ofC if and only ifI⊥={f :C→k|f(I) = 0}is a subalgebra ofC∗. 6) Prove Theorem 2.6.
7) LetAbe a finite-dimensional associative unitalk-algebra.
(a) Prove thatA∗ is a coalgebra. Hint: Use that (A⊗A)∗'A∗⊗A∗.
(b) Prove thatB is a subalgebra ofAif and only ifB⊥ ={f :A→k|f(B) = 0}is a coideal ofA∗. (c) Prove thatIis a two-sided ideal ofAif and only ifI⊥={f :A→k|f(I) = 0}is a subcoalgebra
ofA∗.
8) LetB be ak-vector space endowed with an algebra structure (B, m, u) and a coalgebra structure (B,∆, ε). Prove that ∆ andεare algebra maps if and only ifmanduare coalgebra maps.
9) LetB be a bialgebra,Ia bi-ideal ofB andπ:B→B/Ithe canonical linear map onto the quotient vector space. Then
(a) B/I has a unique bialgebra structure such thatπis a bialgebra map.
(b) Iff :B→B0 is any bialgebra map then Kerf is a bi-ideal.
(c) IfI⊆Kerf then there is a unique bialgebra map ¯f such that the following diagram commutes
B f //
πBBBBB!!
BB
B B0
B/I
f¯
=={
{{ {{ {{ {
10) Let Gbe a finite group and consider the bialgebra structure on kG defined above. Prove that G=G(kG).
11) Let G be a finite group and consider the bialgebra structure on kG defined above. Prove that G(kG) = Algk(kG,k) =G, whereb Gb is the character group ofG.
12) LetGbe a finite group. Prove that (kG)∗'kG as Hopf algebras. Hint: Use thatkG ⊆(kG)∗ via hδg, ehi=δgh for allg, h∈G.
13) LetH be a finite-dimensional Hopf algebra overk. Prove thatH∗is a Hopf algebra.
14) Let H be a Hopf algebra, I a Hopf ideal of H and π:H →H/I the canonical linear map onto the quotient vector space. Then
(a) H/I has a unique Hopf algebra structure such thatπis a Hopf algebra map.
(b) Iff :H→H0 is any Hopf algebra map then Kerf is a Hopf ideal.
(c) IfI⊆Kerfthen there is a unique Hopf algebra map ¯fsuch that the following diagram commutes
H f //
πCCCCC!!
CC
C H0
H/I
f¯
==z
zz zz zz z
15) Let (H, m, u,∆, ε,S) be a Hopf algebra over a field k. Prove that (Hop, mop, u,∆, ε,S−1), (Hcop, m, u,∆cop, ε,S−1) and (Hop,∆, mop, u,∆∆, ε,S) are Hopf algebras.
16) Prove Proposition 2.18.
17) Prove that the group homomorphism
Algk(O(SL2(k)),k)−→ϕ SL2(k), α7→
α(a) α(b) α(c) α(d)
is an isomorphism with inverseψdetermined by
ψ(A)(a) =a11, ψ(A)(b) =a12, ψ(A)(c) =a21 and ψ(A)(d) =a22, for allA= (aij)1≤i,j≤2.
18) LetAbe ak-algebra. Thefinite dual orSweedler dual ofAis given by
A◦={f ∈A∗|f(I) = 0, for some two-sided idealI ofAsuch that dimA/I <∞}.
Let (A, m, u,∆, ε,S) be a Hopf algebra. Prove that A◦ is a Hopf algebra with the structural maps given by
mA◦= ∆∗:A◦⊗A◦→A◦ ∆∗(f⊗g)(a) = (f ⊗g)∆(a),
uA◦= ε∗:k→A◦ ε∗(λ)(a) = λε(a),
∆A◦= m∗:A◦→A◦⊗A◦ m∗(f)(a⊗b) = f(ab), εA◦= u∗:A◦→k u∗(f) = f(1), SA◦= S∗:A◦ →A◦ S∗(f)(a) = f(S(a)),
for all a, b ∈ A, f, g ∈ A◦. In particular, if A is finite-dimensional, then A◦ = A∗ and whence (A∗,∆∗, ε∗, m∗, u∗,S∗) is a Hopf algebra.
3 Quantum groups
From now on we will assume thatk is an algebraically closed field of characteristic zero. In the last section we saw that to to any commutative Hopf algebra corresponds a group, and conversely, to any
group corresponds a commutative Hopf algebra
Groups G /o /o /o /o /o /o /o /o /o //O(G) Commutative Hopf algebras
Algk(A,k)oo o/ o/ o/ o/ o/ o/ o/ o/ A
and this bijection is an equivalence
AlgC(O(G),C) =Goo /o /o /o /o /o /o /o //O(G)
Grothendieck’s philosophy was extended to quantum groups by Drinfel’d, who stated that one should quantizeclassical coordinate rings such asO(G) by deforming them to non-commutative Hopf algebras, and that one should study new Hopf algebras as if they consisted ofnon-commuting functions on a non-existing object, namely aquantum group corresponding to G.
Gq oo /o /o /o /o /o /o /o // Oq(G) noncommutative Hopf algebras
Thus, quantum groups do not exist as objects, only their algebras of functions. As a convention, the function algebras themselves are called quantum groups.
There is no rigorous, universally accepted definition of the term quantum group. However, it is generally agreed that this term includes certain deformation of classical objects associated to algebraic groups or to semisimimple Lie algebras. To date no axiomatic definition of this family of algebras has been given, nor a complete formulation of properties an algebra should satisfy in order to qualify as a quantum analogue of a given classical coordinate ring. Thus, it is a field driven much more by examples than by axioms.
Some authors define quantum groups as non-commutative and non-cocommutative Hopf algebras. In this notes, we will follow Drinfeld’s convention: the category of quantum groups is the opposite category of Hopf algebras. That is, as objects quantum groups are Hopf algebras, but the morphisms are the opposite ones. This is because of the following: if Γ is a subgroup ofG, then there is a Hopf algebra surjectionO(G)O(Γ) between the algebras of functions on them.
Γ,→Goo /o /o /o /o /o /o /o /o //O(G)O(Γ)
Γq ,→Gq oo /o /o /o /o /o /o /o //Oq(G)Oq(Γ)
We define in this chapter the first easiest examples of quantum groups that illustrate the theory, Oq(SL2(k)) andUq(sl2)(k).
3.1 Quantum SL
2Letq∈k×=k r{0}.
Definition 3.1 The algebraOq(M2(k)) is the algebra generated by the elements a, b, c, d satisfying the relations
ba=qab, db=qbd, ca=qac, dc=qcd, bc=cb, ad−da= (q−1−q)bc.
Clearly, when q = 1 we have that O1(M2(k)) = O(M2(k)), and if q 6= 1, then Oq(M2(k)) is not commutative.
Theorem 3.2 (a) There exist algebra maps
∆ :Oq(M2(k))→ Oq(M2(k))⊗ Oq(M2(k)), ε:Oq(M2(k))→k,
uniquely determined by
∆(a) =a⊗a+b⊗c, ∆(b) =a⊗b+b⊗d,
∆(c) =c⊗a+d⊗c, ∆(d) =c⊗b+d⊗d, ε(a) =ε(d) = 1, ε(b) =ε(c) = 0
(b) With these morphisms, the algebra Oq(M2(k))is a bialgebra which is neither commutative nor cocommutative ifq6= 1.
(c) If detq :=ad−q−1bc=da−qbc, then∆(detq) = detq⊗detq andε(detq) = 1, that is,detq is a group-like element in Oq(M2(k)). Moreover, it is central.
Proof. (a) In order to prove that ∆ andε are well-defined algebra maps, it is enough to show that the relations hold under ∆ andε,e.g. ∆(ba) =q∆(ab).
∆(ba) = ∆(b)∆(a) = (a⊗b+b⊗d)(a⊗a+b⊗c)
=a2⊗ba+ab⊗bc+ba⊗da+b2⊗dc,
q∆(ab) =q(a⊗a+b⊗c)(a⊗b+b⊗d) =qa2⊗ab+qab⊗ad+qba⊗cb+qb2⊗cd
=a2⊗qab+ba⊗(da+ (q−1−q)bc) +qba⊗bc+b2⊗qcd
=a2⊗ba+ba⊗da+q−1ba⊗bc−qba⊗bc+qba⊗bc+b2⊗dc
=a2⊗ba+ba⊗da+ab⊗bc+b2⊗dc.
Analogously, one can prove that ∆(db) = q∆(bd), ∆(ca) =q∆(ac), ∆(dc) =q∆(cd), ∆(bc) = ∆(cb) and ∆(ad−da) = (q−1−q)∆(bc), and we leave it as exercise for the reader. For ε it is completely analogous. Indeed,
ε(ba) =ε(b)ε(a) = 0 =qε(ab) =qε(a)ε(b) ε(db) =ε(d)ε(b) = 0 =qε(bd) =qε(b)ε(d) ε(bc) =ε(b)ε(c) = 0 =ε(cb) =ε(c)ε(b) ε(dc) =ε(d)ε(c) = 0 =qε(cd) =qε(c)ε(d) ε(ca) =ε(c)ε(a) = 0 =qε(ac) =qε(a)ε(c)
ε(ad−da) =ε(a)ε(d)−ε(d)ε(a) = 0 = (q−1−q)ε(bc) = (q−1−q)ε(b)ε(c).
(b) Since the coalgebra structure defined onOq(M2(k)) is the same as the one defined onO(Mn(k)), it follows thatOq(M2(k)) is a coalgebra, that is,εis a counit and ∆ is coassociative. Since both maps are algebra maps, it follows that Oq(M2(k)) is indeed a bialgebra. Clearly, it is not commutative if q6= 1, and it is not cocommtuative since ∆(a) =a⊗a+b⊗c6=a⊗a+c⊗b=τ◦∆(a).
(c) Let detq=ad−q−1bc. Then
∆(detq) = ∆(a)∆(d)−q−1∆(b)∆(c)
= (a⊗a+b⊗c)(c⊗b+d⊗d)−q−1(a⊗b+b⊗d)(c⊗a+d⊗c)
=ac⊗ab+ad⊗ad+bc⊗cb+bd⊗cd−q−1bc⊗da−q−1bd⊗dc
−q−1ac⊗ba−q−1ad⊗bc
=ac⊗ab+ad⊗(ad−q−1bc) +bc⊗cb+bd⊗cd−q−1bc⊗da−bd⊗q−1dc−ac⊗q−1ba
=ac⊗ab+ad⊗(ad−q−1bc) +bc⊗cb+bd⊗cd−q−1bc⊗da−bd⊗cd−ac⊗ab
=ad⊗(ad−q−1bc) +bc⊗cb−q−1bc⊗da
=ad⊗(ad−q−1bc) +bc⊗cb−q−1bc⊗(ad−(q−1−q)bc)
=ad⊗(ad−q−1bc) +bc⊗cb−q−1bc⊗ad+q−2bc⊗bc−bc⊗bc
=ad⊗(ad−q−1bc)−q−1bc⊗(ad−q−1bc)
= (ad−q−1bc)⊗(ad−q−1bc) = detq⊗detq.
Clearly, ε(detq) =ε(a)ε(d)−q−1ε(b)ε(c) = 1. Thus, detq is a group-like element. To see that it is central, it is enough to verify it on the generators:
detqa= (ad−q−1bc)a=ada−q−1bca=a(ad−(q−1−q)bc)−q−1q2abc
=a(ad−q−1bc) +qabc−qabc=adetq,
detqb= (ad−q−1bc)b=adb−q−1bcb=q−1qbad−bbc
=b(ad−q−1bc) =bdetq,
detqc= (ad−q−1bc)c=adc−q−1bcc=q−1qbad−cbc
=c(ad−q−1bc) =cdetq,
detqd= (ad−q−1bc)d=add−q−1bcd= (da+ (q−1−q)bc)d−q−1bcd
=dad+q−1bcd−qbcd−q−1bcd=dad−qq−2dbc=ddetq.
Definition 3.3 [Mn] We defineOq(SL2(k)) as thek-algebra given by the quotient Oq(SL2(k)) =Oq(M2(k))/(detq−1),
where (detq−1) is the two-sided ideal ofOq(M2(k)) generated by the element detq−1.
In other words, the algebraOq(SL2(k)) can be presented as thek-algebra generated by the elements a, b, c, dsatisfying the relations
ba=qab, db=qbd, ca=qac, dc=qcd, bc=cb, ad−da= (q−1−q)bc, ad−q−1bc= 1.
Clearly, when q = 1 we have that O1(SL2(k)) = O(SL2(k)), and ifq 6= 1, then Oq(SL2(k)) is not commutative.
Since detq is a central group-like element, the ideal (detq−1) of Oq(M2(k)) is indeed a bi-ideal and thusOq(SL2(k)) is a bialgebra with the comultiplication and counit definesd on the generators as in Oq(M2(k)).
Theorem 3.4 Oq(SL2(k)) is a Hopf algebra with the antipode determined by S(a) S(b)
S(c) S(d)
=
d −qb
−q−1c a
, that is,S(a) =d,S(b) =−qb, S(c) =−q−1candS(d) =a.
Proof. First we have to prove thatS:Oq(SL2(k))→ Oq(SL2(k))opis a well-defined algebra map:
S(ba) =S(a)S(b) =d(−qb) =−qdb=−q2bd=qS(b)S(a) =qS(ab), S(db) =S(b)S(d) = (−qb)a=−q2ab=qS(d)S(b) =qS(bd),
S(ca) =S(a)S(c) =d(−q−1c) =−q−1dc=−cd=qS(c)S(a) =qS(ac), S(dc) =S(c)S(d) = (−q−1c)a=−ac=qS(d)S(c) =qS(cd),
S(bc) =S(c)S(b) = (−q−1c)(−qb) =cb=bcS(b)S(c) =S(cb),
S(ad−da) =S(ad)− S(da) =S(d)S(a)− S(a)S(d) =ad−da= (q−1−q)bc= (q−1−q)cb
= (q−1−q)S(c)S(b) = (q−1−q)S(bc),
S(ad−q−1bc) =S(ad)−q−1S(bc) =S(d)S(a)−q−1S(c)S(b) =ad−q−1q−1qcb
=ad−q−1cb=ad−q−1bc= 1 =S(1).
To prove thatSdefines an antipode forOq(SL2(k)), we have to check equation (3) for the generators.
As for the case ofO(SL2(k)), this is equivalent to prove the following matrix equality a b
c d
S(a) S(b) S(c) S(d)
=
S(a) S(b) S(c) S(d)
a b c d
=
ε(a) ε(b) ε(c) ε(d)
1 0 0 1
,
which follows from the defining relations ofOq(SL2(k)) and we leave it as exercise for the reader.
Remark 3.5 The quantum groupOq(SL2(k)) corresponds to thequantized coordinate ring ofSL2(k) generated by the matrix coefficients.
3.2 Quantum sl
2There is another group associated toSL2(k). In effect,SL2(C) is not only a group but also a smooth manifold,i.e. a Lie group. As such, it has a tangent space at the identity, which is a Lie algebra and it is calledsl2. Moreover, one can see that
sl2={A∈M2(C) : tr(A) = 0}.
The quantum group we introduce in this section corresponds to the deformation in one parameter of the enveloping algebraU(sl2) ofsl2. The deformation uses the classification of semisimple Lie algebras over an algebraically closed field of characteristic zero, done by Cartan and Killing. Thus, the fieldk is an arbitrary field with these properties.
The origins of the subject of quantum groups lie in mathematical physics, where the term quantum comes from. The starting point of the study of this subject lies in the Quantum Inverse Scattering Method, with the aim of solving certain integrable quantum systems. A key ingredient in this method is theQuantum Yang-Baxter Equation(QYBE). While there is no general method for solving the QYBE, it was discovered in the early 1980s that some solutions could be constructed from the representation theory of certain algebras resembling deformations of enveloping algebras of semisimple Lie algebras.
The first such deformation of U(sl2), arose from a paper of Kulish and Reshetikhin [KR]. In the mid-1980s, Drinfeld and Jimbo independently discovered analogous deformations corresponding to arbitrary semisimple Lie algebras [Dr, Dr2, Ji].
Definition 3.6 Letq∈k×,q6=±1. We defineUq(sl2)(k) as thek-algebra generated by the elements E, F, K, K−1 satisfying the relations
KK−1=K−1K= 1, KEK−1=q2E, KF K−1=q−2F, EF−F E= K−K−1 q−q−1 . If no confusion arrives, we denote this algebra simply byUq(sl2). Observe that it is non-commutative.
Moreover, it has the following properties.
Proposition 3.7
(a) Uq(sl2)(k) is a noetherian domain with no zero divisors.
(b) The set{EiFjKl : i, j ∈N0, l∈Z} is a linear basis of Uq(sl2)(k). In particular, Uq(sl2)(k) is infinite-dimensional.
Proof. See [K, Prop. VI.1.4] or [BG, Chp. 1.3].
Remark 3.8 The basis given by part (b) is called a PBW-basis ofUq(sl2)(k).
Theorem 3.9
(a) There exist algebra maps
∆ :Uq(sl2)→Uq(sl2)⊗Uq(sl2), ε:Uq(sl2)→k,
uniquely determined by
∆(K) =K⊗K, ∆(K−1) =K−1⊗K−1,
∆(E) = 1⊗E+E⊗K,
∆(F) =K−1⊗F +F⊗1,
ε(K) =ε(K−1) = 1, ε(E) =ε(F) = 0.
(b) With these morphisms,Uq(sl2) is a bialgebra which is non-commutative and non-cocommutative.
In particular, the powers ofK are group-like elements,E∈PK,1andF ∈P1,K−1. (c) Uq(sl2) is a Hopf algebra with the antipode determined by
S(E) =−EK−1, S(F) =−KF, S(K) =K−1 and S(K−1) =K.
Proof. (a) We first show that ∆ defines an algebra map. For this it is enough to check that the ideal of relations is a coideal, or equivalently, that the following equalities hold
∆(KK−1) = ∆(K−1K) = 1⊗1 = ∆(1), ∆(KEK−1) =q2∆(E),
∆(KF K−1) =q−2∆(F), ∆(EF−F E) = ∆
K−K−1 q−q−1
.
The first relations are clear since
∆(KK−1) = ∆(K)∆(K−1) = (K⊗K)(K−1⊗K−1) =KK−1⊗KK−1= 1⊗1.
For the others we have
∆(KEK−1) = (K⊗K)(1⊗E+E⊗K)(K−1⊗K−1)
= (K⊗KE+KE⊗K2)(K−1⊗K−1)
= 1⊗KEK−1+KEK−1⊗K
= 1⊗q2E+q2E⊗K=q2∆(E).
The relation for F is completely analogous and we leave it as exercise for the reader. For the last relation we have
∆(EF−F E) = ∆(E)∆(F)−∆(F)∆(E)
= (1⊗E+E⊗K)(K−1⊗F+F⊗1)−(K−1⊗F+F⊗1)(1⊗E+E⊗K)
=K−1⊗EF+F⊗E+EK−1⊗KF+EF⊗K−K−1⊗F E−K−1E⊗F E
−F⊗E−F E⊗K
=K−1⊗(EF−F E) + (EF−F E)⊗K+EK−1⊗KF −K−1E⊗F K
=K−1⊗(EF−F E) + (EF−F E)⊗K+q2q−2K−1E⊗F K−K−1E⊗F K
=K−1⊗(EF−F E) + (EF−F E)⊗K
=K−1⊗
K−K−1 q−q−1
+
K−K−1 q−q−1
⊗K
= 1
q−q−1(K−1⊗K−K−1⊗K−1+K⊗K−K−1⊗K)
= 1
q−q−1(K⊗K−K−1⊗K−1)
= ∆
K−K−1 q−q−1
.
Now we check thatεis a well-defined algebra map by showing that the equalities in the relations hold after applyingε:
ε(KK−1) =ε(K)ε(K−1) = 1.1 =ε(1) =ε(K−1)ε(K) =ε(K−1K) ε(KEK−1) =ε(K)ε(E)ε(K−1) = 1.0.1 = 0 =q2ε(E)
ε(KF K−1) =ε(K)ε(F)ε(K−1) = 1.0.1 = 0 =q−2ε(F) ε(EF−F E) =ε(E)ε(F)−ε(F)ε(E) = 0 =ε
K−K−1 q−q−1
= ε(K)−ε(K−1) q−q−1 .
(b) To prove that Uq(sl2) is a bialgebra, we need to show that (Uq(sl2),∆, ε) is a coalgebra, since by (a), we know that ∆ andε are algebra maps. We prove thatεis a counit and ∆ is coassociative by checking the equalities
m(ε⊗id)∆ =m(id⊗ε)∆ = id and (∆⊗id)∆ = (id⊗∆)∆, on the generators. We begin by the counit:
m(ε⊗id)∆(K) =m(ε⊗id)(K⊗K) =m(ε(K)⊗K) =m(1⊗K) =K and m(id⊗ε)∆(K) =m(id⊗ε)(K⊗K) =m(K⊗ε(K)) =m(K⊗1) =K,
m(ε⊗id)∆(K−1) =m(ε⊗id)(K−1⊗K−1) =m(ε(K−1)⊗K−1) =m(1⊗K−1) =K−1 and m(id⊗ε)∆(K−1) =m(id⊗ε)(K−1⊗K−1) =m(K−1⊗ε(K−1)) =m(K−1⊗1) =K−1,
m(ε⊗id)∆(E) =m(ε⊗id)(1⊗E+E⊗K) =m(ε(1)⊗E+ε(E)⊗K) =m(1⊗E) =E and m(id⊗ε)∆(E) =m(id⊗ε)(1⊗E+E⊗K) =m(1⊗ε(E) +E⊗ε(K)) =m(E⊗1) =E, m(ε⊗id)∆(F) =m(ε⊗id)(K−1⊗F+F⊗1) =m(ε(K−1)⊗F+ε(F)⊗1) =m(1⊗F) =F
m(id⊗ε)∆(F) =m(id⊗ε)(K−1⊗F+F⊗1) =m(K−1⊗ε(F) +F⊗ε(1)) =m(F⊗1) =F.
For the coassociativity we have
(∆⊗id)∆(K) = (∆⊗id)(K⊗K) = ∆(K)⊗K=K⊗K⊗K and (id⊗∆)∆(K) = (id⊗∆)(K⊗K) =K⊗∆(K) =K⊗K⊗K,
(∆⊗id)∆(K−1) = (∆⊗id)(K−1⊗K−1) = ∆(K−1)⊗K−1=K−1⊗K−1⊗K−1 and (id⊗∆)∆(K−1) = (id⊗∆)(K−1⊗K−1) =K−1⊗∆(K−1) =K−1⊗K−1⊗K−1,
(∆⊗id)∆(E) = (∆⊗id)(1⊗E+E⊗K) = ∆(1)⊗E+ ∆(E)⊗K=
= 1⊗1⊗E+ 1⊗E⊗K+E⊗K⊗K and (id⊗∆)∆(E) = (id⊗∆)(1⊗E+E⊗K) = 1⊗∆(E) +E⊗∆(K)
= 1⊗1⊗E+ 1⊗E⊗K+E⊗K⊗K,
(∆⊗id)∆(F) = (∆⊗id)(K−1⊗F+F⊗1) = ∆(K−1)⊗F+ ∆(F)⊗1 =
=K−1⊗K−1⊗F+K−1⊗F⊗1 +F⊗1⊗1 and (id⊗∆)∆(F) = (id⊗∆)(K−1⊗F+F⊗1) =K−1⊗∆(F) +F⊗∆(1)
=K−1⊗K−1⊗F+K−1⊗F⊗1 +F⊗1⊗1.
Thus ∆ is coassociative and clearlyUq(sl2) is not cocommutative sinceτ◦∆6= ∆ because
∆(E) = 1⊗E+E⊗K6=E⊗1 +K⊗E=τ◦∆(E).
(c) To prove that S is an antipode, we have to check first that S :Uq(sl2)→Uq(sl2)opis an algebra map and then that equality (3) holds for all generators ofUq(sl2). To show thatS defines an algebra map, we have to verify that the equalities of the relations hold when appyingS, but using the opposite multiplication, for exampleS(KEK−1) =S(K−1)S(E)S(K) =q2S(E), but
S(KEK−1) =S(K−1)S(E)S(K) =K(−EK−1)K−1=−KEK−1K−1=−q2EK−1=q2S(E).
Clearly it holds forKandK−1and the computation forFis completely analogous to the computation above and we leave it as exercise. For the last relation we have
S(EF−F E) =S(F)S(E)− S(E)S(F) = (−KF)(−EK−1)−(−EK−1)(−KF)
=KF EK−1−EF =KF q2K−1E−EF=q−2q2KK−1F E−EF =F E−EF
=−K−K−1
q−q−1 =S(K)− S(K−1) q−q−1 =S
K−K−1 q−q−1
.
Thus, S : Uq(sl2) → Uq(sl2)op is a well-defined algebra map. Now we prove that the equality m(id⊗S)∆ =uε=m(S ⊗id)∆ holds by verifying it on the generators:
m(id⊗S)∆(K) =m(id⊗S)(K⊗K) =m(K⊗ S(K)) =m(K⊗K−1) = 1 and m(S ⊗id)∆(K) =m(S ⊗id)(K⊗K) =m(S(K)⊗K) =m(K−1⊗K) = 1,
m(id⊗S)∆(F) =m(id⊗S)(K−1⊗F+F⊗1) =m(K−1⊗ S(F) +F⊗ S(1))
=m(K−1⊗(−KF) +F⊗1) =K−1(−KF) +F = 0 and m(S ⊗id)∆(F) =m(S ⊗id)(K−1⊗F+F⊗1) =m(S(K−1)⊗F+S(F)⊗1)
=m(K⊗F+ (−KF)⊗1) =KF −KF = 0.
The equalities forK−1and Eare again completely analogous and we leave it as exercise.
3.2.1 Quantum Borel subgroups
The quantum groupsUq(b+) andUq(b−) we introduce here are actually quantum quotients, since they are constructed as Hopf subalgebras of Uq(sl2). Their terminology comes from classical Lie theory, sinceU(b+)⊆U(sl).
These quantum groups are just the subalgebras of Uq(sl2) generated by a subset of the generators, K±1, E in the first case, andK±1, F in the second. In particular, they can be described as algebras as follows
Uq(b+) =k{K, K−1, E: KK−1= 1 =K−1K, KEK−1=q2E}
Uq(b−) =k{K, K−1, F : KK−1= 1 =K−1K, KF K−1=q−2F}.
They are called the Quantum Borel subgroups of Uq(sl2) and the positive and the negative part of Uq(sl2), respectively. They are also usually denoted by Uq(sl2)+ and Uq(sl2)−, or Uq(sl2)≥0 and Uq(sl2)≤0, respectively.
From Theorem 3.9 it follows that both algebras are indeed Hopf subalgebras ofUq(sl2), since ∆(E) = 1⊗E+E⊗K∈Uq(b+)⊗Uq(b+) and ∆(F) =K−1⊗F+F ⊗1∈Uq(b−)⊗Uq(b−). Clearly, the multiplication onUq(sl2) induces a surjective Hopf algebra map
Uq(b+)⊗Uq(b−)Uq(sl2).
Thus, the quantum group may be considered in some sense as adoubleobject. This notion was formally defined by Drinfeld who showed a way of producing solutions of the QYBE from Hopf algebras which areDrinfeld doubles.
3.2.2 Relation betweenOq(SL2) and Uq(sl2)
Up to now, we have introduced the quantum groupsOq(SL2) andUq(sl2). In fact, these objects are deeply related and in some sense, they are dual of each other. In this section we describe in a formal way this relation, which holds for quantum groups associated to arbitrary semisimple Lie algebras.
