quantum groups and hopf algebrasggarcia/encuentros/notas-curso-qg... · 2014-02-06 · 4 quantum...

Quantum Groups and Hopf Algebras

Gaston Andres GarcıaUniversidad Nacional de Cordoba, Argentina

XVIII Latin American Algebra ColloquiumSao Pedro, Brazil

August 3rd to August 8th, 2009

Abstract

These notes correspond to a mini-course for graduate and undergraduate students given inthe XVIII Latin American Algebra Colloquium in Sao Pedro, Brazil. Because of the lenght of thecourse, 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 Hopfalgebras over an algebraically closed field of characteristic zero. The reader who is interestedin this subject may have a look at the bibliography on quantum groups and Hopf algebras andreferences therein. We did not intend to give an exhaustive or comprehensive list of the referencesin 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. Upto date there is no rigorous, universally accepted definition, but it is generally agreed that this termincludes certain deformations in one or more parameters of classical objects associated to algebraicgroups, such as enveloping algebras of semisimple Lie algebras or algebras of regular functions on thecorresponding algebraic groups. As one can relate algebraic groups with commutative Hopf algebrasvia group schemes, it is also agreed that the category of quantum groups should correspond to theopposite category of the category of Hopf algebras. This is why some authors define quantum groupsas 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. Firstin relation with algebraic groups and later as objects of self interest. One of the main open problemsin the theory of Hopf algebras is the classification of Hopf algebras H of a fixed dimension over analgebraically closed field of characteristic zero. Up to now, very few general results are known and theclassification is solved only if the dimension N of H is smaller or equal than 19, if N can be factorizedin a simple way, i.e. N = p, p2, 2p2 with p a prime number, or if the Hopf algebra has additionalproperties such as semisimplicity or pointness. It turns out that there is a deep relation betweensemisimple Hopf algebras and group theory, and between pointed Hopf algebras and Lie theory. Sincewe 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 byquantum groups. They consists of a large family with different structural properties and were usedwith profit to solve the classification problem for fixed dimensions.

After introducing Hopf algebras, together with some basic examples, we give in Section 3 the definitionof the simpliest quantum groups Oq(SL2(k)) and Uq(sl2)(k) over an algebraically closed field k ofcharacteristic zero. If q is a primitive `-th root of unity of k, then one can define the small quantumgroup uq(sl2)(k), which is a finite-dimensional quotient of Uq(sl2)(k) of dimension `3.Finally in Section 4, we show how these small quantum groups and variation of them enter into thescene of the classification problem of Hopf algebras of dimension p2 and pointed Hopf algebras ofdimension p3.

1

2 Quantum Groups and Hopf Algebras

2 Hopf algebras

One of the major threads running through this subject has it roots in a philosophy proposed byGrothendieck, which states that one should study objects by means of the functions on them. Thisallows to relate algebraic objects to geometric objects and vice versa. In this section we outline howthis philosophy leads naturally to the concept of a Hopf algebra.

Let X be a finite set. ThenA = CX = {f : X → C| f function}

is a unital algebra over C of finite dimension. Indeed, A is an algebra over a field k if

(i) A is a vector space over k,

(ii) A has a multiplication

m : A×A→ A, (f, g) 7→ fg, with (fg)(x) = f(x)g(x),

which is associative, i.e. (fg)h = f(gh) for all f, g, h ∈ A.

(iii) A has a unitu : k→ A, λ 7→ λ · 1A,

where 1A = u(1k) is the unit of the algebra and the image of u is contained in the centre of A.

Since m is a bilinear map (i.e. linear in each component), we may consider it as a linear map

m : A⊗A→ A, f ⊗ g 7→ fg, with (fg)(x) = f(x)g(x).

In this case, the associativity can be described by the commutative diagram

A⊗A⊗Am⊗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⊗id::ttttttttt

'$$JJJJJJJJJJ A⊗ k

id⊗uddJJJJJJJJJ

'zztttttttttt

A

Note that in this case, A is commutative, that is (fg)(x) = f(x)g(x) = g(x)f(x) = (gf)(x) for allf, g ∈ A, x ∈ X. Another way of saying this is the following:

Definition 2.1 Let V and W be two k-vector spaces. The flip map τ is the linear map τ : V ⊗W →W ⊗ V given by τ(v ⊗ w) = w ⊗ v for all v ∈ V, w ∈W .

Then A is commutative if and only if m ◦ τ = m in A⊗A.

XVIII Latin American Colloquium 3

Let x ∈ X and define δx ∈ A by

δx(y) = δx,y =

{1 if x 6= y,

0 if x = y.(1)

Then the set {δx}x∈X is a linear basis of A. In particular, dimA = |X|. The multiplication of A isgiven by δxδy = δxy and the unit by 1 =

∑x∈X δx.

Conversely, given a finite-dimensional commutative algebra over C without nilpotent elements, we canassociate to it the set

X = SpecA = {α : A→ C| α is 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 anequivalence

{finite sets} oo ///o/o/o/o/o/o/o {comm. finite-dim. C− alg. without nilp. elem.}

X� // CX

SpecA oo �A

Clearly, if A = CX , then X ⊆ Spec CX , since for any x ∈ X we may define x(α) = α(x) for all α ∈ Aand it holds that x(αβ) = (αβ)(x) = α(x)β(x) = x(α)x(β) for all α, β ∈ A; that is, x is an algebramap. The other equality follows from Hilbert’s Nullstellensatz.

Suppose now that X = G is a finite group. Then we have maps

m : G×G→ G u : {1} → G S : G→ G

(g, h) 7→ gh 1 7→ e g 7→ g−1,

where m is the product of the group, u gives the unit and S gives the inverse. What are the corre-sponding maps in CG? Using these maps we may define in CG 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 arethe transpose maps of m, u and S, respectively. In particular, ∆ is coassociative: since f((gh)k) =f(g(hk)) for all f ∈ 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 all f ∈ CG. This motivates the following definition.From now on, k will denote a field.

Definition 2.2 A k-colgebra is a triple (C,∆, ε), where C is a k-vector space ∆ : C → C ⊗ C andε : 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'

%%JJJJJJJJJJ

k⊗ C C ⊗ k

C ⊗ Cε⊗id

ddJJJJJJJJJ id⊗ε

::ttttttttt


The map ∆ is called coproduct or comultiplication and the map ε is called counit. Usually we refer toa coalgebra only by C, if no confusion arrives. We say that C is cocommutative if τ ◦∆ = ∆ in C.Note that the commutative diagrams are the same diagrams as in the definition of algebra, but withinverted arrows.

Example 2.3 Let X be a non-empty set and consider the k-vector space kX with basis {ex}x∈X .Then kX is a coalgebra with the linear maps defined on the basis elements by

∆(ex) = ex ⊗ ex, ε(ex) = 1 for all x ∈ X.

Indeed, since ∆ and ε are defined on a basis of kX, 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 all x ∈ X. For the counit we have to check that ex = m(ε ⊗ id)∆(ex) = m(id⊗ε)∆(ex) for allx ∈ X. But

m(ε⊗ id)∆(ex) = m(ε⊗ id)(ex ⊗ ex) = m(ε(ex)⊗ ex) = m(1⊗ ex) = ex andm(id⊗ε)∆(ex) = m(id⊗ε)(ex ⊗ ex) = m(ex ⊗ ε(ex)) = m(ex ⊗ 1) = ex.

Example 2.4 Let G be a finite group. Then kG is a coalgebra with the comultiplication and counitgiven by

∆(δg) =∑h∈G

δh ⊗ δh−1g and ε(δg) = δg,e,

for all g ∈ G, where e ∈ G is the identity of the group. Note that ∆ and ε are the linear maps inducedby the group operations m and u. Indeed,

∆(δg)(h⊗ t) = δg(ht) =

{1 if g = ht

0 if g 6= ht=

{1 if h−1g = t,

0 if h−1g 6= t

Since {δg}g∈G is a linear basis of kG, it follows that ∆(δg) =∑h∈G δh ⊗ δh−1g. Analogously, ε(δg) =

δg(e) = δg,e. Since m is associative and u gives the unit, it follows that kG is a coalgebra.

Definition 2.5 Let C and D be two colgebras with comultiplication ∆C and ∆D and counit εC andεD, respectively.

(i) A linear map f : C → D is called a colgebra map if ∆D ◦ f = (f ⊗ f)∆C and εC = 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 and εC(I) = 0.

The following theorem says that coideals can be viewed as kernels of coalgebra maps and vice versa.We leave its proof as exercise.

Theorem 2.6 [Sw, Thm. 1.4.7] Let C be a coalgebra, I a coideal of C and π : C → C/I the canonicallinear map onto the quotient vector space. Then

(a) C/I has a unique coalgebra structure such that π is a coalgebra map. This structure is inducedby

∆C/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 then Ker f is a coideal.


(c) If I ⊆ Ker f then there is a unique coalgebra map f such that the following diagram commutes

Cf //

π !!BBBBBBBB D

C/I

f

=={{{{{{{{

In particular, from the theorem follows that for all coalgebra C, C+ = Ker ε ⊆ C is a coideal of C,since εC is a coalgebra map.

Remark 2.7 In coalgebra theory, one uses usually the Sweedler sigma notation for the comultiplica-tion: if c ∈ C, we denote ∆(c) =

∑i ai ⊗ bi ∈ C ⊗ C by

∆(c) = c(1) ⊗ c(2).

For example, the coassociativity axiom of C given by the equality (∆⊗ id) ◦∆ = (id⊗∆) ◦∆, can bewritten 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 all c ∈ 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,∆ and ε are algebra maps. Indeed,

∆(δgδh)(s⊗ t) = δgδh(st) = δg(st)δh(st) = [∆(δg)(s⊗ t)][∆(δh)(s⊗ t)],

for all s, t ∈ G. This implies that ∆(δgδh) = ∆(δg)∆(δh) for all g, h ∈ G. Moreover, as the unit of kGis given by 1kG =

∑g∈G δg, we have

∆(1kG) = ∆(∑g∈G

δg) =∑g∈G

∆(δg) =∑g,h∈G

δh⊗δh−1g =∑k,h∈G

δh⊗δk = (∑h∈G

δh)⊗(∑k∈G

δk) = 1kG⊗1kG .

That is, ∆ is an algebra map. Analogously, it can be seen that ε is an algebra map and we leave it asexercise for the reader.Furthermore, this happens to be equivalent to m and u being algebra maps. This motivates thefollowing definition.

Definition 2.8 A bialgebra is a k-vector space B endowed with an algebra structure (B,m, u) anda coalgebra structure (B,∆, ε) such that ∆ and ε are algebra maps, or equivalently, m and u arecoalgebra maps. That is, ∆ and ε must satisfy

∆(ab) = (ab)(1) ⊗ (ab)(2) = a(1)b(1) ⊗ a(2)b(2) = ∆(a)∆(b), ∆(1) = 1⊗ 1 andε(ab) = ε(a)ε(b), ε(1) = 1, for all a, 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 → B′

between two bialgebras is a bialgebra map if f is an algebramap and a coalgebra map. A subspace I ⊆ B is called a bi-ideal if it is a two-sided ideal and a coideal.As before, I is a bi-ideal of a bialgebra B if and only if the k-vector space B/I is a bialgebra with thestructure induced by the quotient.


Example 2.9 Let G be a finite group. We have seen in Example 2.3 that the linear space kG withbasis {eg}g∈G is a coalgebra. It also has an algebra structure with unit 1kG = e1 and the multiplicationdefined by egeh = egh for all g, h ∈ G. The algebra kG is usually called the group algebra. Moreover,since

∆(e1) = e1 ⊗ e1 and ∆(egeh) = ∆(egh) = egh ⊗ egh = (eg ⊗ eg)(eh ⊗ eh) = ∆(eg)∆(eh),

it follows that kG is a bialgebra.

Example 2.10 Let G be a finite group. In Example 2.4 we showed that kG has a coalgebra structure.It is easy to see that this coalgebra structure is compatible with the algebra structure defined above,giving kG 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 hasa monoid structure with respect to the multiplication, since not all elements are invertible. LetO(Mn(k)) be the commutative algebra over k generated by the elements {Xij | 1 ≤ i, j ≤ n}. Asalgebra, it is simply the commutative ring of polynomials in n2 variables

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} onMn(k), where Xij is the function defined by the matrix coefficient

Xij(A) = aij for all A = (aij)1≤i,j≤n ∈Mn(k).

If we denote by Eij the matrix with a 1 in the entry (i, j) and zero in all others, the set {Eij}1≤i,j≤nis a linear basis of Mn(k) and the set {Xij}1≤i,j≤n is the correspondig dual basis with

〈Xij , Ekl〉 = δikδjl.

Therefore, O(Mn(k)) is the algebra of regular functions on Mn(k). O(Mn(k)) is a bialgebra with thecoalgebra structure determined by

∆(Xij) =n∑k=1

Xik ⊗Xkj , ε(Xij) = δij for all 1 ≤ i, j ≤ n.

Indeed, since O(Mn(k)) is generated as a free algebra by the elements {Xij | 1 ≤ i, j ≤ n}, to definethe algebra maps ∆ and ε, it suffices to define them on the generators. Moreover, since both mapsare uniquely determined by their values on the generators, it is enought to check the coassociativityand the counit axioms on them. For the coassociativity we have

(∆⊗ id)∆(Xij) = (∆⊗ id)

(n∑k=1

Xik ⊗Xkj

)=

n∑k=1

∆(Xik)⊗Xkj =n∑

k,l=1

Xil ⊗Xlk ⊗Xkj and

(id⊗∆)∆(Xij) = (id⊗∆)

(n∑l=1

Xil ⊗Xlj

)=

n∑l=1

Xil ⊗∆(Xlj) =n∑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∑k=1

Xik ⊗Xkj

)= m

(n∑k=1

ε(Xik)⊗Xkj

)

= m

(n∑k=1

δik ⊗Xkj

)= m(1⊗Xij) = Xij and

m(id⊗ε)∆(Xij) = m(id⊗ε)

(n∑k=1

Xik ⊗Xkj

)= m

(n∑k=1

Xik ⊗ ε(Xkj)

)

= m

(n∑k=1

Xik ⊗ δkj

)= m(Xij ⊗ 1) = Xij ,

for all 1 ≤ i, j ≤ n; which proves that ε is a counit and thus O(Mn(k)) is a bialgebra.


Definition 2.12 Let C be a coalgebra and let c ∈ C.

(i) We say that c is a group-like element if ∆(c) = c ⊗ c and ε(c) = 1. We denote the set ofgroup-like elements by G(C). If C has a bialgebra structure, then G(C) is a group under themultiplication.

(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 of P1,1

are called simply primitive elements.

Examples 2.13 (i) Let G be a finite group and consider the bialgebra structure on kG. ThenG ⊆ G(kG), since ∆(g) = g ⊗ g for all g ∈ G. Moreover, on has that G = G(kG).

(ii) Consider now the bialgebra structure in kG. Then G(kG) = Algk(kG,k) = G, where G is thecharacter group of G.

Let C be a coalgebra and A an algebra. The set Homk(C,A) becomes an algebra under the convolutionproduct given by

(f ∗ g)(c) = f(c(1))g(c(2)) for all f, g ∈ Homk(C,A), c ∈ C.

The unit element in Homk(C,A) is uε with uε(c) = ε(c)1A for all c ∈ C. Note that when A = k, thenHomk(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 algebrawith the convolution product.

(i) An endomorphism S of H is called an antipode 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 if S is an antipode for H. We shall denote itonly by H if no confusion arrives.

Remarks 2.15 (i) A bialgebra does not need necessarily to have an antipode. If it does, it has onlyone, simply because S 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 followingequality for all h ∈ 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. Thecorresponding commutative diagram is the following

H∆

{{wwwwwwwww∆

##GGGGGGGGG

uε

��

H ⊗H

id⊗S��

H ⊗H

S⊗id

��H ⊗H

m##GGGGGGGGG H ⊗H

m{{wwwwwwwww

H

since by definition we must have that m(S⊗ ∈)∆(h) = m(id⊗S)∆(h), for all h ∈ H.


Example 2.16 Let G be a finite group. Then the group algebra kG is a Hopf algebra, where theantipode is defined by S(eg) = eg−1 for all g ∈ G. Indeed, since the elements {eg}g∈G for a linearbasis of kG, S is defined as a linear map by its values on the basis. Let us check that S is an antipodeby showing equality (3) on eg for all g ∈ 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 Let G be a finite group. Then the algebra kG is a Hopf algebra, where the antipodeis defined by S(δg) = δg−1 for all g ∈ 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 that S is an antipode by showingequality (3) on δg for all g ∈ G. Recall that ∆(δg) =

∑h∈G δh ⊗ δh−1g.∑

h∈G

S(δh)δh−1g =∑h∈G

δh−1δh−1g ==∑h∈G

δh−1,h−1gδh−1 =∑h∈G

δg,1δh−1 = ε(δg)∑h∈G

δh−1 = ε(δg)1kG,∑h∈G

δhS(δh−1g) =∑h∈G

δhδg−1h ==∑h∈G

δh,g−1hδh =∑h∈G

δg−1,1δh = ε(δg−1)∑h∈G

δh = ε(δg)1kG.

Proposition 2.18 [Sw, Prop. 4.0.1] Let H be a Hopf algebra with antipode S. Then

(a) S(hk) = S(k)S(h) and S(1) = 1, for all h, k ∈ H. In particular, S defines an algebra mapS : H → Hop.

(b) ∆(S(h)) = S(h(2)) ⊗ S(h(1)) for all h ∈ H. In particular, S : H → Hcop defines a coalgebramap.

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 thatS : B → Bop is an algebra map and equality (3) holds for all bi, i ∈ I.

As for coalgebras an bialgebras, we have the obvious definitions for Hopf algebra maps and Hopfideals.

A linear map f : H → K between two Hopf algebras is called a Hopf algebra map if f is a bialgebramap and f(SH(h)) = SK(f(h)) for all h ∈ H. Actually, it can be proved using the uniqueness ofthe antipode that if f : H → K is a bialgebra map between two Hopf algebras, then necessarily fpreserves the antipode, i.e. f is a Hopf algebra map.

A linear subspace I of a Hopf algebra H is called a Hopf ideal if I is a bi-ideal and S(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+ = Ker ε is a Hopf ideal of H and it is called the augmentation ideal of H.

With this definition, it is straighforward to check that for any finite group G, (kG)∗ ' kG as Hopfalgebras. 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 caneasily 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, where Hop = H as coalgebra but with the opposite multiplication mop(h ⊗ k) =m ◦ τ(h ⊗ k) = m(k ⊗ h) and Hcop = H as algebra but with the opposite comultiplication, that is,∆cop(h) = τ ◦∆(h) = h(2) ⊗ h(1) for all h ∈ H. We leave the proof of these claims as exercise for thereader.

Example 2.21 Recall from Example 2.11 that for n = 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 by ε(a) = 1 = ε(d), ε(b) = ε(c) = 0.

We define now O(SL2) by the commutative algebra generated by the elements a, b, c, d satisfying therelation ad− bc = 1. For short we write

O(SL2) = k[a, b, c, d| ad− bc = 1].

To see that O(SL2) inherits the bialgebra structure of O(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 assaying that the ideal I = O(M2)(ad− bc− 1) generated by the element ad− bc− 1 is a bi-ideal. Thuswe have to prove that ε(ad−bc) = ε(1) = 1 and ∆(ad−bc) = (ad−bc)⊗ (ad−bc) = ∆(1) = 1⊗1 holdin O(Mn). Indeed, let A = O(M2) and denote by π : A → A/I the canonical quotient. By Theorem2.6, 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 that I ⊆ Ker(π ⊗ π)∆. But if t = ad− bc and ∆(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 by ε : A → k and I ⊆ Ker ε sinceε(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).

Thus O(SL2) is a bialgebra. This implies in particular that the determinant t = ad−bc is a group-likeelement in O(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 bc d

)⊗(a bc d

)=(a⊗ a+ b⊗ c a⊗ b+ b⊗ dc⊗ 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 inO(SL2). Actually, O(Mn) is not a Hopf algebra with the bialgebra structure defined above, sincethe determinant t is a group-like element which is not invertible. Notably, adding an inverse t−1 toO(Mn) is enough to give a Hopf algebra structure on the localization O(Mn)[t−1] of O(Mn) at t−1.This Hopf algebra is called O(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 definesO(SLn).


In matrix notation, the coassociativity follows from the equality((a bc d

)⊗(a bc d

))⊗(a bc d

)=(a bc d

)⊗((

a bc d

)⊗(a bc d

)),

and the counit from (a bc d

)(1 00 1

)=(a bc d

)=(

1 00 1

)(a bc 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)op is a well-defined algebra map, and then check equation (3) for the generators.Since S(1) = 1 and S(ad− bc) = S(ad)− S(bc) = S(d)S(a)− S(c)S(b) = ad− (−c)(−b) = ad− cb =ad− bc, it follows that S : O(SL2)→ O(SL2)op is a well-defined algebra map.

To check equation (3) for the generators is equivalent to prove the following matrix equality(a bc d

)(S(a) S(b)S(c) S(d)

)=(S(a) S(b)S(c) S(d)

)(a bc d

)=(ε(a) ε(b)ε(c) ε(d)

)(1 00 1

),

which follows from the equality ad− bc = 1 in O(SL2) and we leave it as exercise.

Remark 2.22 Recall that SL2(k) is the subgroup of matrices of GL2(k) given by

SL2(k) = {A ∈ k2×2 : detA = 1}.

Then, O(SL2(k)) is the commutative algebra of rational functions on SL2(k) generated by the matrixcoefficients via

a(A) = a11, b(A) = a12, c(A) = a21 and d(A) = a22,

for all A = (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 algebramaps from O(SL2(k)) to k, by

A(a) = a11, A(b) = a12, A(c) = a21, A(d) = a22 and A(1) = 1.

It is well-defined since A(ad− bc) = A(a)A(d)−A(b)A(c) = a11a22− a12a21 = detA = 1. Conversely,every element α of Algk(O(SL2(k)),k) defines a matrix in SL2(k) by(

α(a) α(b)α(c) α(d)

),

and it holds that α(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 algebrasof functions on these groups. We end this section with the following theorem that states that if thefield k is algebraically closed of characteristic zero, then all commutative Hopf algebras arise in thisway.

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, whereG is the finite group given by G = Spec(H) = Algk(H,k).

(b) Let H be a commutative Hopf algebra, then H is isomorphic to the algebra of regular functionsO(G) on a (pro) algebraic group G.


2.1 Exercises

1) Prove that the set {δx}x∈X defined in (1) is a linear basis of A = kX and it is an algebra with themultiplication given by δxδy = δxy for all x, y ∈ X and the unit by 1 =

∑x∈X δx.

2) Let G be a finite group. Prove that kG is a coalgebra whose dimension is equal to the order of thegroup.3) Let C be a k-vector space with basis {cm| m ∈ N ∪ {0}}. Prove that C is a coalgebra withcomultiplication ∆ and counit ε defined for all m ∈ N ∪ {0} by

∆(cm) =m∑i=0

ci ⊗ cm−i, ε(cm) = δ0,m.

4) Let C be a k-vector space with basis {s, c}. Prove that C is a coalgebra with comultiplication ∆and counit ε defined by

∆(s) = s⊗ c+ c⊗ s, ε(s) = 0,∆(c) = c⊗ c− s⊗ s, ε(s) = 1.

5) Let C be coalgebra over k.

(a) Prove that the dual space C∗ = {f : C → k| f is linear} is an algebra with the multiplicationand unit defined by

(f · g)(c) = f(c(1))g(c(2)) and 1(c) = ε(c) for all f, g ∈ C∗, c ∈ C,

where ∆(c) = c(1) ⊗ c(2) is the comultiplication of c ∈ C.

(b) Prove that D is a subcoalgebra of C if and only if D⊥ = {f : C → k| f(D) = 0} is a two-sidedideal of C∗.

(c) Prove that I is a coideal of C if and only if I⊥ = {f : C → k| f(I) = 0} is a subalgebra of C∗.

6) Prove Theorem 2.6.7) Let A be a finite-dimensional associative unital k-algebra.

(a) Prove that A∗ is a coalgebra. Hint: Use that (A⊗A)∗ ' A∗ ⊗A∗.

(b) Prove that B is a subalgebra of A if and only if B⊥ = {f : A→ k| f(B) = 0} is a coideal of A∗.

(c) Prove that I is a two-sided ideal of A if and only if I⊥ = {f : A→ k| f(I) = 0} is a subcoalgebraof A∗.

8) Let B be a k-vector space endowed with an algebra structure (B,m, u) and a coalgebra structure(B,∆, ε). Prove that ∆ and ε are algebra maps if and only if m and u are coalgebra maps.9) Let B be a bialgebra, I a bi-ideal of B and π : B → B/I the canonical linear map onto the quotientvector space. Then

(a) B/I has a unique bialgebra structure such that π is a bialgebra map.

(b) If f : B → B′ is any bialgebra map then Ker f is a bi-ideal.

(c) If I ⊆ Ker f then there is a unique bialgebra map f such that the following diagram commutes

Bf //

π !!BBBBBBBB B′

B/I

f

=={{{{{{{{


10) Let G be a finite group and consider the bialgebra structure on kG defined above. Prove thatG = G(kG).11) Let G be a finite group and consider the bialgebra structure on kG defined above. Prove thatG(kG) = Algk(kG,k) = G, where G is the character group of G.12) Let G be a finite group. Prove that (kG)∗ ' kG as Hopf algebras. Hint: Use that kG ⊆ (kG)∗ via〈δg, eh〉 = δgh for all g, h ∈ G.13) Let H be a finite-dimensional Hopf algebra over k. Prove that H∗ 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 ontothe quotient vector space. Then

(a) H/I has a unique Hopf algebra structure such that π is a Hopf algebra map.

(b) If f : H → H ′ is any Hopf algebra map then Ker f is a Hopf ideal.

(c) If I ⊆ Ker f then there is a unique Hopf algebra map f such that the following diagram commutes

Hf //

π !!CCCCCCCC H ′

H/I

f

==zzzzzzzz

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),

α 7→(α(a) α(b)α(c) α(d)

)is an isomorphism with inverse ψ determined by

ψ(A)(a) = a11, ψ(A)(b) = a12, ψ(A)(c) = a21 and ψ(A)(d) = a22,

for all A = (aij)1≤i,j≤2.18) Let A be a k-algebra. The finite dual or Sweedler dual of A is given by

A◦ = {f ∈ A∗| f(I) = 0, for some two-sided ideal I of A such that dimA/I <∞}.

Let (A,m, u,∆, ε,S) be a Hopf algebra. Prove that A◦ is a Hopf algebra with the structural mapsgiven 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 that k is an algebraically closed field of characteristic zero. In the lastsection 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) Aoo o/ o/ o/ o/ o/ o/ o/ o/

and this bijection is an equivalence

AlgC(O(G),C) = G oo ///o/o/o/o/o/o/o O(G)

Grothendieck’s philosophy was extended to quantum groups by Drinfel’d, who stated that one shouldquantize classical 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 of non-commuting functions on anon-existing object, namely a quantum 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, thefunction algebras themselves are called quantum groups.There is no rigorous, universally accepted definition of the term quantum group. However, it isgenerally agreed that this term includes certain deformation of classical objects associated to algebraicgroups or to semisimimple Lie algebras. To date no axiomatic definition of this family of algebrashas been given, nor a complete formulation of properties an algebra should satisfy in order to qualifyas a quantum analogue of a given classical coordinate ring. Thus, it is a field driven much more byexamples than by axioms.Some authors define quantum groups as non-commutative and non-cocommutative Hopf algebras. Inthis notes, we will follow Drinfeld’s convention: the category of quantum groups is the opposite categoryof Hopf algebras. That is, as objects quantum groups are Hopf algebras, but the morphisms are theopposite ones. This is because of the following: if Γ is a subgroup of G, then there is a Hopf algebrasurjection O(G) � O(Γ) between the algebras of functions on them.

Γ ↪→ G oo ///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)) and Uq(sl2)(k).

3.1 Quantum SL2

Let q ∈ k× = k r {0}.

Definition 3.1 The algebra Oq(M2(k)) is the algebra generated by the elements a, b, c, d satisfyingthe 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 notcommutative.

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 norcocommutative if q 6= 1.

(c) If detq := ad− q−1bc = da− qbc, then ∆(detq) = detq ⊗detq and ε(detq) = 1, that is, detq is agroup-like element in Oq(M2(k)). Moreover, it is central.

Proof. (a) In order to prove that ∆ and ε are well-defined algebra maps, it is enough to show thatthe relations hold under ∆ and ε, 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 completelyanalogous. 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 on Oq(M2(k)) is the same as the one defined on O(Mn(k)),it follows that Oq(M2(k)) is a coalgebra, that is, ε is a counit and ∆ is coassociative. Since both mapsare algebra maps, it follows that Oq(M2(k)) is indeed a bialgebra. Clearly, it is not commutative ifq 6= 1, and it is not cocommtuative since ∆(a) = a⊗ a+ b⊗ c 6= 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 iscentral, 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 define Oq(SL2(k)) as the k-algebra given by the quotient

Oq(SL2(k)) = Oq(M2(k))/(detq − 1),

where (detq − 1) is the two-sided ideal of Oq(M2(k)) generated by the element detq − 1.

In other words, the algebra Oq(SL2(k)) can be presented as the k-algebra generated by the elementsa, b, c, d satisfying 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 if q 6= 1, then Oq(SL2(k)) is notcommutative.

Since detq is a central group-like element, the ideal (detq −1) of Oq(M2(k)) is indeed a bi-ideal andthus Oq(SL2(k)) is a bialgebra with the comultiplication and counit definesd on the generators as inOq(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−1c and S(d) = a.

Proof. First we have to prove that S : Oq(SL2(k))→ Oq(SL2(k))op is 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 that S defines an antipode for Oq(SL2(k)), we have to check equation (3) for the generators.As for the case of O(SL2(k)), this is equivalent to prove the following matrix equality(

a bc d

)(S(a) S(b)S(c) S(d)

)=(S(a) S(b)S(c) S(d)

)(a bc d

)=(ε(a) ε(b)ε(c) ε(d)

)(1 00 1

),


which follows from the defining relations of Oq(SL2(k)) and we leave it as exercise for the reader.

Remark 3.5 The quantum group Oq(SL2(k)) corresponds to the quantized coordinate ring of SL2(k)generated by the matrix coefficients.

3.2 Quantum sl2

There is another group associated to SL2(k). In effect, SL2(C) is not only a group but also a smoothmanifold, i.e. a Lie group. As such, it has a tangent space at the identity, which is a Lie algebra andit is called sl2. 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 ofthe enveloping algebra U(sl2) of sl2. The deformation uses the classification of semisimple Lie algebrasover an algebraically closed field of characteristic zero, done by Cartan and Killing. Thus, the field kis an arbitrary field with these properties.

The origins of the subject of quantum groups lie in mathematical physics, where the term quantumcomes from. The starting point of the study of this subject lies in the Quantum Inverse ScatteringMethod, with the aim of solving certain integrable quantum systems. A key ingredient in this method isthe Quantum 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 representationtheory 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 themid-1980s, Drinfeld and Jimbo independently discovered analogous deformations corresponding toarbitrary semisimple Lie algebras [Dr, Dr2, Ji].

Definition 3.6 Let q ∈ k×, q 6= ±1. We define Uq(sl2)(k) as the k-algebra generated by the elementsE,F,K,K−1 satisfying the relations

KK−1 = K−1K = 1, KEK−1 = q2E, KFK−1 = q−2F, EF − FE =K −K−1

q − q−1.

If no confusion arrives, we denote this algebra simply by Uq(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 {EiF jKl : i, j ∈ N0, l ∈ Z} is a linear basis of Uq(sl2)(k). In particular, Uq(sl2)(k) isinfinite-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 of Uq(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 of K are group-like elements, E ∈ PK,1 and F ∈ 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 idealof relations is a coideal, or equivalently, that the following equalities hold

∆(KK−1) = ∆(K−1K) = 1⊗ 1 = ∆(1), ∆(KEK−1) = q2∆(E),

∆(KFK−1) = q−2∆(F ), ∆(EF − FE) = ∆(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 lastrelation we have

∆(EF − FE) = ∆(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 ⊗ FE −K−1E ⊗ FE− F ⊗ E − FE ⊗K

= K−1 ⊗ (EF − FE) + (EF − FE)⊗K + EK−1 ⊗KF −K−1E ⊗ FK= K−1 ⊗ (EF − FE) + (EF − FE)⊗K + q2q−2K−1E ⊗ FK −K−1E ⊗ FK= K−1 ⊗ (EF − FE) + (EF − FE)⊗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 that ε is a well-defined algebra map by showing that the equalities in the relations holdafter applying ε:

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(KFK−1) = ε(K)ε(F )ε(K−1) = 1.0.1 = 0 = q−2ε(F )

ε(EF − FE) = 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 ∆ and ε are algebra maps. We prove that ε is a counit and ∆ is coassociative bychecking 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 andm(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 andm(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 clearly Uq(sl2) is not cocommutative since τ ◦∆ 6= ∆ because

∆(E) = 1⊗ E + E ⊗K 6= 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)op is an algebramap and then that equality (3) holds for all generators of Uq(sl2). To show that S defines an algebramap, we have to verify that the equalities of the relations hold when appying S, but using the oppositemultiplication, for example S(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 for K and K−1 and the computation for F is completely analogous to the computationabove and we leave it as exercise. For the last relation we have

S(EF − FE) = S(F )S(E)− S(E)S(F ) = (−KF )(−EK−1)− (−EK−1)(−KF )

= KFEK−1 − EF = KFq2K−1E − EF = q−2q2KK−1FE − EF = FE − 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 equalitym(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 for K−1 and E are again completely analogous and we leave it as exercise.

3.2.1 Quantum Borel subgroups

The quantum groups Uq(b+) and Uq(b−) we introduce here are actually quantum quotients, since theyare constructed as Hopf subalgebras of Uq(sl2). Their terminology comes from classical Lie theory,since U(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, and K±1, F in the second. In particular, they can be described as algebrasas 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,KFK−1 = q−2F}.

They are called the Quantum Borel subgroups of Uq(sl2) and the positive and the negative part ofUq(sl2), respectively. They are also usually denoted by Uq(sl2)+ and Uq(sl2)−, or Uq(sl2)≥0 andUq(sl2)≤0, respectively.

From Theorem 3.9 it follows that both algebras are indeed Hopf subalgebras of Uq(sl2), since ∆(E) =1 ⊗ E + E ⊗K ∈ Uq(b+) ⊗ Uq(b+) and ∆(F ) = K−1 ⊗ F + F ⊗ 1 ∈ Uq(b−) ⊗ Uq(b−). Clearly, themultiplication on Uq(sl2) induces a surjective Hopf algebra map

Uq(b+)⊗ Uq(b−) � Uq(sl2).

Thus, the quantum group may be considered in some sense as a double object. This notion was formallydefined by Drinfeld who showed a way of producing solutions of the QYBE from Hopf algebras whichare Drinfeld doubles.

3.2.2 Relation between Oq(SL2) and Uq(sl2)

Up to now, we have introduced the quantum groups Oq(SL2) and Uq(sl2). In fact, these objects aredeeply related and in some sense, they are dual of each other. In this section we describe in a formalway this relation, which holds for quantum groups associated to arbitrary semisimple Lie algebras.


Definition 3.10 [Tk2] Given two bialgebras U and H, a Hopf pairing between them is a bilinearform

〈−,−〉 : H × U → k,

such that for all u, v ∈ U , h, k ∈ H, it holds

〈h, uv〉 = 〈h(1), u〉〈h(2), v〉,〈hk, u〉 = 〈h, u(1)〉〈k, u(2)〉,〈1, u〉 = ε(u), 〈h, 1〉 = ε(h).

If moreover U and H are Hopf algebras, from the equalities above it follows that 〈S(h), u〉 = 〈h,S(u)〉,for all u ∈ U , h ∈ H.

Let ϕ and ψ be the linear maps from U to H∗ and from H to U∗ given by

ϕ : U → H∗, ϕ(u)(h) = 〈h, u〉 for all u ∈ U, h ∈ H,ψ : H → U∗, ψ(u)(h) = 〈h, u〉, for all h ∈ H,u ∈ U.

If ϕ and ψ are injective, we say that the Hopf pairing is perfect. The following theorem states theduality between the quantum groups defined above.

Theorem 3.11 [K, Thm. VII.4.4], [BG, I.9.23]. There is a Hopf pairing between Oq(SL2(k)) andUq(sl2)(k) given by

〈a,K〉 = q, 〈a,K−1〉 = q−1, 〈a,E〉 = 0, 〈a, F 〉 = 0,〈b,K〉 = 0, 〈b,K−1〉 = 0, 〈b, E〉 = 1, 〈b, F 〉 = 0,〈c,K〉 = 0, 〈c,K−1〉 = 0, 〈c, E〉 = 0, 〈c, F 〉 = 1,〈a,K〉 = q, 〈a,K−1〉 = q−1, 〈a,E〉 = 0, 〈a, F 〉 = 0,〈d,K〉 = q−1, 〈d,K−1〉 = q, 〈d,E〉 = 0, 〈d, F 〉 = 0.

If q is not a root of unity, the Hopf pairing is perfect. In particular Oq(SL2(k)) ↪→ Uq(sl2)(k)∗ andUq(sl2)(k) ↪→ Oq(SL2(k))∗.

Remark 3.12 It holds in fact that the image of ψ : Oq(SL2(k)) → Uq(sl2)(k)∗ is the finite dualUq(sl2)(k)◦ of Uq(sl2)(k), that is Oq(SL2(k)) ' Uq(sl2)(k)◦ as Hopf algebras. This statement holdsin general for simply connected semisiple Lie groups G with corresponding semisimple Lie algebrasg, i.e. Oq(G)(k) ' Uq(g)(k)◦. However, it is not true that Oq(G)(k)◦ ' Uq(g)(k). See [Tk] for adetailled discussion on the case G = SL2.

3.3 Small quantum groups

From now on we assume that q is a primitive root of unity of order ` 6= 1, ` > 2. In this section weintroduce the Frobenius-Lusztig kernels uq(sl2), which are finite-dimensional Hopf algebras. They areconstructed as quotients of Uq(sl2) by the two-sided ideal generated by some central elements. Forthis reason they are called small quantum groups. We begin first by describing these central elements.

Lemma 3.13 The elements E`, F `, K` and K−` are central in Uq(sl2).

Proof. It is a straightforward computation and we leave it as exercise. For example, K` is centralsince it commutes with all the generators of Uq(sl2):

K`E = q2`EK` = EK`

K`F = q−2`FK` = FK`.


For E` the computations are similar. Indeed, KE` = q2`E`K = E`K and

E`F = E`−1

(FE +

K −K−1

q − q−1

)= E`−1FE +

1q − q−1

E`−1(K −K−1)

= E`−2

(FE +

K −K−1

q − q−1

)E +

1q − q−1

E`−1(K −K−1)

= E`−2FE2 +q2

q − q−1E`−1K − q−2

q − q−1E`−1K−1 +

1q − q−1

E`−1(K −K−1)

= E`−2FE2 +q2 + 1q − q−1

E`−1K − q−2 + 1q − q−1

E`−1K−1

...

= FE` +q2` + q2(`−1) + q2 + 1

q − q−1E`−1K − q−2` + q−2(`−1) + q−2 + 1

q − q−1E`−1K−1

= FE`,

since q−2` + q−2(`−1) + q−2 + 1 = 0 = q2` + q2(`−1) + q2 + 1 because q is an `-th root of unity.

Since the elements K` − 1, E`, F ` are central in Uq(sl2), the ideal I generated by these elements is atwo-sided ideal. Moreover, from the quantum binomial formula it follows

∆(K` − 1) = (K` − 1)⊗K` + 1⊗ (K` − 1),

∆(E`) = 1⊗ E` + E` ⊗K`,

∆(F `) = K−` ⊗ F ` + F ` ⊗ 1,

see Exercises 10), 11) and 12). Hence, ∆(I) ⊆ I ⊗ Uq(sl2) + Uq(sl2) ⊗ I. Since ε(K` − 1) = 0 =ε(E`) = ε(F `), it follows that I is a bi-ideal. Furthermore, it is a Hopf ideal since

S(K` − 1) = K−` − 1 = K−`(1−K`) ∈ IS(E`) = S(E)` = (−EK−1)` = (−1)`q−2(1+2+···+`−1)E`K−` = (−1)`q−`(`−1)E`K−`

= (−1)`E`K−` ∈ IS(F `) = S(F )` = (−KF )` = q2(1+2+···+`−1)K`F ` = q`(`−1)K`F `

= K`F ` ∈ I.

The Frobenius-Lusztig Kernel is defined as the Hopf algebra given by the quotient

uq(sl2) = Uq(sl2)/I.

It is also called the restricted enveloping algebra of sl2. It can be also presented as the algebragenerated by the elements {K,E, F} satisfying the relations

K` = 1, E` = 0 = F `, KEK−1 = q2E, KFK−1 = q−2F, and EF−FE =K −K−1

q − q−1.

Lemma 3.14 dim uq(sl2) = `3.

Proof. (Sketch) As the set {EiF jKm : i, j ∈ N,m ∈ Z} is a linear basis of Uq(sl2), it follows thatthe set {EiF jKm : 0 ≤ i, j,m < `} is a linear basis for uq(sl2). It is clear that they generate uq(sl2)as a linear space. To see that they are linearly independent one may look at the representation theoryor use the Diamond Lemma, see for example [AS2].


Remarks 3.15 (i) By definition, the comultiplication in uq(sl2) is determined by

∆(K) = K ⊗K, ∆(E) = 1⊗ E + E ⊗K and ∆(F ) = K−1 ⊗ F + F ⊗ 1.

In particular, we have that S(u) = KuK−1 for all u ∈ uq(sl2).

(ii) If q is a root of unity, the Hopf algebra Oq(SL2) contains a central Hopf subalgebra B isomorphicto O(SL2), which is given by

B = k[a`, b`, c`, d` : a`d` − b`c` = 1] ⊆ Oq(SL2).

Moreover, one has the central exact sequence of Hopf algebras

k→ O(SL2)→ Oq(SL2)→ uq(sl2)∗ → k.

There is another small quantum group, which is related to the quantum Borel subgroup Uq(b+) ofUq(sl2). Just take the ideal J of Uq(b+) generated by the central elements K` − 1 and E`. Again,this ideal J is a Hopf ideal and one has the finite-dimensional Hopf algebra

uq(b+) = Uq(b+)/J,

which can be presented as an algebra as follows

uq(b+) = k{K,E : K` = 1, E` = 0,KEK−1 = q2E}.

The Hopf algebra structure is determined by

∆(K) = K ⊗K, ∆(E) = 1⊗ E + E ⊗K,ε(K) = 1, ε(E) = 0S(K) = K−1, S(E) = −xg.

Lemma 3.16

(a) dim uq(b+) = `2.

(b) There is an injective Hopf algebra map uq(b+) ↪→ uq(sl2).

Proof. (b) is clear and (a) follows from the fact that {EiKm : 0 ≤ i,m < `} is a basis of uq(b+).

Remark 3.17 These type of Hopf algebras where defined by Sweedler in the case ` = 2 and Taft inthe case ` > 2. They are called now the Sweedler and Taft algebras, respectively, and are presentedas follows:

Tq = k{g, x : g` = 1, x` = 0, gxg−1 = qx}.

The Hopf algebra structure is determined by

∆(g) = g ⊗ g, ∆(x) = 1⊗ x+ x⊗ g,ε(g) = 1, ε(x) = 0

S(g) = g−1, S(x) = −xg.

In particular, S2(h) = ghg−1 for all h ∈ Tq. With the notation used before we have uq(b+) ' Tq2 .

3.4 Exercises

1) Prove that the algebra map S : Oq(SL2(k))→ Oq(SL2(k))op defined by(S(a) S(b)S(c) S(d)

)=(

d −qb−q−1c a

)


is an antipode for Oq(SL2(k)).2) Let t be an indeterminate and define the quantum general linear group by

Oq(GL2(k)) = Oq(M2(k))[t]/(tdetq − 1).

Prove that Oq(GL2(k)) is a Hopf algebra with the coalgebra structure induced by the quotient andthe antipode given by (

S(a) S(b)S(c) S(d)

)= t

(d −qb

−q−1c a

).

3) Coaction on the quantum plane. The quantum plane is define as the k-algebra kq[x, y] givenby

kq[x, y] = k{x, y| yx = qxy}.

There exists a linear map

ρ :kq[x, y]→ Oq(SL2(k))⊗ kq[x, y],x 7→ a⊗ x+ b⊗ y,y 7→ c⊗ x+ d⊗ y,

which in matrix notation can be written as

ρ

(xy

)=(a bc d

)⊗(xy

).

Prove that ρ defines a coaction of Oq(SL2(k)) on kq[x, y], that is, the following equality holds

(id⊗ρ)ρ = (∆⊗ id)ρ.

Moreover, ρ is an algebra map which implies that kq[x, y] is a Oq(SL2(k))-comodule algebra.4) Prove that Uq(b+) and Uq(b−) are Hopf subalgebras of Uq(sl2).5) Let U+ be the subalgebra of Uq(sl2) generated by E, U− the subalgebra generated by F and U0

the subalgebra generated by K and K−1. Prove that there is a linear isomorphism

U+ ⊗ U0 ⊗ U− ' Uq(sl2)

This property is called the triangular decomposition and resembles the same property in the classicalcase.6) Prove that S2(u) = KuK−1 for all u ∈ Uq(sl2).7) Chevalley involution. Prove that there is a unique algebra automorphism w of Uq(sl2) determinedby

w(E) = F, w(F ) = E and w(K) = K−1,

which satisfies w2 = id.8) Limit as q → 1. We could have presented Uq(sl2) as the algebra generated by the elementsE,F,K,K−1 and L satisfying the relations

KEK−1 = q2E, KFK−1 = q−2F, EF − FE = L, KK−1 = K−1K = 1,

(q − q−1)L = K −K−1.

The presentation given by these 8 relations has the advantage that it makes sense when q = 1. LetU(sl2) be the algebra given by

U(sl2) = k{H,E, F : HE − EH = 2E,HF − FH = −2F,EF − FE = H}

Prove that U(sl2) ' U1(sl2)/(K − 1). Hint: Prove first that the relations above imply when q 6= ±1

LE − EL = q(EK +K−1E), LF − FL = −q(KF + FK−1)

Then prove that U1(sl2) = U(sl2)[K]/(K2 − 1) and then U(sl2) ' U1(sl2)/(K − 1).


9) Let q be a primitive root of unity of order `, with ` > 2. Prove that the elements E`, F `, K` andK−` are central in Uq(sl2).10) In the polynomial algebra Z[q], q an indeterminate, we consider the q-binomial coecients(

ni

)q

=(n)q!

(n− i)q!(i)q!,

where (n)q! = (n)q(n− 1)q · · · (1)q and (n)q = 1 + q + q2 + · · ·qn−1 for all n ∈ N and 0 ≤ i ≤ n.

(a) Prove the identity

qk(nk

)q

+(

nk − 1

)q

=(n+ 1k

)q

for all 0 ≤ k ≤ n. (4)

(b) Prove by induction that(ni

)q

∈ Z[q], for all n ∈ N, 0 ≤ i ≤ n.

11) Quantum binomial formula. If A is an associative algebra over k and q ∈ k, then(ni

)q

denotes the specialization of(ni

)q

at q. Prove that if x, y ∈ A are two elements that q-commute, i.e.

xy = qyx, then the following formula holds for every n ∈ N:

(x+ y)n =n∑i=0

(ni

)q

yixn−i. (5)

12) Let q be a primitive `-th root of unity. Then by definition we have that(`i

)q

= 0 for all 0 < i < `.

Let E,F be the the generators of Uq(sl2). Prove using the quantum binomial formula that

∆(E`) = 1⊗ E` + E` ⊗K` and ∆(F `) = K−` ⊗ F ` + F ` ⊗ 1.

13) Let q be a primitive `-th root of unity and consider the Taft algebra Tq defined above.

(a) Prove that Tq ' Tq′ if and only if q = q′.

(b) Prove that Tq is self-dual, that is, Tq ' T ∗q .

4 Quantum groups and the classification problem of finite-dimensional Hopf algebras

In this last section we sketch how quantum groups get into the scene of the classification problem.First we need some definitions.

Definition 4.1 (i) Let H be a Hopf algebra. We say that H is semisimple if it is semisimple asalgebra; that is, if its Jacobson radical is zero.

(ii) A coalgebra is called simple if it does not contain non-trivial subcoalgebras. It is called cosemisim-ple if it is the sum of simple subcoalgebras.

The following theorem is due to several author, see [LR1], [LR2], [LR3], [R, Prop. 2], [OSch1] and[OSch2]. For a complete proof see [Sch].

Theorem 4.2 Let H be a finite-dimensional Hopf algebra over an algebraically closed field k ofcharacteristic zero. The following are equivalent


(a) H is semisimple.

(b) H is cosemisimple.

(c) S2 = idH .

(d) TrS2 6= 0.

From now on we assume that all Hopf algebras are finite-dimensional and the field is algebraicallyclosed of characteristic zero.

Examples 4.3 (i) Let G be a finite group. Then kG and kG are semisimple Hopf algebras. Forinstance, in kG we have that S(eg) = eg−1 for all g ∈ G. This implies that S2(eg) = eg for all g ∈ G.Since {eg}g∈G is a linear basis of kG, it follows that S2 = id. One can see that kG is also semisimpleby a direct computation or just use that (kG)∗ ' kG.

(ii) The Taft algebra Tq is not semisimple since S2(x) = gxg−1 = qx 6= x.

(iii) The Frobenius-Lusztig kernel uq(sl2) is not semisimple if q2 6= 2, since S2(E) = KEK−1 = q2E.

Definition 4.4 (i) The coradical of a coalgebra C is the sum of all simple subcoalgebras and it isdenoted by C0. Clearly, C0 is a cosemisimple subcoalgebra of C.

(ii) Let H be a Hopf algebra. We say that H is pointed if all simple subcoalgebras are one-dimensional.In particular, this implies that H0 = kG(H).

Example 4.5 The Taft algebras Tq and the Frobenius-Lusztig kernels uq(sl2) are pointed.

The study of finite-dimensional Hopf algebras goes through two different directions: the semisimplecase and the non-semisimple case.

semisimple

finite-dim. Hopf alg.

44iiiiiiiiiiiiiiiiiiii

$$IIIIIIIIIIIIIIIIIIII

pointed

non-semisimple

55kkkkkkkkkkkkkkk

))SSSSSSSSSSSSSSS

non-pointed

In the non-semisimple case, a particular subclass was intensively studied: the pointed ones. Forexample, Andruskiewitsch and Schneider [AS3] proved that all Hopf algebras which are pointed non-semisimple and whose coradical is a finite abelian group G such that |G| is not divisible by a primenumber smaller or equal to 7, are variations of Frobenius-Lusztig kernels.

Up to now, the smallest dimension where the classification is unknown is 20.

Let H be a Hopf algebra and let p be a prime number. The classification is known if dimH = p, p2

and 2p, 2p2 with p odd. We describe now the classification for the cases p, p2 and some known resultson dimension p3.The classification of the Hopf algebras over k of dimension p was obtained by Zhu. It turns out thatthere are only group algebras of cyclic groups of order p.


Theorem 4.6 [Z] Let H be a Hopf algebra of dimension p. Then H is isomorphic to the groupalgebra k[Z/(p)].

The classification of Hopf algebras of dimension p2 is due to several authors. The semisimple oneswere classified by Masuoka, using that all semisimple Hopf algebras of dimension pn, n ∈ N have acentral group-like element. Andruskiewitsch and Schneider prove that the only pointed Hopf algebrasof dimension p2 are the Taft algebras. Finally, Ng completed the classification by showing that a Hopfalgebra of dimension p2 is either semisimple or pointed (non-semisimple).

Theorem 4.7 [Mk1], [AS1], [Ng] Let H be a Hopf algebra of dimension p2. Then H is isomorphicto one and only one of the following Hopf algebras:

(a) k[Z/(p2)],

(b) k[Z/(p)× Z/(p)]

(c) The Taft algebra Tq with qp = 1, q 6= ±1.

We close these notes with the classification of semisimple and pointed non-semisimple Hopf algebrasof dimension p3. Hopf algebras of dimension 8 were classified by Williams [W] in her PhD Thesis.Masuoka [Mk2] and Stefan [St] gave later a different proof of this result. In general, the classificationproblem of Hopf algebras of dimension p3 remains open. Nevertheless, the classification is known forthe semisimple case and the pointed non-semisimple case. Suppose that p is an odd prime. SemisimpleHopf algebras of dimension p3 were classified by Masuoka [Mk1]; there are exactly p+ 8 isomorphismclasses:

(a) Three group algebras of abelian groups.

(b) Two group algebras of non-abelian groups and their duals.

(c) p + 1 self-dual Hopf algebras which are neither commutative nor cocommutative and they aregiven by the extensions of k[Z/(p)× Z/(p)] by k[Z/(p)].

Note that in this case, there are semisimple Hopf algebras which are non-trivial, i.e. not isomorphicto a group algebra or the dual of a group algebra.

Non-semisimple pointed Hopf algebras of dimension p3 were classified by [AS2], [CD] and [SvO], bydifferent methods. It turns out that they consist of variations of small quantum groups. The explicitlist is the following, where q ∈ Gp r {1}:

(d) The tensor-product Hopf algebra T (q)⊗ k[Z/(p)].

(e) T (q) := k < g, x| gxg−1 = q1/px, gp2

= 1, xp = 0 > (q1/p a p-th root of q), with comultiplication∆(x) = x⊗ gp + 1⊗ x, ∆(g) = g ⊗ g.

(f) T (q) := k < g, x| gxg−1 = qx, gp2

= 1, xp = 0 >, with comultiplication ∆(x) = x ⊗ g + 1 ⊗x, ∆(g) = g ⊗ g.

(g) r(q) := k < g, x| gxg−1 = qx, gp2

= 1, xp = 1 − gp >, with comultiplication ∆(x) =x⊗ g + 1⊗ x, ∆(g) = g ⊗ g.

(h) The Frobenius-Lusztig kernel uq(sl2) := k < g, x, y| gxg−1 = q2x, gyg−1 = q−2y, gp = 1,xp = 0, yp = 0, xy − yx = g − g−1 >, with comultiplication ∆(x) = x ⊗ g + 1 ⊗ x, ∆(y) =y ⊗ 1 + g−1 ⊗ y, ∆(g) = g ⊗ g.


(i) The book Hopf algebra h(q,m) := k < g, x, y| gxg−1 = qx, gyg−1 = qmy, gp = 1, xp =0, yp = 0, xy−yx = 0 >, m ∈ Z/(p)r{0}, with comultiplication ∆(x) = x⊗g+1⊗x, ∆(y) =y ⊗ 1 + gm ⊗ y, ∆(g) = g ⊗ g.

Furthermore, there are two examples of non-semisimple but also non-pointed Hopf algebras of dimen-sion p3, namely

(j) The dual of the Frobenius-Lusztig kernel, uq(sl2)∗.

(k) The dual of the case (g), r(q)∗.

There are no isomorphisms between different Hopf algebras in the list. Moreover, the Hopf algebras incases (d), . . . , (k) are not isomorphic for different values of q ∈ Gpr{1}, except for the book algebras,where h(q,m) is isomorphic to h(q−m

2,m−1). In particular, the Hopf algebra T (q) does not depend,

modulo isomorphisms, upon the choice of the p-th root of q.

It is a conjecture that any Hopf algebra H of dimension p3 is semisimple or pointed or its dual ispointed, that is, H is one of the Hopf algebras of the list (a), . . . , (k). In [G] this conjecture is provedunder additional assumptions.

4.1 Exercises

1) Let G be a finite group. Prove that the Hopf algebra kG is semisimple.2) Let q be a primitive root of unity. Prove that the Hopf algebras Tq and uq(sl2) are pointed andnon-semisimple.3) Prove using the duality between algebras and coalgebras that a finite-dimensional Hopf algebra His pointed if and only if all simple H-modules are one-dimensional.

Aknowledgments

The author would like to thank the organizers of the XVIII Latin American Algebra Colloquium forgiving him the possibility to give this mini-course on this wonderful subject.

References

Hopf algebras

[A] E. Abe, Hopf algebras, Cambridge Tracts in Mathematics, 74, Cambridge University Press,Cambridge-New York, 1980. xii+284 pp.

[An] N. Andruskiewitsch, ‘About finite dimensional Hopf algebras’, Notes of the Lectures at theschool ‘Quantum Symmetries in the Theorical Physics and Mathematics’, Bariloche, January2000.

[AS1] N. Andruskiewitsch and H-J. Schneider, ‘Hopf algebras of order p2 and braided Hopfalgebras of order p’, J. Algebra 199 (1998), no. 2, 430–454.

[AS2] N. Andruskiewitsch and H-J. Schneider, ‘Lifting of quantum linear spaces and pointedHopf algebras of order p3’, J. Algebra 209 (1998), no. 2, 658–691.

[AS3] N. Andruskiewitsch y H-J. Schneider, ‘On the classification of finite-dimensional pointedHopf algebras’. Ann. Math., to appear.

[CD] S. Caenepeel and S. Dascalescu, ‘Pointed Hopf algebras of dimension p3’, J. Algebra 209(1998), no. 2, 622–634.

[DNR] S. Dascalescu, S. Nastasescu and S. Raianu, Hopf algebras. An introduction, Monographsand Textbooks in Pure and Applied Mathematics, 235. Marcel Dekker, Inc., New York, 2001.x+401 pp.


[G] G. A. Garcıa, ‘On Hopf algebras of dimension p3’, Tsukuba J. Math. 29 (2005), no. 1, 259–284.

[LR1] R. Larson and D. Radford, ‘An associative orthogonal bilinear form for Hopf algebras’Amer. J. Math. 91 (1969), 75–94.

[LR2] R. Larson and D. Radford, ‘Semisimple cosemisimple Hopf algebras’, Amer. J. Math. 110(1988), no. 1, 187–195.

[LR3] R. Larson and D. Radford, ‘Finite dimensional cosemisimple Hopf algebras in characteristic0 are semisimple’, J. Algebra 117 (1988), no. 2, 267–289.

[Mk1] A. Masuoka, ‘The pn theorem for semisimple Hopf algebras’, Proc. Amer. Math. Soc. 124(1996), no. 3, 735–737.

[Mk2] A. Masuoka, ‘Semisimple Hopf algebras of dimension 6, 8’, Israel. J. Math. 92 (1995), no.1-3, 361–373.

[Mk3] A. Masuoka, ‘Examples of almost commutative Hopf algebras which are not coquasitriangu-lar’, Hopf algebras, Lecture Notes in Pure and Appl. Math., 237 (2004), Dekker, New York,185–191.

[Mo] S. Montgomery, Hopf Algebras y their Actions on Rings, CBMS Reg. Conf. Ser. Math. 82(1993). American Mathematical Society (AMS).

[Ng] S-H. Ng, ‘Non-semisimple Hopf algebras of Dimension p2’, J. Algebra 255 (2002), no. 1,182–197.

[OSch1] U. Oberst and H-J. Schneider, ‘Untergruppen formeller Gruppen von endlichem Index’,J. Algebra 31 (1974), 10–44.

[OSch2] U. Oberst and H-J. Schneider, ‘Uber Untergruppen endlicher algebraischer Gruppen’,Manuscripta Math. 8 (1973), 217–241.

[R] D. Radford, ‘The Trace Functions and Hopf algebras’, J. Algebra 163 (1994), no. 3, 583–622.

[Sch] H-J. Schneider, ‘Lectures on Hopf algebras’ (1995), Trabajos de Matematica 31/95 (FaMAF,1995). Available at http://www.famaf.unc.edu.ar/series/BMat22-31.htm.

[St] D. Stefan, ‘Hopf algebras of low dimension’ J. Algebra 211 (1999), no. 1, 343–361.

[SvO] D. Stefan and F. van Oystaeyen, ‘Hochschild cohomology and coradical filtration of pointedcoalgebras: Applications’, J. Algebra 210 (1998), no. 2, 535–556.

[Sw] M. Sweedler, ‘Hopf algebras’, Benjamin, New York, 1969.

[Tk2] M. Takeuchi, ‘Matched pairs of groups and bismash products of Hopf algebras’, Comm.Algebra 9 (1981), no. 8, 841–882.

[W] R. Williams, ‘Finite dimensional Hopf algebras’, Ph. D. Thesis, Florida State University(1988).

[Z] Y. Zhu, ‘Hopf algebras of prime dimension’ Internat. Math. Res. Notices 1 (1994), 53–59.

Quantum groups

[BG] K. A. Brown and K. R. Goodearl, Lectures on Algebraic Quantum Groups, AdvancedCourses in Mathematics - CRM Barcelona. Basel: Birkhauser.

[CP] V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, Cam-bridge, 1995. xvi+651 pp.

[Dr2] V. Drinfeld, ‘Quantum groups’, Proc. Int. Congr. Math., Berkeley 1986, Vol. 1 (1987), 798-820.


[Dr] V. Drinfeld, ‘Hopf algebras and the quantum Yang-Baxter equation’ ,(Russian) Dokl. Akad.Nauk SSSR 283 (1985), no. 5, 1060–1064.

[J] J.C. Jantzen, Lectures on Quantum Groups, Graduate Studies in Mathematics. 6 (1996).Providence, RI: American Mathematical Society (AMS).

[Ji] M. Jimbo, ‘A q-difference analogue of U(g) and the Yang-Baxter equation’, Lett. Math. Phys.10 (1985), no. 1, 63–69.

[K] C. Kassel, Quantum Groups, Graduate Texts in Mathematics. 155 (1995). New York, NY:Springer-Verlag.

[KS] A. Klimyk and K. Schumdgen, Quantum groups and their representations, Texts and Mono-graphs in Physics. Springer-Verlag, Berlin, 1997.

[KR] P.P. Kulish and Ju. N. Reshetikhin, ‘Quantum linear problem for the sine-Gordon equationand higher representations’ (Russian), Questions in quantum field theory and statistical physics,2. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 101 (1981), 101–110, 207.

[L3] G. Lusztig, Introduction to quantum groups, Progress in Mathematics, 110 (1993). BirkhuserBoston, Inc., Boston, MA.

[Mj] S. Majid, Foundations of quantum group theory, Cambridge University Press, Cambridge,1995. x+607 pp.

[Mj2] S. Majid, A quantum groups primer, London Mathematical Society Lecture Note Series, 292.Cambridge University Press, Cambridge, 2002. x+169 pp.

[Mn] Y. Manin, Quantum groups and noncommutative geometry, Universit de Montral, Centre deRecherches Mathmatiques, Montreal, QC, 1988. vi+91 pp.

[Tk] M. Takeuchi, ‘A short course on quantum matrices’, Notes taken by Bernd Strber, Math.Sci. Res. Inst. Publ., 43, New directions in Hopf algebras, 383–435, Cambridge Univ. Press,Cambridge, 2002.

FCEFyN-FaMAF-CIEM,Universidad Nacional de CordobaMedina Allende s/nCiudad Universitaria5000 CordobaRepublica Argentinae-mail: [email protected]://www.mate.uncor.edu/∼ggarcia/

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