A brief introduction to quantum groups Pavel Etingof MIT · 2020. 5. 6. · A brief introduction to quantum groups Pavel Etingof MIT May 5, 2020 1. Introduction ... school, Drinfeld,

A brief introduction to quantum groups

Pavel Etingof

MIT

May 5, 2020

1

Introduction

The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics.

Sincethen, it’s grown into a vast subject with deep links to many areas:

representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and

examples of quantum groups and briefly describe some of theapplications.

2

Introduction

The theory of quantum groups developed in mid 1980s(Faddeev’s school, Drinfeld, Jimbo) from attempts to constructand understand solutions of the quantum Yang-Baxter equationarising in quantum field theory and statistical mechanics. Sincethen, it’s grown into a vast subject with deep links to many areas:



3

Introduction


representation theory

the Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and


4

Introduction


representation theorythe Langlands program

low-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and


5

Introduction


representation theorythe Langlands programlow-dimensional topology

category theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and


6

Introduction


representation theorythe Langlands programlow-dimensional topologycategory theory

enumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and


7

Introduction


representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometry

quantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and


8

Introduction


representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computation

algebraic combinatoricsconformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and


9

Introduction


representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatorics

conformal field theoryintegrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and


10

Introduction


representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theory

integrable systemsintegrable probabilityThe goal of this talk is to review some of the main ideas and


11

Introduction


representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systems

integrable probabilityThe goal of this talk is to review some of the main ideas and


12

Introduction


representation theorythe Langlands programlow-dimensional topologycategory theoryenumerative geometryquantum computationalgebraic combinatoricsconformal field theoryintegrable systemsintegrable probability

The goal of this talk is to review some of the main ideas andexamples of quantum groups and briefly describe some of theapplications.

13

Introduction



examples of quantum groups and briefly describe some of theapplications. 14

Hopf algebras

Classical physics:

X – a space of states;A = O(X ) – the algebra of observables, i.e., (say, complex)functions on X (a commutative, associative algebra).Example: X = R2. Then the coordinate x and momentum pare examples of observables (elements of A).

Quantum physics:A is deformed to a non-commutative (but still associative)algebra A~ with quantization parameter ~, e.g., theHeisenberg uncertainty relation [p, x ] = −i~.

15

Hopf algebras

Classical physics:X – a space of states;

A = O(X ) – the algebra of observables, i.e., (say, complex)functions on X (a commutative, associative algebra).Example: X = R2. Then the coordinate x and momentum pare examples of observables (elements of A).


16

Hopf algebras

Classical physics:X – a space of states;A = O(X ) – the algebra of observables, i.e., (say, complex)functions on X (a commutative, associative algebra).

Example: X = R2. Then the coordinate x and momentum pare examples of observables (elements of A).


17

Hopf algebras

Classical physics:X – a space of states;A = O(X ) – the algebra of observables, i.e., (say, complex)functions on X (a commutative, associative algebra).Example: X = R2. Then the coordinate x and momentum pare examples of observables (elements of A).


18

Hopf algebras


Quantum physics:

A is deformed to a non-commutative (but still associative)algebra A~ with quantization parameter ~, e.g., theHeisenberg uncertainty relation [p, x ] = −i~.

19

Hopf algebras


Quantum physics:A is deformed to a non-commutative (but still associative)algebra A~ with quantization parameter ~,

e.g., theHeisenberg uncertainty relation [p, x ] = −i~.

20

Hopf algebras



21

Hopf algebras, ctd.

What if X = G is a group?

A group is equipped with an associative product, which has aunit and all elements are invertible:

m : G × G → G , m(x , y) = xy , x(yz) = (xy)z ,

∃ e : ∀g ∈ G : eg = ge = g ,

∀ g ∃ g−1 : gg−1 = g−1g = e.

Thus the algebra A = O(G ) of functions on a (say, finite)group has a natural structure of a coalgebra. Namely, decomposingO(G × G ) as O(G )⊗O(G ), one gets comultiplication, orcoproduct on A from the multiplication m in G :

∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).

Here we used Sweedler’s notation for the coproduct ∆f = f(1)⊗ f(2)where summation is implied, i.e., f(1) ⊗ f(2) is really

∑i f

i(1) ⊗ f i(2).

22

Hopf algebras, ctd.

What if X = G is a group?A group is equipped with an associative product, which has a

unit and all elements are invertible:


∃ e : ∀g ∈ G : eg = ge = g ,

∀ g ∃ g−1 : gg−1 = g−1g = e.


∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).


∑i f

i(1) ⊗ f i(2).

23

Hopf algebras, ctd.




∃ e : ∀g ∈ G : eg = ge = g ,

∀ g ∃ g−1 : gg−1 = g−1g = e.


∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).


∑i f

i(1) ⊗ f i(2).

24

Hopf algebras, ctd.




∃ e : ∀g ∈ G : eg = ge = g ,

∀ g ∃ g−1 : gg−1 = g−1g = e.

Thus the algebra A = O(G ) of functions on a (say, finite)group has a natural structure of a coalgebra.

Namely, decomposingO(G × G ) as O(G )⊗O(G ), one gets comultiplication, orcoproduct on A from the multiplication m in G :

∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).


∑i f

i(1) ⊗ f i(2).

25

Hopf algebras, ctd.




∃ e : ∀g ∈ G : eg = ge = g ,

∀ g ∃ g−1 : gg−1 = g−1g = e.


∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).


∑i f

i(1) ⊗ f i(2).

26

Hopf algebras, ctd.




∃ e : ∀g ∈ G : eg = ge = g ,

∀ g ∃ g−1 : gg−1 = g−1g = e.


∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).


∑i f

i(1) ⊗ f i(2).

27

Hopf algebras, ctd.




∃ e : ∀g ∈ G : eg = ge = g ,

∀ g ∃ g−1 : gg−1 = g−1g = e.


∆ : A→ A⊗ A : (∆f )(x , y) = f (xy) = f(1)(x)⊗ f(2)(y).


∑i f

i(1) ⊗ f i(2).

28

Hopf algebras ctd.

It is clear that• ∆ is an algebra homomorphism.

The algebra A also has a natural counit and antipode obtainedfrom the unit and inversion in G :

ε : A→ C : ε(f ) = f (e),

S : A→ A : S(f )(x) = f (x−1).

The following properties could be easily checked:

∆ is coassociative, i.e., (id⊗∆) ◦∆ = (∆⊗ id) ◦∆;(ε⊗ id) ◦∆ = (id⊗ ε) ◦∆ = id;µ ◦ (S ⊗ id) ◦∆(x) = µ ◦ (id⊗ S) ◦∆(x) = ε(x),where µ : A⊗ A→ A is the multiplication.

Definition

A quantum group or Hopf algebra is a unital associative algebra A(not necessary commutative) which is equipped with ∆, ε,S andhas the properties listed above.

29

Hopf algebras ctd.

It is clear that• ∆ is an algebra homomorphism.The algebra A also has a natural counit and antipode obtained

from the unit and inversion in G :

ε : A→ C : ε(f ) = f (e),

S : A→ A : S(f )(x) = f (x−1).



Definition


30

Hopf algebras ctd.



ε : A→ C : ε(f ) = f (e),

S : A→ A : S(f )(x) = f (x−1).



Definition


31

Hopf algebras ctd.



ε : A→ C : ε(f ) = f (e),

S : A→ A : S(f )(x) = f (x−1).


∆ is coassociative, i.e., (id⊗∆) ◦∆ = (∆⊗ id) ◦∆;

(ε⊗ id) ◦∆ = (id⊗ ε) ◦∆ = id;µ ◦ (S ⊗ id) ◦∆(x) = µ ◦ (id⊗ S) ◦∆(x) = ε(x),where µ : A⊗ A→ A is the multiplication.

Definition


32

Hopf algebras ctd.



ε : A→ C : ε(f ) = f (e),

S : A→ A : S(f )(x) = f (x−1).


∆ is coassociative, i.e., (id⊗∆) ◦∆ = (∆⊗ id) ◦∆;(ε⊗ id) ◦∆ = (id⊗ ε) ◦∆ = id;

µ ◦ (S ⊗ id) ◦∆(x) = µ ◦ (id⊗ S) ◦∆(x) = ε(x),where µ : A⊗ A→ A is the multiplication.

Definition


33

Hopf algebras ctd.



ε : A→ C : ε(f ) = f (e),

S : A→ A : S(f )(x) = f (x−1).


∆ is coassociative, i.e., (id⊗∆) ◦∆ = (∆⊗ id) ◦∆;(ε⊗ id) ◦∆ = (id⊗ ε) ◦∆ = id;µ ◦ (S ⊗ id) ◦∆(x) = µ ◦ (id⊗ S) ◦∆(x) = ε(x),

where µ : A⊗ A→ A is the multiplication.

Definition


34

Hopf algebras ctd.



ε : A→ C : ε(f ) = f (e),

S : A→ A : S(f )(x) = f (x−1).



Definition


35

Hopf algebras ctd.



ε : A→ C : ε(f ) = f (e),

S : A→ A : S(f )(x) = f (x−1).



Definition


36

Hopf algebras ctd.

Note that this definition makes sense over any field and evenover a commutative ring.

Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms. We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we canview the unit of A as a linear map ι : C→ A.

Proposition

1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.2. ε is an algebra homomorphism.3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).4. ε and S are uniquely determined by ∆.5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).

37

Hopf algebras ctd.

Note that this definition makes sense over any field and evenover a commutative ring. Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms.

We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we canview the unit of A as a linear map ι : C→ A.

Proposition


38

Hopf algebras ctd.

Note that this definition makes sense over any field and evenover a commutative ring. Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms. We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u.

Finally, note that we canview the unit of A as a linear map ι : C→ A.

Proposition


39

Hopf algebras ctd.

Note that this definition makes sense over any field and evenover a commutative ring. Usually it is also assumed that S isinvertible (we will do so below); this does not follow from theabove axioms. We will also use the notation ∆op = P ◦∆ where Pis the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we canview the unit of A as a linear map ι : C→ A.

Proposition


40

Hopf algebras ctd.


Proposition

1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.

2. ε is an algebra homomorphism.3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).4. ε and S are uniquely determined by ∆.5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).

41

Hopf algebras ctd.


Proposition

1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.2. ε is an algebra homomorphism.

3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).4. ε and S are uniquely determined by ∆.5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).

42

Hopf algebras ctd.


Proposition

1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.2. ε is an algebra homomorphism.3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).

4. ε and S are uniquely determined by ∆.5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).

43

Hopf algebras ctd.


Proposition

1. If A is a finite dimensional Hopf algebra then A∗ is also, withthe operations of A∗ being dual to the operations of A.2. ε is an algebra homomorphism.3. S is an algebra and coalgebra antihomomorphism, i.e.,S(xy) = S(y)S(x) and ∆(S(x)) = (S ⊗ S)(∆op(x)).4. ε and S are uniquely determined by ∆.

5. If A is commutative or cocommutative (i.e., ∆ = ∆op) thenS2 = id (even without the assumption that S is invertible).

44

Hopf algebras ctd.


Proposition


45

Examples of Hopf algebras

Example

1 A = O(G ), G is finite. Then A is commutative.

2 A = O(G ) (the algebra of regular functions), G is an affinealgebraic group, A is commutative.

3 A = CG - the group algebra,∆(g) = g ⊗ g , S(g) = g−1, ε(g) = 1, A is cocommutative:∆ = ∆op.

4 g a Lie algebra, A = U(g) (the universal enveloping algebra),∆(x) = x ⊗ 1 + 1⊗ x , S(x) = −x , ε(x) = 0 for x ∈ g, A iscocommutative.

46

Examples of Hopf algebras

Example

1 A = O(G ), G is finite. Then A is commutative.

2 A = O(G ) (the algebra of regular functions), G is an affinealgebraic group, A is commutative.

3 A = CG - the group algebra,∆(g) = g ⊗ g , S(g) = g−1, ε(g) = 1, A is cocommutative:∆ = ∆op.

4 g a Lie algebra, A = U(g) (the universal enveloping algebra),∆(x) = x ⊗ 1 + 1⊗ x , S(x) = −x , ε(x) = 0 for x ∈ g, A iscocommutative.

47

Quantum SL2

Let q ∈ C, q 6= 0,±1. The quantum group Uq(sl2) is generatedby e, f ,K±1 with relations

KeK−1 = q2e, KfK−1 = q−2f , ef − fe =K − K−1

q − q−1.

The coproduct and counit are defined by

∆e = e ⊗ K + 1⊗ e, ∆f = f ⊗ 1 + K−1 ⊗ f , ∆K = K ⊗ K ,

ε(e) = 0, ε(f ) = 0, ε(K ) = 1.

The antipode can then be found from the Hopf algebra axioms:

µ ◦ (S ⊗ 1) ◦∆(e) = ε(e) ⇒ S(e) = −eK−1.Similarly one obtains S(f ) = −Kf , S(K ) = K−1.

This is a deformation of U(sl2) because if one sets K = qh andsends q → 1, one recovers the sl2 relations.

This example shows that S2 6= id in general: we haveS2(x) = KxK−1.

48

Quantum SL2



q − q−1.


∆e = e ⊗ K + 1⊗ e, ∆f = f ⊗ 1 + K−1 ⊗ f , ∆K = K ⊗ K ,

ε(e) = 0, ε(f ) = 0, ε(K ) = 1.





49

Quantum SL2



q − q−1.


∆e = e ⊗ K + 1⊗ e, ∆f = f ⊗ 1 + K−1 ⊗ f , ∆K = K ⊗ K ,

ε(e) = 0, ε(f ) = 0, ε(K ) = 1.





50

Quantum SL2



q − q−1.


∆e = e ⊗ K + 1⊗ e, ∆f = f ⊗ 1 + K−1 ⊗ f , ∆K = K ⊗ K ,

ε(e) = 0, ε(f ) = 0, ε(K ) = 1.


µ ◦ (S ⊗ 1) ◦∆(e) = ε(e) ⇒ S(e) = −eK−1.

Similarly one obtains S(f ) = −Kf , S(K ) = K−1.This is a deformation of U(sl2) because if one sets K = qh and

sends q → 1, one recovers the sl2 relations.This example shows that S2 6= id in general: we have

S2(x) = KxK−1.

51

Quantum SL2



q − q−1.


∆e = e ⊗ K + 1⊗ e, ∆f = f ⊗ 1 + K−1 ⊗ f , ∆K = K ⊗ K ,

ε(e) = 0, ε(f ) = 0, ε(K ) = 1.





52

Quantum SL2



q − q−1.


∆e = e ⊗ K + 1⊗ e, ∆f = f ⊗ 1 + K−1 ⊗ f , ∆K = K ⊗ K ,

ε(e) = 0, ε(f ) = 0, ε(K ) = 1.





53

Representations of Uq(sl2)

Assume that q is not a root of unity.

Then the representationtheory of Uq(sl2) is very similar to the representation theory of sl2.

Proposition

Finite dimensional representations of Uq(sl2) are semisimple.

So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.

Proposition

There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1, with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n} such that Kf jv = qn−2j f jv ,

ef jv = [j ]q[n − j + 1]qfj−1v , where [k]q := qk−q−k

q−q−1 .

54


Assume that q is not a root of unity. Then the representationtheory of Uq(sl2) is very similar to the representation theory of sl2.

Proposition



Proposition



q−q−1 .

55



Proposition


So it remains to classify the irreducible f.d. representations V .

We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.

Proposition



q−q−1 .

56



Proposition


So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q.

E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.

Proposition



q−q−1 .

57



Proposition


So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I.

However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.Thus it suffices to classify irreducibles of type I.

Proposition



q−q−1 .

58



Proposition


So it remains to classify the irreducible f.d. representations V .We say that V is of type I if the eigenvalues of K on V are integerpowers of q. E.g., the character χ : Uq(sl2)→ C given byχ(e) = χ(f ) = 0, χ(K ) = −1 is not of type I. However, if V is notof type I then it has the form V = V+ ⊗ χ where V+ is of type I.

Thus it suffices to classify irreducibles of type I.

Proposition



q−q−1 .

59



Proposition



Proposition



q−q−1 .

60



Proposition



Proposition

There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1,

with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n} such that Kf jv = qn−2j f jv ,


q−q−1 .

61



Proposition



Proposition

There is exactly one type I irreducible representation Vn of Uq(sl2)of each positive dimension n + 1, with generator v with ev = 0,Kv = qnv and basis {f jv , 0 ≤ j ≤ n}

such that Kf jv = qn−2j f jv ,


q−q−1 .

62



Proposition



Proposition



q−q−1 .

63

Tensor products of representations of Hopf algebras

For a group and a Lie algebra the representation categoriesRepG and Rep g are endowed with tensor products:

πV : G → AutV , πW : G → AutW ⇒

πV⊗W (g) = πV (g)⊗ πW (g) ∀g ∈ G .

πV : g→ AutV , πW : g→ AutW ⇒

πV⊗W (x) = πV (x)⊗ 1 + 1⊗ πW (x) ∀x ∈ g.

The formula for the tensor product of representations of a Hopfalgebra A is a straightforward generalization:

πV⊗W (x) = (πV ⊗ πW )(∆(x)) = πV (x(1))⊗ πW (x(2)) ∀x ∈ A.

So one can regard the category C = RepA of representations of Aas a category equipped with a tensor product bifunctor

⊗ : C × C → C : (X ,Y ) 7→ X ⊗ Y .

64




πV⊗W (g) = πV (g)⊗ πW (g) ∀g ∈ G .


πV⊗W (x) = πV (x)⊗ 1 + 1⊗ πW (x) ∀x ∈ g.




⊗ : C × C → C : (X ,Y ) 7→ X ⊗ Y .

65




πV⊗W (g) = πV (g)⊗ πW (g) ∀g ∈ G .


πV⊗W (x) = πV (x)⊗ 1 + 1⊗ πW (x) ∀x ∈ g.




⊗ : C × C → C : (X ,Y ) 7→ X ⊗ Y .

66




πV⊗W (g) = πV (g)⊗ πW (g) ∀g ∈ G .


πV⊗W (x) = πV (x)⊗ 1 + 1⊗ πW (x) ∀x ∈ g.




⊗ : C × C → C : (X ,Y ) 7→ X ⊗ Y .

67




πV⊗W (g) = πV (g)⊗ πW (g) ∀g ∈ G .


πV⊗W (x) = πV (x)⊗ 1 + 1⊗ πW (x) ∀x ∈ g.




⊗ : C × C → C : (X ,Y ) 7→ X ⊗ Y .

68




πV⊗W (g) = πV (g)⊗ πW (g) ∀g ∈ G .


πV⊗W (x) = πV (x)⊗ 1 + 1⊗ πW (x) ∀x ∈ g.




⊗ : C × C → C : (X ,Y ) 7→ X ⊗ Y .

69

Tensor products of representations of Hopf algebras, ctd.

This product also has a unit (the trivial 1-dimensionalrepresentation):

1 = C : π1(a) = ε(a), 1⊗ X ∼= X ⊗ 1 ∼= X ∀X ∈ C.Finally, the tensor product is associative on isomorphism classes:

(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).

Thus C is a category with a unital tensor product associative up toan isomorphism.

However, this notion is not very useful; about such categoriesone can say very little, if anything at all. On the other hand, innatural examples a lot more structure is present, which is just alittle bit less obvious.

More precisely, a much better notion is obtained if, according tothe general yoga of category theory, we don’t just say simply that(X ⊗Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ), but make this isomorphism a part ofthe data and impose coherence conditions on this data.This leadsto the notion of a monoidal category.

70



1 = C : π1(a) = ε(a), 1⊗ X ∼= X ⊗ 1 ∼= X ∀X ∈ C.

Finally, the tensor product is associative on isomorphism classes:

(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).




71




(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).




72




(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).




73




(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).




74




(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).


However, this notion is not very useful; about such categoriesone can say very little, if anything at all.

On the other hand, innatural examples a lot more structure is present, which is just alittle bit less obvious.


75




(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).




76




(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).



More precisely, a much better notion is obtained if, according tothe general yoga of category theory, we don’t just say simply that(X ⊗Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ), but make this isomorphism a part ofthe data and impose coherence conditions on this data.

This leadsto the notion of a monoidal category.

77




(X ⊗ Y )⊗ Z ∼= X ⊗ (Y ⊗ Z ).




78

Monoidal categories

Namely, we should equip C with an associativity isomorphism

αXYZ : (X ⊗ Y )⊗ Z∼−→ X ⊗ (Y ⊗ Z )

functorial in X ,Y ,Z , which satisfies the pentagon identity

X (Y (ZT ))

(XY )(ZT )

((XY )Z )T (X (YZ ))T

X ((YZ )T )

where the arrows are induced by α and we have omitted the ⊗signs for brevity. The relation is that the diagram commutes.

79

Monoidal categories


αXYZ : (X ⊗ Y )⊗ Z∼−→ X ⊗ (Y ⊗ Z )


X (Y (ZT ))

(XY )(ZT )

((XY )Z )T (X (YZ ))T

X ((YZ )T )


80

Monoidal categories


αXYZ : (X ⊗ Y )⊗ Z∼−→ X ⊗ (Y ⊗ Z )

functorial in X ,Y ,Z ,

which satisfies the pentagon identity

X (Y (ZT ))

(XY )(ZT )

((XY )Z )T (X (YZ ))T

X ((YZ )T )


81

Monoidal categories


αXYZ : (X ⊗ Y )⊗ Z∼−→ X ⊗ (Y ⊗ Z )


X (Y (ZT ))

(XY )(ZT )

((XY )Z )T (X (YZ ))T

X ((YZ )T )


82

Monoidal categories


αXYZ : (X ⊗ Y )⊗ Z∼−→ X ⊗ (Y ⊗ Z )


X (Y (ZT ))

(XY )(ZT )

((XY )Z )T (X (YZ ))T

X ((YZ )T )

where the arrows are induced by α and we have omitted the ⊗signs for brevity.

The relation is that the diagram commutes.

83

Monoidal categories


αXYZ : (X ⊗ Y )⊗ Z∼−→ X ⊗ (Y ⊗ Z )


X (Y (ZT ))

(XY )(ZT )

((XY )Z )T (X (YZ ))T

X ((YZ )T )


84

Monoidal categories, ctd.

We should also require the existence of a unit object 1

with anisomorphism

ι : 1⊗ 1 ∼= 1

such that the functors 1⊗ and ⊗1 are autoequivalences of C.

Definition

A category C with such structures and properties is called amonoidal category.

We see that for a Hopf algebra A, the category RepA is amonoidal category, with αXYZ being the natural isomorphism(X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z ), sending (x ⊗ y)⊗ z to x ⊗ (y ⊗ z).In a similar way, comodules over A (i.e., spaces V with a linearmap ρ : V → A⊗ V defining an action of the algebra A∗ on V )form a monoidal category ComodA. In fact, if dimA <∞ thenComodA = RepA∗. Also, if G is an algebraic group then analgebraic representation of G is the same thing as a finitedimensional O(G )-comodule.

85


We should also require the existence of a unit object 1 with anisomorphism

ι : 1⊗ 1 ∼= 1


Definition



86



ι : 1⊗ 1 ∼= 1


Definition



87



ι : 1⊗ 1 ∼= 1


Definition



88



ι : 1⊗ 1 ∼= 1


Definition


We see that for a Hopf algebra A, the category RepA is amonoidal category, with αXYZ being the natural isomorphism(X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z ), sending (x ⊗ y)⊗ z to x ⊗ (y ⊗ z).

In a similar way, comodules over A (i.e., spaces V with a linearmap ρ : V → A⊗ V defining an action of the algebra A∗ on V )form a monoidal category ComodA. In fact, if dimA <∞ thenComodA = RepA∗. Also, if G is an algebraic group then analgebraic representation of G is the same thing as a finitedimensional O(G )-comodule.

89



ι : 1⊗ 1 ∼= 1


Definition


We see that for a Hopf algebra A, the category RepA is amonoidal category, with αXYZ being the natural isomorphism(X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z ), sending (x ⊗ y)⊗ z to x ⊗ (y ⊗ z).In a similar way, comodules over A (i.e., spaces V with a linearmap ρ : V → A⊗ V defining an action of the algebra A∗ on V )form a monoidal category ComodA.

In fact, if dimA <∞ thenComodA = RepA∗. Also, if G is an algebraic group then analgebraic representation of G is the same thing as a finitedimensional O(G )-comodule.

90



ι : 1⊗ 1 ∼= 1


Definition


We see that for a Hopf algebra A, the category RepA is amonoidal category, with αXYZ being the natural isomorphism(X ⊗ Y )⊗ Z → X ⊗ (Y ⊗ Z ), sending (x ⊗ y)⊗ z to x ⊗ (y ⊗ z).In a similar way, comodules over A (i.e., spaces V with a linearmap ρ : V → A⊗ V defining an action of the algebra A∗ on V )form a monoidal category ComodA. In fact, if dimA <∞ thenComodA = RepA∗.

Also, if G is an algebraic group then analgebraic representation of G is the same thing as a finitedimensional O(G )-comodule.

91



ι : 1⊗ 1 ∼= 1


Definition



92

Duality in monoidal categories

Let us now discuss duality for representations of Hopf algebras,which generalizes duality for group and Lie algebrarepresentations.

The dual representations X ∗, ∗X ∈ RepA forX ∈ RepA are both the dual vector space to X with the actionsdefined as follows:

πX∗(a) = πX (S(a))∗ - left dual, π∗X (a) = πX (S−1(a))∗ - right dual.

For the pair X ,X ∗ there is the evaluation morphism

X ∗ ⊗ X → 1

(the usual pairing). For finite dimensional representations there isalso the coevaluation morphism

1→ X ⊗ X ∗.

and a pair of functorial isomorphisms

(∗X )∗ = X , ∗(X ∗) = X .

93


Let us now discuss duality for representations of Hopf algebras,which generalizes duality for group and Lie algebrarepresentations.The dual representations X ∗, ∗X ∈ RepA forX ∈ RepA are both the dual vector space to X with the actionsdefined as follows:



X ∗ ⊗ X → 1


1→ X ⊗ X ∗.


(∗X )∗ = X , ∗(X ∗) = X .

94





X ∗ ⊗ X → 1


1→ X ⊗ X ∗.


(∗X )∗ = X , ∗(X ∗) = X .

95





X ∗ ⊗ X → 1

(the usual pairing).

For finite dimensional representations there isalso the coevaluation morphism

1→ X ⊗ X ∗.


(∗X )∗ = X , ∗(X ∗) = X .

96





X ∗ ⊗ X → 1


1→ X ⊗ X ∗.


(∗X )∗ = X , ∗(X ∗) = X .

97





X ∗ ⊗ X → 1


1→ X ⊗ X ∗.


(∗X )∗ = X , ∗(X ∗) = X .

98





X ∗ ⊗ X → 1


1→ X ⊗ X ∗.


(∗X )∗ = X , ∗(X ∗) = X .

99





X ∗ ⊗ X → 1


1→ X ⊗ X ∗.


(∗X )∗ = X , ∗(X ∗) = X .

100

Duality in monoidal categories, ctd.

Let’s axiomatize this in the setting of monoidal categories.

Definition

An object Y of a monoidal category C is a left dual to X , denotedY = X ∗, if there exist evaluation and coevaluation morphisms

ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,

such that the following morphisms are the identities:

Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X

αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,

Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )

α−1YXY−−−−→ (Y ⊗ X )⊗ Y

ev⊗1−−−−→ Y .

If Y is a left dual to X then we have a functorial isomorphismHom(Z ,Y ) ∼= Hom(Z ⊗ X , 1). By the Yoneda lemma, this impliesthat the left dual, if exists, is unique up to a unique isomorphism.

101



Definition


ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,


Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X

αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,

Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )

α−1YXY−−−−→ (Y ⊗ X )⊗ Y

ev⊗1−−−−→ Y .


102



Definition


ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,


Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X

αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,

Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )

α−1YXY−−−−→ (Y ⊗ X )⊗ Y

ev⊗1−−−−→ Y .


103



Definition


ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,


Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X

αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,

Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )

α−1YXY−−−−→ (Y ⊗ X )⊗ Y

ev⊗1−−−−→ Y .


104



Definition


ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,


Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X

αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,

Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )

α−1YXY−−−−→ (Y ⊗ X )⊗ Y

ev⊗1−−−−→ Y .


105



Definition


ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,


Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X

αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,

Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )

α−1YXY−−−−→ (Y ⊗ X )⊗ Y

ev⊗1−−−−→ Y .

If Y is a left dual to X then we have a functorial isomorphismHom(Z ,Y ) ∼= Hom(Z ⊗ X , 1).

By the Yoneda lemma, this impliesthat the left dual, if exists, is unique up to a unique isomorphism.

106



Definition


ev : Y ⊗ X → 1, coev : 1→ X ⊗ Y ,


Xcoev⊗1−−−−→ (X ⊗ Y )⊗ X

αXYX−−−−→ X ⊗ (Y ⊗ X )1⊗ev−−−−→ X ,

Y1⊗coev−−−−→ Y ⊗ (X ⊗ Y )

α−1YXY−−−−→ (Y ⊗ X )⊗ Y

ev⊗1−−−−→ Y .


107


Definition

An object Z = ∗X is the right dual to X if X ∼= Z ∗.

As the left dual, the right dual is unique up to a uniqueisomorphism if exists.

Definition

An object X is rigid if it has both the left and the right dual. Acategory C is rigid if all its objects are rigid.

Thus, if A is a Hopf algebra then the category Repf A of finitedimensional representations of A is a rigid monoidal category.

108


Definition



Definition



109


Definition



Definition

An object X is rigid if it has both the left and the right dual.

Acategory C is rigid if all its objects are rigid.


110


Definition



Definition



111


Definition



Definition



112

Examples of rigid monoidal categories

Example

Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G .

The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).

Example

The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).

113


Example

Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1.

ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).

Example


114


Example

Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category.

We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).

Example


115


Example

Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces).

This category makes sense for anygroup G (not necessarily finite).

Example


116


Example

Let G be a finite group, A = O(G ), then Repf A is spanned by1-dimensional representations parametrized by g ∈ G . The tensorproduct is defined by g ⊗ h = gh, and g∗ = ∗g = g−1. ThusRepf A is a rigid monoidal category. We denote it by Vec(G )(G -graded vector spaces). This category makes sense for anygroup G (not necessarily finite).

Example


117


Example


Example

The previous example has the following twisted version.

Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).

118


Example


Example

The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.

Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).

119


Example


Example

The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G .

If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).We will denote thiscategory Vec(G , α).

120


Example


Example

The previous example has the following twisted version. Letαg ,h,k : (g ⊗ h)⊗ k = ghk → g ⊗ (h ⊗ k) = ghk, i.e. αg ,h,k ∈ C∗.Then α satisfies the pentagon identity ⇔ α is a 3-cocycle of thegroup G . If so then this equips C = Vec(G ) with another structureof a rigid monoidal category (with the same tensor product functorbut different associativity isomorphism).

We will denote thiscategory Vec(G , α).

121


Example


Example


122

Monoidal functors

Let C,D be monoidal categories.

Definition

A functor F : C → D is a monoidal functor if F (1C) ∼= 1D and F isequipped with a functorial (in X ,Y ) isomorphismJX ,Y : F (X )⊗ F (Y )

∼−→ F (X ⊗ Y ) which makes the diagram

(F (X )⊗ F (Y ))⊗ F (Z )αD−−−−→ F (X )⊗ (F (Y )⊗ F (Z ))yJX ,Y⊗idZ

yidX⊗JY ,Z

F (X ⊗ Y )⊗ F (Z ) F (X )⊗ F (Y ⊗ Z )yJX⊗Y ,Z

yJX ,Y⊗Z

F ((X ⊗ Y )⊗ Z )F (αC)−−−−→ F (X ⊗ (Y ⊗ Z ))

commutative. A monoidal functor is an equivalence of monoidalcategories if it is an equivalence of categories.

123

Monoidal functors


Definition


∼−→ F (X ⊗ Y )

which makes the diagram


yidX⊗JY ,Z


yJX ,Y⊗Z

F ((X ⊗ Y )⊗ Z )F (αC)−−−−→ F (X ⊗ (Y ⊗ Z ))


124

Monoidal functors


Definition




yidX⊗JY ,Z


yJX ,Y⊗Z

F ((X ⊗ Y )⊗ Z )F (αC)−−−−→ F (X ⊗ (Y ⊗ Z ))

commutative.

A monoidal functor is an equivalence of monoidalcategories if it is an equivalence of categories.

125

Monoidal functors


Definition




yidX⊗JY ,Z


yJX ,Y⊗Z

F ((X ⊗ Y )⊗ Z )F (αC)−−−−→ F (X ⊗ (Y ⊗ Z ))


126

Monoidal functors ctd.

The notion of monoidal equivalence is useful because monoidalcategories that are monoidally equivalent are “the same for allpractical purposes”.

Example

If α, β are two 3-cocycles on G then the identity functor

F = Id : Vec(G , α)→ Vec(G , β), g 7→ g

is monoidal with Jg ,h ∈ C∗ : g ⊗ h→ g ⊗ h iff dJ = α/β, where dis the differential in the standard complex of G with coefficients inC∗. In particular, F admits a monoidal structure if and only if thecohomology classes of α and β are the same.This shows thatVec(G , α) is equivalent to Vec(G ) iff α is trivial in H3(G ,C∗).

127



Example




128



Example



is monoidal with Jg ,h ∈ C∗ : g ⊗ h→ g ⊗ h iff dJ = α/β, where dis the differential in the standard complex of G with coefficients inC∗.

In particular, F admits a monoidal structure if and only if thecohomology classes of α and β are the same.This shows thatVec(G , α) is equivalent to Vec(G ) iff α is trivial in H3(G ,C∗).

129



Example



is monoidal with Jg ,h ∈ C∗ : g ⊗ h→ g ⊗ h iff dJ = α/β, where dis the differential in the standard complex of G with coefficients inC∗. In particular, F admits a monoidal structure if and only if thecohomology classes of α and β are the same.

This shows thatVec(G , α) is equivalent to Vec(G ) iff α is trivial in H3(G ,C∗).

130



Example




131

The universal R-matrix of Uq(sl2)

If A is a Hopf algebra and X ,Y ∈ RepH then X ⊗Y � Y ⊗Xin general, as ∆ 6= ∆op (e.g. g ⊗ h � h ⊗ g for g , h ∈ Vec(G )).

However, sometimes ∆ 6= ∆op but still X ⊗ Y ∼= Y ⊗ X .

Example

For A = Uq(sl2) where qn 6= 1 define the universal R-matrix

R = qh⊗h2

∞∑k=0

qk(k−1)

2(q − q−1)k

[k]q!ek ⊗ f k , where [k]q! = [1]q...[k]q.

This is an infinite series, but it makes sense as an operator onX ⊗ Y for any finite dimensional type I representations X ,Y ,

because the sum terminates. Here qh⊗h2 (x ⊗ y) = q

λµ2 x ⊗ y if x , y

have weights λ, µ, i.e., Kx = qλx ,Ky = qµy .

Theorem (Drinfeld)

The operator c = P ◦ R defines an isomorphism of representationsc : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)Ron X ⊗ Y for a ∈ Uq(sl2).

132


If A is a Hopf algebra and X ,Y ∈ RepH then X ⊗Y � Y ⊗Xin general, as ∆ 6= ∆op (e.g. g ⊗ h � h ⊗ g for g , h ∈ Vec(G )).However, sometimes ∆ 6= ∆op but still X ⊗ Y ∼= Y ⊗ X .

Example


R = qh⊗h2

∞∑k=0

qk(k−1)

2(q − q−1)k






Theorem (Drinfeld)


133



Example


R = qh⊗h2

∞∑k=0

qk(k−1)

2(q − q−1)k

[k]q!ek ⊗ f k ,

where [k]q! = [1]q...[k]q.





Theorem (Drinfeld)


134



Example


R = qh⊗h2

∞∑k=0

qk(k−1)

2(q − q−1)k






Theorem (Drinfeld)


135



Example


R = qh⊗h2

∞∑k=0

qk(k−1)

2(q − q−1)k



because the sum terminates.

Here qh⊗h2 (x ⊗ y) = q



Theorem (Drinfeld)


136



Example


R = qh⊗h2

∞∑k=0

qk(k−1)

2(q − q−1)k






Theorem (Drinfeld)


137



Example


R = qh⊗h2

∞∑k=0

qk(k−1)

2(q − q−1)k






Theorem (Drinfeld)

The operator c = P ◦ R defines an isomorphism of representationsc : X ⊗ Y → Y ⊗ X .

In other words, we have R∆(a) = ∆op(a)Ron X ⊗ Y for a ∈ Uq(sl2).

138



Example


R = qh⊗h2

∞∑k=0

qk(k−1)

2(q − q−1)k






Theorem (Drinfeld)


139



Example


R = qh⊗h2

∞∑k=0

qk(k−1)

2(q − q−1)k






Theorem (Drinfeld)


140

The Drinfeld center

A prototypical example of a monoidal category whereX ⊗ Y ∼= Y ⊗ X is the Drinfeld center of a monoidal category C.

Definition

The Drinfeld center Z(C) of C is the category of pairs (Y , ϕ)where Y ∈ C and ϕ : Y⊗?→?⊗ Y is a functorial isomorphismgiven by ϕX : Y ⊗ X

∼−→ X ⊗ Y ∀X ∈ C, satisfying the followingcommutative diagram:

Y ⊗ (X1 ⊗ X2)ϕX1⊗X2−−−−→ (X1 ⊗ X2)⊗ Yyα−1

YX1X2

xα−1X1X2Y

(Y ⊗ X1)⊗ X2 X1 ⊗ (X2 ⊗ Y )yϕX1⊗id

xid⊗ϕX2

(X1 ⊗ Y )⊗ X2

αX1YX2−−−−→ X1 ⊗ (Y ⊗ X2)

Morphisms of such pairs are morphisms in C which preserve ϕ.

141

The Drinfeld center


Definition



Y ⊗ (X1 ⊗ X2)ϕX1⊗X2−−−−→ (X1 ⊗ X2)⊗ Yyα−1

YX1X2

xα−1X1X2Y

(Y ⊗ X1)⊗ X2 X1 ⊗ (X2 ⊗ Y )yϕX1⊗id

xid⊗ϕX2

(X1 ⊗ Y )⊗ X2

αX1YX2−−−−→ X1 ⊗ (Y ⊗ X2)


142

The Drinfeld center


Definition



Y ⊗ (X1 ⊗ X2)ϕX1⊗X2−−−−→ (X1 ⊗ X2)⊗ Yyα−1

YX1X2

xα−1X1X2Y

(Y ⊗ X1)⊗ X2 X1 ⊗ (X2 ⊗ Y )yϕX1⊗id

xid⊗ϕX2

(X1 ⊗ Y )⊗ X2

αX1YX2−−−−→ X1 ⊗ (Y ⊗ X2)


143

The Drinfeld center


Definition



Y ⊗ (X1 ⊗ X2)ϕX1⊗X2−−−−→ (X1 ⊗ X2)⊗ Yyα−1

YX1X2

xα−1X1X2Y

(Y ⊗ X1)⊗ X2 X1 ⊗ (X2 ⊗ Y )yϕX1⊗id

xid⊗ϕX2

(X1 ⊗ Y )⊗ X2

αX1YX2−−−−→ X1 ⊗ (Y ⊗ X2)

Morphisms of such pairs are morphisms in C which preserve ϕ.144

The Drinfeld center, ctd.

The Drinfeld center Z(C) has a natural monoidal structuredefined by (Y , ϕ)⊗ (Z , ψ) = (Y ⊗ Z , η),

where (suppressing α)

ηX = (φX⊗idZ )◦(idY⊗ψX ) : Y⊗Z⊗X → Y⊗X⊗Z → X⊗Y⊗Z .

Note also that we have a monoidal forgetful functor Z (C)→ C,(Y , ϕ) 7→ Y . Moreover, if Y ,Z ∈ Z(C) then there are two ways

Y ⊗ ZcYZ ,c

−1ZY−−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ ,

cZY = ψY . This gives an action of the braid group Bn on V⊗n forV ∈ Z(C). Recall that Bn = π1(Cn\diagonals/Sn) =

〈s1, ..., sn−1 | si sj = sjsi if |i − j | ≥ 2, si si+1si = si+1si si+1〉

Proposition

There is an action of Bn on V⊗n is defined by

ρ : Bn → Aut(V⊗n), ρ(si ) = ci ,i+1 = (cV ,V )i ,i+1

145


The Drinfeld center Z(C) has a natural monoidal structuredefined by (Y , ϕ)⊗ (Z , ψ) = (Y ⊗ Z , η), where (suppressing α)



Y ⊗ ZcYZ ,c




Proposition



146




Note also that we have a monoidal forgetful functor Z (C)→ C,(Y , ϕ) 7→ Y .

Moreover, if Y ,Z ∈ Z(C) then there are two ways

Y ⊗ ZcYZ ,c




Proposition



147





Y ⊗ ZcYZ ,c


cZY = ψY .

This gives an action of the braid group Bn on V⊗n forV ∈ Z(C). Recall that Bn = π1(Cn\diagonals/Sn) =


Proposition



148





Y ⊗ ZcYZ ,c


cZY = ψY . This gives an action of the braid group Bn on V⊗n forV ∈ Z(C).

Recall that Bn = π1(Cn\diagonals/Sn) =


Proposition



149





Y ⊗ ZcYZ ,c




Proposition



150





Y ⊗ ZcYZ ,c




Proposition



151





Y ⊗ ZcYZ ,c




Proposition



152





Y ⊗ ZcYZ ,c




Proposition



153


Proof.

This follows from the hexagon relations for X ,Y ,Z ∈ Z(C):

X ⊗ Y ⊗ Z → Y ⊗ X ⊗ Z↘ ↓

Y ⊗ Z ⊗ X

X ⊗ Y ⊗ Z → X ⊗ Z ⊗ Y↘ ↓

Z ⊗ X ⊗ Y

where we suppress α and the maps are given by c .

They are call “hexagon relations” because they would havebeen hexagons had we not suppressed α.

This motivates the definition of a braided monoidal category.

154


Proof.


X ⊗ Y ⊗ Z → Y ⊗ X ⊗ Z↘ ↓

Y ⊗ Z ⊗ X

X ⊗ Y ⊗ Z → X ⊗ Z ⊗ Y↘ ↓

Z ⊗ X ⊗ Y




155


Proof.


X ⊗ Y ⊗ Z → Y ⊗ X ⊗ Z↘ ↓

Y ⊗ Z ⊗ X

X ⊗ Y ⊗ Z → X ⊗ Z ⊗ Y↘ ↓

Z ⊗ X ⊗ Y




156


Proof.


X ⊗ Y ⊗ Z → Y ⊗ X ⊗ Z↘ ↓

Y ⊗ Z ⊗ X

X ⊗ Y ⊗ Z → X ⊗ Z ⊗ Y↘ ↓

Z ⊗ X ⊗ Y




157

Braided monoidal categories

Definition

A braided monoidal category is a monoidal category endowed witha functorial isomorphism c : ⊗ → ⊗op, cX ,Y : X ⊗ Y → Y ⊗ X

which satisfies the hexagon relations.

Thus we obtain

Theorem

The Drinfeld center Z(C) is a braided monoidal category.

Moreover, every braided monoidal category C is a braidedsubcategory of its Drinfeld center using the inclusion ι : C → Z(C)given by X 7→ (X , cX?). In this sense the Drinfeld center is aprototypical example of a braided category.

Theorem

The category of finite dimensional (type I) representations ofUq(sl2) is a braided monoidal category, with c = P ◦ R.

158


Definition

A braided monoidal category is a monoidal category endowed witha functorial isomorphism c : ⊗ → ⊗op, cX ,Y : X ⊗ Y → Y ⊗ Xwhich satisfies the hexagon relations.

Thus we obtain

Theorem



Theorem


159


Definition


Thus we obtain

Theorem



Theorem


160


Definition


Thus we obtain

Theorem



Theorem


161

Braided monoidal categories ctd.

Note that as a consequence R satisfies the QuantumYang-Baxter equation

R12R13R23 = R23R13R12

(this follows from the relation s1s2s1 = s2s1s2).

Remark

A braided category is called symmetric if cXY cYX = idX⊗Y for allX ,Y . For example, the categories RepG and Rep g aresymmetric. However, Z(C) is usually not symmetric, andRepf Uq(sl2) isn’t either.

We’ll explain that the theorem on Uq(sl2), and in fact theconstruction of R, are consequences of the theorem about theDrinfeld center. To this end, let us compute the Drinfeld center ofthe category RepA of representations of a Hopf algebra A.

162



R12R13R23 = R23R13R12


Remark

A braided category is called symmetric if cXY cYX = idX⊗Y for allX ,Y .

For example, the categories RepG and Rep g aresymmetric. However, Z(C) is usually not symmetric, andRepf Uq(sl2) isn’t either.


163



R12R13R23 = R23R13R12


Remark

A braided category is called symmetric if cXY cYX = idX⊗Y for allX ,Y . For example, the categories RepG and Rep g aresymmetric.

However, Z(C) is usually not symmetric, andRepf Uq(sl2) isn’t either.


164



R12R13R23 = R23R13R12


Remark


We’ll explain that the theorem on Uq(sl2), and in fact theconstruction of R, are consequences of the theorem about theDrinfeld center.

To this end, let us compute the Drinfeld center ofthe category RepA of representations of a Hopf algebra A.

165



R12R13R23 = R23R13R12


Remark



166

Yetter-Drinfeld modules

Let Y ∈ Z(RepA).

Then the mapϕA : IndACY = Y ⊗ A→ A⊗ Y defines a comodule structureτ = ϕA|Y : Y → A⊗ Y . The compatibility condition between thiscomodule structure and the A-module structure on Y is

τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).

where y ∈ Y , τ(y) = y(1) ⊗ y(2) and

a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).

Definition

A Yetter-Drinfeld module over A is an A-module Y which is also anA-comodule with τ : Y → A⊗Y satisfying the above compatibilitycondition. The category of such modules is denoted YD(A).

Proposition

One has Z(RepA) ∼= YD(A).

167


Let Y ∈ Z(RepA). Then the mapϕA : IndACY = Y ⊗ A→ A⊗ Y defines a comodule structureτ = ϕA|Y : Y → A⊗ Y .

The compatibility condition between thiscomodule structure and the A-module structure on Y is

τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).

where y ∈ Y , τ(y) = y(1) ⊗ y(2) and

a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).

Definition


Proposition


168


Let Y ∈ Z(RepA). Then the mapϕA : IndACY = Y ⊗ A→ A⊗ Y defines a comodule structureτ = ϕA|Y : Y → A⊗ Y . The compatibility condition between thiscomodule structure and the A-module structure on Y is

τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).

where y ∈ Y , τ(y) = y(1) ⊗ y(2) and

a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).

Definition


Proposition


169



τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).

where y ∈ Y , τ(y) = y(1) ⊗ y(2)

and

a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).

Definition


Proposition


170



τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).

where y ∈ Y , τ(y) = y(1) ⊗ y(2) and

a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).

Definition


Proposition


171



τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).

where y ∈ Y , τ(y) = y(1) ⊗ y(2) and

a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).

Definition

A Yetter-Drinfeld module over A is an A-module Y which is also anA-comodule with τ : Y → A⊗Y satisfying the above compatibilitycondition.

The category of such modules is denoted YD(A).

Proposition


172



τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).

where y ∈ Y , τ(y) = y(1) ⊗ y(2) and

a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).

Definition


Proposition


173



τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).

where y ∈ Y , τ(y) = y(1) ⊗ y(2) and

a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).

Definition


Proposition


174



τ(ay) = a(1)y(1)S(a(2))⊗ y(2)a(3).

where y ∈ Y , τ(y) = y(1) ⊗ y(2) and

a ∈ A, (1⊗∆) ◦∆(a) = a(1) ⊗ a(2) ⊗ a(3).

Definition


Proposition


175

Quantum double

If A is finite dimensional, an A-comodule is the same as anA∗-module,

so the category YD(A) can be realized as the categoryof modules over some algebra generated by A and A∗ with somecommutation relation between them. Thus we get

Proposition

For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as avector space to A⊗ A∗ such that there is a monoidal equivalence

YD(A) ∼= RepD(A).

The Hopf algebra D(A) is called the (Drinfeld) quantum doubleof A. The algebras A and A∗op (with opposite coproduct) are Hopfsubalgebras of D(A), and the commutation relations between themhave the form ba = (a(1), b(1))(a(3), b(3))a(2)S

−1(b(2)),a ∈ A, b ∈ A∗.

176

Quantum double

If A is finite dimensional, an A-comodule is the same as anA∗-module, so the category YD(A) can be realized as the categoryof modules over some algebra generated by A and A∗ with somecommutation relation between them.

Thus we get

Proposition


YD(A) ∼= RepD(A).


−1(b(2)),a ∈ A, b ∈ A∗.

177

Quantum double

If A is finite dimensional, an A-comodule is the same as anA∗-module, so the category YD(A) can be realized as the categoryof modules over some algebra generated by A and A∗ with somecommutation relation between them. Thus we get

Proposition

For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as avector space to A⊗ A∗

such that there is a monoidal equivalence

YD(A) ∼= RepD(A).


−1(b(2)),a ∈ A, b ∈ A∗.

178

Quantum double


Proposition


YD(A) ∼= RepD(A).


−1(b(2)),a ∈ A, b ∈ A∗.

179

Quantum double


Proposition


YD(A) ∼= RepD(A).

The Hopf algebra D(A) is called the (Drinfeld) quantum doubleof A.

The algebras A and A∗op (with opposite coproduct) are Hopfsubalgebras of D(A), and the commutation relations between themhave the form ba = (a(1), b(1))(a(3), b(3))a(2)S

−1(b(2)),a ∈ A, b ∈ A∗.

180

Quantum double


Proposition


YD(A) ∼= RepD(A).


−1(b(2)),a ∈ A, b ∈ A∗.

181

Quantum double, ctd.

It is natural to ask how to express the braided structure ofYD(A) in terms of D(A).

This is addressed by the followingtheorem.

Theorem (Drinfeld)

The braiding of RepD(A) = YD(A) is given by the formulac = P ◦ R where R ∈ A⊗ A∗op ⊂ D(A)⊗ D(A) is given by theuniversal R-matrix

R =∑i

ai ⊗ a∗i ,

where ai is basis of A and a∗i the dual basis of A∗. In particular, Rsatisfies the quantum Yang-Baxter equation

R12R13R23 = R23R13R12 ∈ D(A)3.

Moreover, the commutation relation between A and A∗ is uniquelydetermined by this equation.

182


It is natural to ask how to express the braided structure ofYD(A) in terms of D(A). This is addressed by the followingtheorem.

Theorem (Drinfeld)


R =∑i

ai ⊗ a∗i ,


R12R13R23 = R23R13R12 ∈ D(A)3.


183



Theorem (Drinfeld)

The braiding of RepD(A) = YD(A) is given by the formulac = P ◦ R

where R ∈ A⊗ A∗op ⊂ D(A)⊗ D(A) is given by theuniversal R-matrix

R =∑i

ai ⊗ a∗i ,


R12R13R23 = R23R13R12 ∈ D(A)3.


184



Theorem (Drinfeld)


R =∑i

ai ⊗ a∗i ,


R12R13R23 = R23R13R12 ∈ D(A)3.


185



Theorem (Drinfeld)


R =∑i

ai ⊗ a∗i ,

where ai is basis of A and a∗i the dual basis of A∗.

In particular, Rsatisfies the quantum Yang-Baxter equation

R12R13R23 = R23R13R12 ∈ D(A)3.


186



Theorem (Drinfeld)


R =∑i

ai ⊗ a∗i ,


R12R13R23 = R23R13R12 ∈ D(A)3.


187



Theorem (Drinfeld)


R =∑i

ai ⊗ a∗i ,


R12R13R23 = R23R13R12 ∈ D(A)3.


188

Examples

Example

Let A = CG where G is a finite group.

Then D(A) = CG nO(G ),where G acts on itself by conjugation (i.e., the commutationrelation for the double gives the conjugation action). We haveR =

∑g∈G g ⊗ δg , where δg (h) = δgh, g , h ∈ G .

Example

Let ` be an odd positive integer, q a primitive root of unity of order`. Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1, andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K . ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1

q−q−1 and coproduct given

by the above formulas and ∆(f ) = f ⊗ 1 + K−1 ⊗ f .

189

Examples

Example

Let A = CG where G is a finite group. Then D(A) = CG nO(G ),where G acts on itself by conjugation (i.e., the commutationrelation for the double gives the conjugation action).

We haveR =


Example




190

Examples

Example

Let A = CG where G is a finite group. Then D(A) = CG nO(G ),where G acts on itself by conjugation (i.e., the commutationrelation for the double gives the conjugation action). We haveR =

∑g∈G g ⊗ δg ,

where δg (h) = δgh, g , h ∈ G .

Example




191

Examples

Example



Example

Let ` be an odd positive integer, q a primitive root of unity of order`.

Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1, andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K . ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1



192

Examples

Example



Example

Let ` be an odd positive integer, q a primitive root of unity of order`. Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1,

andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K . ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1



193

Examples

Example



Example

Let ` be an odd positive integer, q a primitive root of unity of order`. Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1, andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K .

ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1



194

Examples

Example



Example

Let ` be an odd positive integer, q a primitive root of unity of order`. Let A be the Taft Hopf algebra (of dimension `2), generated bye,K±1 with the relations Ke = q2eK , e` = 0, K ` = 1, andcoproduct defined by ∆(e) = e ⊗K + 1⊗ e, ∆(K ) = K ⊗K . ThenD(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where

D0 is aHopf algebra of dimension `3 generated by e, f ,K±1 with relationsas above and Kf = q−2fK , ef − fe = K−K−1



195

Examples

Example



Example


q−q−1

and coproduct given


196

Examples

Example



Example




197

Small quantum group

Note that D0 is nothing but the quotient of Uq(sl2) by therelations E ` = F ` = K ` − 1 = 0.

It is called the small quantumgroup and was introduced by G. Lusztig (for any simple Liealgebra), and is denoted by uq(sl2). Note that by virtue of theconstruction D0 has a universal R-matrix, which can be computedto be

R = qh⊗h2

`−1∑k=0

qk(k−1)

2(q − q−1)k

[k]q!ek ⊗ f k ,

i.e., it is just the truncation of the formula for Uq(sl2) for generalq. In fact, the punchline is that Uq(sl2) for general q can also beconstructed by an infinite dimensional version of the quantumdouble construction, which naturally produces both thecommutation relation between e and f and the R-matrix!

198

Small quantum group

Note that D0 is nothing but the quotient of Uq(sl2) by therelations E ` = F ` = K ` − 1 = 0. It is called the small quantumgroup and was introduced by G. Lusztig (for any simple Liealgebra), and is denoted by uq(sl2).

Note that by virtue of theconstruction D0 has a universal R-matrix, which can be computedto be

R = qh⊗h2

`−1∑k=0

qk(k−1)

2(q − q−1)k

[k]q!ek ⊗ f k ,


199

Small quantum group

Note that D0 is nothing but the quotient of Uq(sl2) by therelations E ` = F ` = K ` − 1 = 0. It is called the small quantumgroup and was introduced by G. Lusztig (for any simple Liealgebra), and is denoted by uq(sl2). Note that by virtue of theconstruction D0 has a universal R-matrix, which can be computedto be

R = qh⊗h2

`−1∑k=0

qk(k−1)

2(q − q−1)k

[k]q!ek ⊗ f k ,


200

Small quantum group


R = qh⊗h2

`−1∑k=0

qk(k−1)

2(q − q−1)k

[k]q!ek ⊗ f k ,


201

Small quantum group


R = qh⊗h2

`−1∑k=0

qk(k−1)

2(q − q−1)k

[k]q!ek ⊗ f k ,

i.e., it is just the truncation of the formula for Uq(sl2) for generalq.

In fact, the punchline is that Uq(sl2) for general q can also beconstructed by an infinite dimensional version of the quantumdouble construction, which naturally produces both thecommutation relation between e and f and the R-matrix!

202

Small quantum group


R = qh⊗h2

`−1∑k=0

qk(k−1)

2(q − q−1)k

[k]q!ek ⊗ f k ,


203

Higher rank quantum groups

Quantum groups attached to any simple Lie algebra can beconstructed similarly.

Given a Cartan matrix (aij) withsymmetrizing numbers di ∈ Z+ (i.e., diaij is symmetric), we startwith the Hopf algebra Uq(b+) (quantum Borel) generated byei ,K

±1i with relations

Kiej = qdiaij ejKi , [Ki ,Kj ] = 0,

with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei . Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi with commutation

relations [ei , fj ] = δijKi−K−1

i

qdi−q−di.

204


Quantum groups attached to any simple Lie algebra can beconstructed similarly. Given a Cartan matrix (aij) withsymmetrizing numbers di ∈ Z+ (i.e., diaij is symmetric), we startwith the Hopf algebra Uq(b+) (quantum Borel) generated byei ,K

±1i with relations




i

qdi−q−di.

205



±1i with relations




i

qdi−q−di.

206



±1i with relations


with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei

and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei . Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi with commutation


i

qdi−q−di.

207



±1i with relations


with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei .

Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi with commutation


i

qdi−q−di.

208



±1i with relations


with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei . Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.

This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi with commutation


i

qdi−q−di.

209



±1i with relations


with coproduct defined by ∆(Ki ) = Ki ⊗ Ki ,∆(ei ) = ei ⊗ Ki + 1⊗ ei and also quantum Serre relations, whichare the most constraining relations you can impose preserving theHopf algebra structure without killing any of the ei . Then thequantum group Uq(g) is the quantum double of Uq(b+)(understood appropriately) modded out by the “redundant copy ofthe maximal torus” created by the quantum double construction.This algebra has additional generators fi also satisfying quantumSerre relations and ∆(fi ) = fi ⊗ 1 + K−1i ⊗ fi

with commutation


i

qdi−q−di.

210



±1i with relations




i

qdi−q−di.

211

Jones polynomial

These commutation relations, as well as the R-matrix (which isnow much more complicated) are produced automatically by thedouble construction. In fact, this works more generally, for anysymmetrizable Kac-Moody algebra.

Example

In conclusion let us point out a connection to the Jonespolynomial.

It is well known that any knot K can be obtained byclosing up a braid b. On the other hand, as we have learned, bacts as an operator on the space V⊗n, where V = V1 is the2-dimensional representation of the quantum sl2.

Theorem

The trace of b · K in V⊗n (called the quantum trace of b) is theJones polynomial of K (up to normalization).

For Vi for i > 1 one gets the colored Jones polynomial, and forother Lie algebras – more complex invariants of knots called theReshetikhin-Turaev invariants.

212

Jones polynomial


Example

In conclusion let us point out a connection to the Jonespolynomial. It is well known that any knot K can be obtained byclosing up a braid b.

On the other hand, as we have learned, bacts as an operator on the space V⊗n, where V = V1 is the2-dimensional representation of the quantum sl2.

Theorem



213

Jones polynomial


Example

In conclusion let us point out a connection to the Jonespolynomial. It is well known that any knot K can be obtained byclosing up a braid b. On the other hand, as we have learned, bacts as an operator on the space V⊗n, where V = V1 is the2-dimensional representation of the quantum sl2.

Theorem



214

Jones polynomial


Example


Theorem



215

Jones polynomial


Example


Theorem


For Vi for i > 1 one gets the colored Jones polynomial, and forother Lie algebras – more complex invariants of knots called theReshetikhin-Turaev invariants. 216

Thank you!

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A brief introduction to quantum groups Pavel Etingof MIT · 2020. 5. 6. · A brief introduction to quantum groups Pavel Etingof MIT May 5, 2020 1. Introduction ... school, Drinfeld,