A brief introduction to quantum groups

Pavel Etingof

MIT

May 5, 2020

1 Since then, it’s grown into a vast subject with deep links to many areas: representation theory the Langlands program low-dimensional topology category theory enumerative geometry quantum computation algebraic combinatorics conformal field theory integrable systems integrable probability The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications.

Introduction

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

2 representation theory the Langlands program low-dimensional topology category theory enumerative geometry quantum computation algebraic combinatorics conformal field theory integrable systems integrable probability The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications.

Introduction

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

3 the Langlands program low-dimensional topology category theory enumerative geometry quantum computation algebraic combinatorics conformal field theory integrable systems integrable probability The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications.

Introduction

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

4 low-dimensional topology category theory enumerative geometry quantum computation algebraic combinatorics conformal field theory integrable systems integrable probability The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications.

Introduction

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

5 category theory enumerative geometry quantum computation algebraic combinatorics conformal field theory integrable systems integrable probability The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications.

Introduction

The theory of quantum groups developed in mid 1980s (Faddeev’s school, Drinfeld, Jimbo) from attempts to construct and understand solutions of the quantum Yang-Baxter equation arising in quantum field theory and statistical mechanics. Since then, it’s grown into a vast subject with deep links to many areas: representation theory the Langlands program low-dimensional topology

6 enumerative geometry quantum computation algebraic combinatorics conformal field theory integrable systems integrable probability The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications.

Introduction

The theory of quantum groups developed in mid 1980s (Faddeev’s school, Drinfeld, Jimbo) from attempts to construct and understand solutions of the quantum Yang-Baxter equation arising in quantum field theory and statistical mechanics. Since then, it’s grown into a vast subject with deep links to many areas: representation theory the Langlands program low-dimensional topology category theory

7 quantum computation algebraic combinatorics conformal field theory integrable systems integrable probability The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications.

Introduction

The theory of quantum groups developed in mid 1980s (Faddeev’s school, Drinfeld, Jimbo) from attempts to construct and understand solutions of the quantum Yang-Baxter equation arising in quantum field theory and statistical mechanics. Since then, it’s grown into a vast subject with deep links to many areas: representation theory the Langlands program low-dimensional topology category theory enumerative geometry

8 algebraic combinatorics conformal field theory integrable systems integrable probability The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications.

Introduction

The theory of quantum groups developed in mid 1980s (Faddeev’s school, Drinfeld, Jimbo) from attempts to construct and understand solutions of the quantum Yang-Baxter equation arising in quantum field theory and statistical mechanics. Since then, it’s grown into a vast subject with deep links to many areas: representation theory the Langlands program low-dimensional topology category theory enumerative geometry quantum computation

9 conformal field theory integrable systems integrable probability The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications.

Introduction

The theory of quantum groups developed in mid 1980s (Faddeev’s school, Drinfeld, Jimbo) from attempts to construct and understand solutions of the quantum Yang-Baxter equation arising in quantum field theory and statistical mechanics. Since then, it’s grown into a vast subject with deep links to many areas: representation theory the Langlands program low-dimensional topology category theory enumerative geometry quantum computation algebraic combinatorics

10 integrable systems integrable probability The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications.

Introduction

The theory of quantum groups developed in mid 1980s (Faddeev’s school, Drinfeld, Jimbo) from attempts to construct and understand solutions of the quantum Yang-Baxter equation arising in quantum field theory and statistical mechanics. Since then, it’s grown into a vast subject with deep links to many areas: representation theory the Langlands program low-dimensional topology category theory enumerative geometry quantum computation algebraic combinatorics conformal field theory

11 integrable probability The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications.

Introduction

The theory of quantum groups developed in mid 1980s (Faddeev’s school, Drinfeld, Jimbo) from attempts to construct and understand solutions of the quantum Yang-Baxter equation arising in quantum field theory and statistical mechanics. Since then, it’s grown into a vast subject with deep links to many areas: representation theory the Langlands program low-dimensional topology category theory enumerative geometry quantum computation algebraic combinatorics conformal field theory integrable systems

12 The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications.

Introduction

The theory of quantum groups developed in mid 1980s (Faddeev’s school, Drinfeld, Jimbo) from attempts to construct and understand solutions of the quantum Yang-Baxter equation arising in quantum field theory and statistical mechanics. Since then, it’s grown into a vast subject with deep links to many areas: representation theory the Langlands program low-dimensional topology category theory enumerative geometry quantum computation algebraic combinatorics conformal field theory integrable systems integrable probability

13 Introduction

The theory of quantum groups developed in mid 1980s (Faddeev’s school, Drinfeld, Jimbo) from attempts to construct and understand solutions of the quantum Yang-Baxter equation arising in quantum field theory and statistical mechanics. Since then, it’s grown into a vast subject with deep links to many areas: representation theory the Langlands program low-dimensional topology category theory enumerative geometry quantum computation algebraic combinatorics conformal field theory integrable systems integrable probability The goal of this talk is to review some of the main ideas and examples of quantum groups and briefly describe some of the applications. 14 X – a space of states; A = O(X ) – the algebra of observables, i.e., (say, complex) functions on X (a commutative, associative algebra). 2 Example: X = R . Then the coordinate x and momentum p are 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., the Heisenberg uncertainty relation[ p, x] = −i~.

Hopf algebras

Classical physics:

15 A = O(X ) – the algebra of observables, i.e., (say, complex) functions on X (a commutative, associative algebra). 2 Example: X = R . Then the coordinate x and momentum p are 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., the Heisenberg uncertainty relation[ p, x] = −i~.

Hopf algebras

Classical physics: X – a space of states;

16 2 Example: X = R . Then the coordinate x and momentum p are 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., the Heisenberg uncertainty relation[ p, x] = −i~.

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).

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

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). 2 Example: X = R . Then the coordinate x and momentum p are examples of observables (elements of A).

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

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). 2 Example: X = R . Then the coordinate x and momentum p are examples of observables (elements of A). Quantum physics:

19 e.g., the Heisenberg uncertainty relation[ p, x] = −i~.

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). 2 Example: X = R . Then the coordinate x and momentum p are examples of observables (elements of A). Quantum physics: A is deformed to a non-commutative (but still associative) algebra A~ with quantization parameter ~,

20 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). 2 Example: X = R . Then the coordinate x and momentum p are 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., the Heisenberg uncertainty relation[ p, x] = −i~.

21 A group is equipped with an associative product, which has a unit 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, decomposing O(G × G) as O(G) ⊗ O(G), one gets comultiplication, or coproduct 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) P i i where summation is implied, i.e., f(1) ⊗ f(2) is really i f(1) ⊗ f(2).

Hopf algebras, ctd.

What if X = G is a group?

22 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, decomposing O(G × G) as O(G) ⊗ O(G), one gets comultiplication, or coproduct 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) P i i where summation is implied, i.e., f(1) ⊗ f(2) is really i f(1) ⊗ f(2).

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:

23 Thus the algebra A = O(G) of functions on a (say, finite) group has a natural structure of a coalgebra. Namely, decomposing O(G × G) as O(G) ⊗ O(G), one gets comultiplication, or coproduct 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) P i i where summation is implied, i.e., f(1) ⊗ f(2) is really i f(1) ⊗ f(2).

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:

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.

24 Namely, decomposing O(G × G) as O(G) ⊗ O(G), one gets comultiplication, or coproduct 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) P i i where summation is implied, i.e., f(1) ⊗ f(2) is really i f(1) ⊗ f(2).

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:

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.

25 ∆ : 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) P i i where summation is implied, i.e., f(1) ⊗ f(2) is really i f(1) ⊗ f(2).

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:

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, decomposing O(G × G) as O(G) ⊗ O(G), one gets comultiplication, or coproduct on A from the multiplication m in G:

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

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:

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, decomposing O(G × G) as O(G) ⊗ O(G), one gets comultiplication, or coproduct on A from the multiplication m in G:

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

27 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:

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, decomposing O(G × G) as O(G) ⊗ O(G), one gets comultiplication, or coproduct 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) P i i where summation is implied, i.e., f(1) ⊗ f(2) is really i f(1) ⊗ f(2). 28 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).

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 and has the properties listed above.

Hopf algebras ctd.

It is clear that • ∆ is an algebra homomorphism.

29 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 and has the properties listed above.

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:

30 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 and has the properties listed above.

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).

31 (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 and has the properties listed above.

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).

The following properties could be easily checked: ∆ is coassociative, i.e., (id ⊗ ∆) ◦ ∆ = (∆ ⊗ id) ◦ ∆;

32 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 and has the properties listed above.

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).

The following properties could be easily checked: ∆ is coassociative, i.e., (id ⊗ ∆) ◦ ∆ = (∆ ⊗ id) ◦ ∆; (ε ⊗ id) ◦ ∆ = (id ⊗ ε) ◦ ∆ = id;

33 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 and has the properties listed above.

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).

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),

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

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).

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.

35 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).

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 and has the properties listed above. 36 Usually it is also assumed that S is invertible (we will do so below); this does not follow from the above axioms. We will also use the notation∆ op = P ◦ ∆ where P is the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we can view 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, with the 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) then S2 = id (even without the assumption that S is invertible).

Hopf algebras ctd.

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

37 We will also use the notation∆ op = P ◦ ∆ where P is the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we can view 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, with the 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) then S2 = id (even without the assumption that S is invertible).

Hopf algebras ctd.

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

38 Finally, note that we can view 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, with the 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) then S2 = id (even without the assumption that S is invertible).

Hopf algebras ctd.

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

39 Proposition 1. If A is a finite dimensional Hopf algebra then A∗ is also, with the 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) then S2 = id (even without the assumption that S is invertible).

Hopf algebras ctd.

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

40 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) then S2 = id (even without the assumption that S is invertible).

Hopf algebras ctd.

Note that this definition makes sense over any field and even over a commutative ring. Usually it is also assumed that S is invertible (we will do so below); this does not follow from the above axioms. We will also use the notation∆ op = P ◦ ∆ where P is the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we can view 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, with the operations of A∗ being dual to the operations of A.

41 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) then S2 = id (even without the assumption that S is invertible).

Hopf algebras ctd.

Note that this definition makes sense over any field and even over a commutative ring. Usually it is also assumed that S is invertible (we will do so below); this does not follow from the above axioms. We will also use the notation∆ op = P ◦ ∆ where P is the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we can view 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, with the operations of A∗ being dual to the operations of A. 2. ε is an algebra homomorphism.

42 4. ε and S are uniquely determined by ∆. 5. If A is commutative or cocommutative (i.e., ∆ = ∆op) then S2 = id (even without the assumption that S is invertible).

Hopf algebras ctd.

Note that this definition makes sense over any field and even over a commutative ring. Usually it is also assumed that S is invertible (we will do so below); this does not follow from the above axioms. We will also use the notation∆ op = P ◦ ∆ where P is the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we can view 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, with the 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)).

43 5. If A is commutative or cocommutative (i.e., ∆ = ∆op) then S2 = id (even without the assumption that S is invertible).

Hopf algebras ctd.

Note that this definition makes sense over any field and even over a commutative ring. Usually it is also assumed that S is invertible (we will do so below); this does not follow from the above axioms. We will also use the notation∆ op = P ◦ ∆ where P is the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we can view 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, with the 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 ∆.

44 Hopf algebras ctd.

Note that this definition makes sense over any field and even over a commutative ring. Usually it is also assumed that S is invertible (we will do so below); this does not follow from the above axioms. We will also use the notation∆ op = P ◦ ∆ where P is the permutation: P(u ⊗ v) = v ⊗ u. Finally, note that we can view 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, with the 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) then S2 = id (even without the assumption that S is invertible).

45 2 A = O(G) (the algebra of regular functions), G is an affine algebraic 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 is cocommutative.

Examples of Hopf algebras

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

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 affine algebraic 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 is cocommutative.

47 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. h This is a deformation of U(sl2) because if one sets K = q and sends q → 1, one recovers the sl2 relations. This example shows that S2 =6 id in general: we have S2(x) = KxK −1.

Quantum SL2

Let q ∈ C, q =6 0, ±1. The quantum group Uq(sl2) is generated by e, f , K ±1 with relations K − K −1 KeK −1 = q2e, KfK −1 = q−2f , ef − fe = . q − q−1

48 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. h This is a deformation of U(sl2) because if one sets K = q and sends q → 1, one recovers the sl2 relations. This example shows that S2 =6 id in general: we have S2(x) = KxK −1.

Quantum SL2

Let q ∈ C, q =6 0, ±1. The quantum group Uq(sl2) is generated by e, f , K ±1 with relations K − K −1 KeK −1 = q2e, KfK −1 = q−2f , ef − fe = . q − q−1 The coproduct and counit are defined by ∆e = e ⊗ K + 1 ⊗ e, ∆f = f ⊗ 1 + K −1 ⊗ f , ∆K = K ⊗ K,

49 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. h This is a deformation of U(sl2) because if one sets K = q and sends q → 1, one recovers the sl2 relations. This example shows that S2 =6 id in general: we have S2(x) = KxK −1.

Quantum SL2

Let q ∈ C, q =6 0, ±1. The quantum group Uq(sl2) is generated by e, f , K ±1 with relations K − K −1 KeK −1 = q2e, KfK −1 = q−2f , ef − fe = . 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.

50 Similarly one obtains S(f ) = −Kf , S(K) = K −1. h This is a deformation of U(sl2) because if one sets K = q and sends q → 1, one recovers the sl2 relations. This example shows that S2 =6 id in general: we have S2(x) = KxK −1.

Quantum SL2

Let q ∈ C, q =6 0, ±1. The quantum group Uq(sl2) is generated by e, f , K ±1 with relations K − K −1 KeK −1 = q2e, KfK −1 = q−2f , ef − fe = . 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.

51 h This is a deformation of U(sl2) because if one sets K = q and sends q → 1, one recovers the sl2 relations. This example shows that S2 =6 id in general: we have S2(x) = KxK −1.

Quantum SL2

Let q ∈ C, q =6 0, ±1. The quantum group Uq(sl2) is generated by e, f , K ±1 with relations K − K −1 KeK −1 = q2e, KfK −1 = q−2f , ef − fe = . 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.

52 Quantum SL2

Let q ∈ C, q =6 0, ±1. The quantum group Uq(sl2) is generated by e, f , K ±1 with relations K − K −1 KeK −1 = q2e, KfK −1 = q−2f , ef − fe = . 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. h This is a deformation of U(sl2) because if one sets K = q and sends q → 1, one recovers the sl2 relations. This example shows that S2 =6 id in general: we have S2(x) = KxK −1. 53 Then the representation theory 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 integer powers of q. E.g., the character χ : Uq(sl2) → C given by χ(e) = χ(f ) = 0, χ(K) = −1 is not of type I. However, if V is not of 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 j v, 0 ≤ j ≤ n} such that Kf j v = qn−2j f j v, j j−1 qk −q−k ef v = [j]q[n − j + 1]qf v, where [k]q := q−q−1 .

Representations of Uq(sl2) Assume that q is not a root of unity.

54 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 integer powers of q. E.g., the character χ : Uq(sl2) → C given by χ(e) = χ(f ) = 0, χ(K) = −1 is not of type I. However, if V is not of 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 j v, 0 ≤ j ≤ n} such that Kf j v = qn−2j f j v, j j−1 qk −q−k ef v = [j]q[n − j + 1]qf v, where [k]q := q−q−1 .

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

55 We say that V is of type I if the eigenvalues of K on V are integer powers of q. E.g., the character χ : Uq(sl2) → C given by χ(e) = χ(f ) = 0, χ(K) = −1 is not of type I. However, if V is not of 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 j v, 0 ≤ j ≤ n} such that Kf j v = qn−2j f j v, j j−1 qk −q−k ef v = [j]q[n − j + 1]qf v, where [k]q := q−q−1 .

Representations of Uq(sl2) Assume that q is not a root of unity. Then the representation theory 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 .

56 E.g., the character χ : Uq(sl2) → C given by χ(e) = χ(f ) = 0, χ(K) = −1 is not of type I. However, if V is not of 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 j v, 0 ≤ j ≤ n} such that Kf j v = qn−2j f j v, j j−1 qk −q−k ef v = [j]q[n − j + 1]qf v, where [k]q := q−q−1 .

Representations of Uq(sl2) Assume that q is not a root of unity. Then the representation theory 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 integer powers of q.

57 However, if V is not of 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 j v, 0 ≤ j ≤ n} such that Kf j v = qn−2j f j v, j j−1 qk −q−k ef v = [j]q[n − j + 1]qf v, where [k]q := q−q−1 .

Representations of Uq(sl2) Assume that q is not a root of unity. Then the representation theory 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 integer powers of q. E.g., the character χ : Uq(sl2) → C given by χ(e) = χ(f ) = 0, χ(K) = −1 is not of type I.

58 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 j v, 0 ≤ j ≤ n} such that Kf j v = qn−2j f j v, j j−1 qk −q−k ef v = [j]q[n − j + 1]qf v, where [k]q := q−q−1 .

Representations of Uq(sl2) Assume that q is not a root of unity. Then the representation theory 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 integer powers of q. E.g., the character χ : Uq(sl2) → C given by χ(e) = χ(f ) = 0, χ(K) = −1 is not of type I. However, if V is not of type I then it has the form V = V+ ⊗ χ where V+ is of type I.

59 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 j v, 0 ≤ j ≤ n} such that Kf j v = qn−2j f j v, j j−1 qk −q−k ef v = [j]q[n − j + 1]qf v, where [k]q := q−q−1 .

Representations of Uq(sl2) Assume that q is not a root of unity. Then the representation theory 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 integer powers of q. E.g., the character χ : Uq(sl2) → C given by χ(e) = χ(f ) = 0, χ(K) = −1 is not of type I. However, if V is not of 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.

60 with generator v with ev = 0, Kv = qnv and basis {f j v, 0 ≤ j ≤ n} such that Kf j v = qn−2j f j v, j j−1 qk −q−k ef v = [j]q[n − j + 1]qf v, where [k]q := q−q−1 .

Representations of Uq(sl2) Assume that q is not a root of unity. Then the representation theory 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 integer powers of q. E.g., the character χ : Uq(sl2) → C given by χ(e) = χ(f ) = 0, χ(K) = −1 is not of type I. However, if V is not of 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,

61 such that Kf j v = qn−2j f j v, j j−1 qk −q−k ef v = [j]q[n − j + 1]qf v, where [k]q := q−q−1 .

Representations of Uq(sl2) Assume that q is not a root of unity. Then the representation theory 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 integer powers of q. E.g., the character χ : Uq(sl2) → C given by χ(e) = χ(f ) = 0, χ(K) = −1 is not of type I. However, if V is not of 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 j v, 0 ≤ j ≤ n}

62 Representations of Uq(sl2) Assume that q is not a root of unity. Then the representation theory 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 integer powers of q. E.g., the character χ : Uq(sl2) → C given by χ(e) = χ(f ) = 0, χ(K) = −1 is not of type I. However, if V is not of 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 j v, 0 ≤ j ≤ n} such that Kf j v = qn−2j f j v, j j−1 qk −q−k ef v = [j]q[n − j + 1]qf v, where [k]q := q−q−1 . 63 πV : G → Aut V , πW : G → Aut W ⇒

πV ⊗W (g) = πV (g) ⊗ πW (g) ∀g ∈ G.

πV : g → Aut V , πW : g → Aut W ⇒

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

The formula for the tensor product of representations of a Hopf algebra 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 = Rep A of representations of A as a category equipped with a tensor product bifunctor ⊗ : C × C → C :(X , Y ) 7→ X ⊗ Y .

Tensor products of representations of Hopf algebras

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

64 πV : g → Aut V , πW : g → Aut W ⇒

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

The formula for the tensor product of representations of a Hopf algebra 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 = Rep A of representations of A as a category equipped with a tensor product bifunctor ⊗ : C × C → C :(X , Y ) 7→ X ⊗ Y .

Tensor products of representations of Hopf algebras

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

πV : G → Aut V , πW : G → Aut W ⇒

πV ⊗W (g) = πV (g) ⊗ πW (g) ∀g ∈ G.

65 The formula for the tensor product of representations of a Hopf algebra 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 = Rep A of representations of A as a category equipped with a tensor product bifunctor ⊗ : C × C → C :(X , Y ) 7→ X ⊗ Y .

Tensor products of representations of Hopf algebras

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

πV : G → Aut V , πW : G → Aut W ⇒

πV ⊗W (g) = πV (g) ⊗ πW (g) ∀g ∈ G.

πV : g → Aut V , πW : g → Aut W ⇒

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

66 πV ⊗W (x) = (πV ⊗ πW )(∆(x)) = πV (x(1)) ⊗ πW (x(2)) ∀x ∈ A. So one can regard the category C = Rep A of representations of A as a category equipped with a tensor product bifunctor ⊗ : C × C → C :(X , Y ) 7→ X ⊗ Y .

Tensor products of representations of Hopf algebras

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

πV : G → Aut V , πW : G → Aut W ⇒

πV ⊗W (g) = πV (g) ⊗ πW (g) ∀g ∈ G.

πV : g → Aut V , πW : g → Aut W ⇒

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

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

67 So one can regard the category C = Rep A of representations of A as a category equipped with a tensor product bifunctor ⊗ : C × C → C :(X , Y ) 7→ X ⊗ Y .

Tensor products of representations of Hopf algebras

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

πV : G → Aut V , πW : G → Aut W ⇒

πV ⊗W (g) = πV (g) ⊗ πW (g) ∀g ∈ G.

πV : g → Aut V , πW : g → Aut W ⇒

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

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

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

68 Tensor products of representations of Hopf algebras

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

πV : G → Aut V , πW : G → Aut W ⇒

πV ⊗W (g) = πV (g) ⊗ πW (g) ∀g ∈ G.

πV : g → Aut V , πW : g → Aut W ⇒

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

The formula for the tensor product of representations of a Hopf algebra 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 = Rep A of representations of A as a category equipped with a tensor product bifunctor ⊗ : C × C → C :(X , Y ) 7→ X ⊗ Y .

69 ∼ ∼ 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 to an isomorphism. However, this notion is not very useful; about such categories one can say very little, if anything at all. On the other hand, in natural examples a lot more structure is present, which is just a little bit less obvious. More precisely, a much better notion is obtained if, according to the general yoga of category theory, we don’t just say simply that (X ⊗ Y ) ⊗ Z =∼ X ⊗ (Y ⊗ Z), but make this isomorphism a part of the data and impose coherence conditions on this data.This leads to the notion of a monoidal category.

Tensor products of representations of Hopf algebras, ctd.

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

70 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 to an isomorphism. However, this notion is not very useful; about such categories one can say very little, if anything at all. On the other hand, in natural examples a lot more structure is present, which is just a little bit less obvious. More precisely, a much better notion is obtained if, according to the general yoga of category theory, we don’t just say simply that (X ⊗ Y ) ⊗ Z =∼ X ⊗ (Y ⊗ Z), but make this isomorphism a part of the data and impose coherence conditions on this data.This leads to the notion of a monoidal category.

Tensor products of representations of Hopf algebras, ctd.

This product also has a unit (the trivial 1-dimensional representation): ∼ ∼ 1 = C : π1(a) = ε(a), 1 ⊗ X = X ⊗ 1 = X ∀ X ∈ C.

71 (X ⊗ Y ) ⊗ Z =∼ X ⊗ (Y ⊗ Z). Thus C is a category with a unital tensor product associative up to an isomorphism. However, this notion is not very useful; about such categories one can say very little, if anything at all. On the other hand, in natural examples a lot more structure is present, which is just a little bit less obvious. More precisely, a much better notion is obtained if, according to the general yoga of category theory, we don’t just say simply that (X ⊗ Y ) ⊗ Z =∼ X ⊗ (Y ⊗ Z), but make this isomorphism a part of the data and impose coherence conditions on this data.This leads to the notion of a monoidal category.

Tensor products of representations of Hopf algebras, ctd.

This product also has a unit (the trivial 1-dimensional representation): ∼ ∼ 1 = C : π1(a) = ε(a), 1 ⊗ X = X ⊗ 1 = X ∀ X ∈ C. Finally, the tensor product is associative on isomorphism classes:

72 Thus C is a category with a unital tensor product associative up to an isomorphism. However, this notion is not very useful; about such categories one can say very little, if anything at all. On the other hand, in natural examples a lot more structure is present, which is just a little bit less obvious. More precisely, a much better notion is obtained if, according to the general yoga of category theory, we don’t just say simply that (X ⊗ Y ) ⊗ Z =∼ X ⊗ (Y ⊗ Z), but make this isomorphism a part of the data and impose coherence conditions on this data.This leads to the notion of a monoidal category.

Tensor products of representations of Hopf algebras, ctd.

This product also has a unit (the trivial 1-dimensional representation): ∼ ∼ 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).

73 However, this notion is not very useful; about such categories one can say very little, if anything at all. On the other hand, in natural examples a lot more structure is present, which is just a little bit less obvious. More precisely, a much better notion is obtained if, according to the general yoga of category theory, we don’t just say simply that (X ⊗ Y ) ⊗ Z =∼ X ⊗ (Y ⊗ Z), but make this isomorphism a part of the data and impose coherence conditions on this data.This leads to the notion of a monoidal category.

Tensor products of representations of Hopf algebras, ctd.

This product also has a unit (the trivial 1-dimensional representation): ∼ ∼ 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 to an isomorphism.

74 On the other hand, in natural examples a lot more structure is present, which is just a little bit less obvious. More precisely, a much better notion is obtained if, according to the general yoga of category theory, we don’t just say simply that (X ⊗ Y ) ⊗ Z =∼ X ⊗ (Y ⊗ Z), but make this isomorphism a part of the data and impose coherence conditions on this data.This leads to the notion of a monoidal category.

Tensor products of representations of Hopf algebras, ctd.

This product also has a unit (the trivial 1-dimensional representation): ∼ ∼ 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 to an isomorphism. However, this notion is not very useful; about such categories one can say very little, if anything at all.

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

Tensor products of representations of Hopf algebras, ctd.

This product also has a unit (the trivial 1-dimensional representation): ∼ ∼ 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 to an isomorphism. However, this notion is not very useful; about such categories one can say very little, if anything at all. On the other hand, in natural examples a lot more structure is present, which is just a little bit less obvious.

76 This leads to the notion of a monoidal category.

Tensor products of representations of Hopf algebras, ctd.

This product also has a unit (the trivial 1-dimensional representation): ∼ ∼ 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 to an isomorphism. However, this notion is not very useful; about such categories one can say very little, if anything at all. On the other hand, in natural examples a lot more structure is present, which is just a little bit less obvious. More precisely, a much better notion is obtained if, according to the general yoga of category theory, we don’t just say simply that (X ⊗ Y ) ⊗ Z =∼ X ⊗ (Y ⊗ Z), but make this isomorphism a part of the data and impose coherence conditions on this data.

77 Tensor products of representations of Hopf algebras, ctd.

This product also has a unit (the trivial 1-dimensional representation): ∼ ∼ 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 to an isomorphism. However, this notion is not very useful; about such categories one can say very little, if anything at all. On the other hand, in natural examples a lot more structure is present, which is just a little bit less obvious. More precisely, a much better notion is obtained if, according to the general yoga of category theory, we don’t just say simply that (X ⊗ Y ) ⊗ Z =∼ X ⊗ (Y ⊗ Z), but make this isomorphism a part of the data and impose coherence conditions on this data.This leads to the notion of a monoidal category. 78 ∼ αXYZ :(X ⊗ Y ) ⊗ Z −→ X ⊗ (Y ⊗ Z) functorial in X , Y , Z, which satisfies the pentagon identity X (Y (ZT ))

(XY )(ZT ) X ((YZ)T )

((XY )Z)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.

Monoidal categories

Namely, we should equip C with an associativity isomorphism

79 functorial in X , Y , Z, which satisfies the pentagon identity X (Y (ZT ))

(XY )(ZT ) X ((YZ)T )

((XY )Z)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.

Monoidal categories

Namely, we should equip C with an associativity isomorphism ∼ αXYZ :(X ⊗ Y ) ⊗ Z −→ X ⊗ (Y ⊗ Z)

80 which satisfies the pentagon identity X (Y (ZT ))

(XY )(ZT ) X ((YZ)T )

((XY )Z)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.

Monoidal categories

Namely, we should equip C with an associativity isomorphism ∼ αXYZ :(X ⊗ Y ) ⊗ Z −→ X ⊗ (Y ⊗ Z) functorial in X , Y , Z,

81 X (Y (ZT ))

(XY )(ZT ) X ((YZ)T )

((XY )Z)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.

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

82 The relation is that the diagram commutes.

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 ) X ((YZ)T )

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

where the arrows are induced by α and we have omitted the ⊗ signs for brevity. 83 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 ) X ((YZ)T )

((XY )Z)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. 84 with an isomorphism 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 a monoidal category.

We see that for a Hopf algebra A, the category RepA is a monoidal 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 linear map ρ : V → A ⊗ V defining an action of the algebra A∗ on V ) form a monoidal category ComodA. In fact, if dim A < ∞ then ComodA = Rep A∗. Also, if G is an algebraic group then an algebraic representation of G is the same thing as a finite dimensional O(G)-comodule.

Monoidal categories, ctd.

We should also require the existence of a unit object 1

85 such that the functors 1⊗ and ⊗1 are autoequivalences of C. Definition A category C with such structures and properties is called a monoidal category.

We see that for a Hopf algebra A, the category RepA is a monoidal 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 linear map ρ : V → A ⊗ V defining an action of the algebra A∗ on V ) form a monoidal category ComodA. In fact, if dim A < ∞ then ComodA = Rep A∗. Also, if G is an algebraic group then an algebraic representation of G is the same thing as a finite dimensional O(G)-comodule.

Monoidal categories, ctd.

We should also require the existence of a unit object 1 with an isomorphism ι : 1 ⊗ 1 =∼ 1

86 Definition A category C with such structures and properties is called a monoidal category.

We see that for a Hopf algebra A, the category RepA is a monoidal 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 linear map ρ : V → A ⊗ V defining an action of the algebra A∗ on V ) form a monoidal category ComodA. In fact, if dim A < ∞ then ComodA = Rep A∗. Also, if G is an algebraic group then an algebraic representation of G is the same thing as a finite dimensional O(G)-comodule.

Monoidal categories, ctd.

We should also require the existence of a unit object 1 with an isomorphism ι : 1 ⊗ 1 =∼ 1 such that the functors 1⊗ and ⊗1 are autoequivalences of C.

87 We see that for a Hopf algebra A, the category RepA is a monoidal 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 linear map ρ : V → A ⊗ V defining an action of the algebra A∗ on V ) form a monoidal category ComodA. In fact, if dim A < ∞ then ComodA = Rep A∗. Also, if G is an algebraic group then an algebraic representation of G is the same thing as a finite dimensional O(G)-comodule.

Monoidal categories, ctd.

We should also require the existence of a unit object 1 with an isomorphism 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 a monoidal category.

88 In a similar way, comodules over A (i.e., spaces V with a linear map ρ : V → A ⊗ V defining an action of the algebra A∗ on V ) form a monoidal category ComodA. In fact, if dim A < ∞ then ComodA = Rep A∗. Also, if G is an algebraic group then an algebraic representation of G is the same thing as a finite dimensional O(G)-comodule.

Monoidal categories, ctd.

We should also require the existence of a unit object 1 with an isomorphism 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 a monoidal category.

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

89 In fact, if dim A < ∞ then ComodA = Rep A∗. Also, if G is an algebraic group then an algebraic representation of G is the same thing as a finite dimensional O(G)-comodule.

Monoidal categories, ctd.

We should also require the existence of a unit object 1 with an isomorphism 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 a monoidal category.

We see that for a Hopf algebra A, the category RepA is a monoidal 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 linear map ρ : V → A ⊗ V defining an action of the algebra A∗ on V ) form a monoidal category ComodA.

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

Monoidal categories, ctd.

We should also require the existence of a unit object 1 with an isomorphism 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 a monoidal category.

We see that for a Hopf algebra A, the category RepA is a monoidal 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 linear map ρ : V → A ⊗ V defining an action of the algebra A∗ on V ) form a monoidal category ComodA. In fact, if dim A < ∞ then ComodA = Rep A∗.

91 Monoidal categories, ctd.

We should also require the existence of a unit object 1 with an isomorphism 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 a monoidal category.

We see that for a Hopf algebra A, the category RepA is a monoidal 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 linear map ρ : V → A ⊗ V defining an action of the algebra A∗ on V ) form a monoidal category ComodA. In fact, if dim A < ∞ then ComodA = Rep A∗. Also, if G is an algebraic group then an algebraic representation of G is the same thing as a finite dimensional O(G)-comodule. 92 The dual representations X ∗, ∗X ∈ Rep A for X ∈ Rep A are both the dual vector space to X with the actions defined as follows:

∗ −1 ∗ πX ∗ (a) = πX (S(a)) - left dual,π ∗X (a) = πX (S (a)) - right dual. For the pair X , X ∗ there is the evaluation morphism X ∗ ⊗ X → 1 (the usual pairing). For finite dimensional representations there is also the coevaluation morphism 1 → X ⊗ X ∗. and a pair of functorial isomorphisms (∗X )∗ = X , ∗(X ∗) = X .

Duality in monoidal categories

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

93 ∗ −1 ∗ πX ∗ (a) = πX (S(a)) - left dual,π ∗X (a) = πX (S (a)) - right dual. For the pair X , X ∗ there is the evaluation morphism X ∗ ⊗ X → 1 (the usual pairing). For finite dimensional representations there is also the coevaluation morphism 1 → X ⊗ X ∗. and a pair of functorial isomorphisms (∗X )∗ = X , ∗(X ∗) = X .

Duality in monoidal categories

Let us now discuss duality for representations of Hopf algebras, which generalizes duality for group and Lie algebra representations.The dual representations X ∗, ∗X ∈ Rep A for X ∈ Rep A are both the dual vector space to X with the actions defined as follows:

94 For the pair X , X ∗ there is the evaluation morphism X ∗ ⊗ X → 1 (the usual pairing). For finite dimensional representations there is also the coevaluation morphism 1 → X ⊗ X ∗. and a pair of functorial isomorphisms (∗X )∗ = X , ∗(X ∗) = X .

Duality in monoidal categories

Let us now discuss duality for representations of Hopf algebras, which generalizes duality for group and Lie algebra representations.The dual representations X ∗, ∗X ∈ Rep A for X ∈ Rep A are both the dual vector space to X with the actions defined as follows:

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

95 For finite dimensional representations there is also the coevaluation morphism 1 → X ⊗ X ∗. and a pair of functorial isomorphisms (∗X )∗ = X , ∗(X ∗) = X .

Duality in monoidal categories

Let us now discuss duality for representations of Hopf algebras, which generalizes duality for group and Lie algebra representations.The dual representations X ∗, ∗X ∈ Rep A for X ∈ Rep A are both the dual vector space to X with the actions defined as follows:

∗ −1 ∗ πX ∗ (a) = πX (S(a)) - left dual,π ∗X (a) = πX (S (a)) - right dual. For the pair X , X ∗ there is the evaluation morphism X ∗ ⊗ X → 1 (the usual pairing).

96 1 → X ⊗ X ∗. and a pair of functorial isomorphisms (∗X )∗ = X , ∗(X ∗) = X .

Duality in monoidal categories

Let us now discuss duality for representations of Hopf algebras, which generalizes duality for group and Lie algebra representations.The dual representations X ∗, ∗X ∈ Rep A for X ∈ Rep A are both the dual vector space to X with the actions defined as follows:

∗ −1 ∗ πX ∗ (a) = πX (S(a)) - left dual,π ∗X (a) = πX (S (a)) - right dual. For the pair X , X ∗ there is the evaluation morphism X ∗ ⊗ X → 1 (the usual pairing). For finite dimensional representations there is also the coevaluation morphism

97 and a pair of functorial isomorphisms (∗X )∗ = X , ∗(X ∗) = X .

Duality in monoidal categories

Let us now discuss duality for representations of Hopf algebras, which generalizes duality for group and Lie algebra representations.The dual representations X ∗, ∗X ∈ Rep A for X ∈ Rep A are both the dual vector space to X with the actions defined as follows:

∗ −1 ∗ πX ∗ (a) = πX (S(a)) - left dual,π ∗X (a) = πX (S (a)) - right dual. For the pair X , X ∗ there is the evaluation morphism X ∗ ⊗ X → 1 (the usual pairing). For finite dimensional representations there is also the coevaluation morphism 1 → X ⊗ X ∗.

98 (∗X )∗ = X , ∗(X ∗) = X .

Duality in monoidal categories

Let us now discuss duality for representations of Hopf algebras, which generalizes duality for group and Lie algebra representations.The dual representations X ∗, ∗X ∈ Rep A for X ∈ Rep A are both the dual vector space to X with the actions defined as follows:

∗ −1 ∗ πX ∗ (a) = πX (S(a)) - left dual,π ∗X (a) = πX (S (a)) - right dual. For the pair X , X ∗ there is the evaluation morphism X ∗ ⊗ X → 1 (the usual pairing). For finite dimensional representations there is also the coevaluation morphism 1 → X ⊗ X ∗. and a pair of functorial isomorphisms

99 Duality in monoidal categories

Let us now discuss duality for representations of Hopf algebras, which generalizes duality for group and Lie algebra representations.The dual representations X ∗, ∗X ∈ Rep A for X ∈ Rep A are both the dual vector space to X with the actions defined as follows:

∗ −1 ∗ πX ∗ (a) = πX (S(a)) - left dual,π ∗X (a) = πX (S (a)) - right dual. For the pair X , X ∗ there is the evaluation morphism X ∗ ⊗ X → 1 (the usual pairing). For finite dimensional representations there is also the coevaluation morphism 1 → X ⊗ X ∗. and a pair of functorial isomorphisms (∗X )∗ = X , ∗(X ∗) = X .

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

ev : Y ⊗ X → 1, coev : 1 → X ⊗ Y ,

such that the following morphisms are the identities:

coev⊗1 α 1⊗ev X −−−−→ (X ⊗ Y ) ⊗ X −−−−→XYX X ⊗ (Y ⊗ X ) −−−−→ X , −1 1⊗coev α ev⊗1 Y −−−−→ Y ⊗ (X ⊗ Y ) −−−−→YXY (Y ⊗ X ) ⊗ Y −−−−→ Y .

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

Duality in monoidal categories, ctd.

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

101 ev : Y ⊗ X → 1, coev : 1 → X ⊗ Y ,

such that the following morphisms are the identities:

coev⊗1 α 1⊗ev X −−−−→ (X ⊗ Y ) ⊗ X −−−−→XYX X ⊗ (Y ⊗ X ) −−−−→ X , −1 1⊗coev α ev⊗1 Y −−−−→ Y ⊗ (X ⊗ Y ) −−−−→YXY (Y ⊗ X ) ⊗ Y −−−−→ Y .

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

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 , denoted Y = X ∗, if there exist evaluation and coevaluation morphisms

102 such that the following morphisms are the identities:

coev⊗1 α 1⊗ev X −−−−→ (X ⊗ Y ) ⊗ X −−−−→XYX X ⊗ (Y ⊗ X ) −−−−→ X , −1 1⊗coev α ev⊗1 Y −−−−→ Y ⊗ (X ⊗ Y ) −−−−→YXY (Y ⊗ X ) ⊗ Y −−−−→ Y .

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

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 , denoted Y = X ∗, if there exist evaluation and coevaluation morphisms

ev : Y ⊗ X → 1, coev : 1 → X ⊗ Y ,

103 coev⊗1 α 1⊗ev X −−−−→ (X ⊗ Y ) ⊗ X −−−−→XYX X ⊗ (Y ⊗ X ) −−−−→ X , −1 1⊗coev α ev⊗1 Y −−−−→ Y ⊗ (X ⊗ Y ) −−−−→YXY (Y ⊗ X ) ⊗ Y −−−−→ Y .

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

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 , denoted Y = X ∗, if there exist evaluation and coevaluation morphisms

ev : Y ⊗ X → 1, coev : 1 → X ⊗ Y ,

such that the following morphisms are the identities:

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

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 , denoted Y = X ∗, if there exist evaluation and coevaluation morphisms

ev : Y ⊗ X → 1, coev : 1 → X ⊗ Y ,

such that the following morphisms are the identities:

coev⊗1 α 1⊗ev X −−−−→ (X ⊗ Y ) ⊗ X −−−−→XYX X ⊗ (Y ⊗ X ) −−−−→ X , −1 1⊗coev α ev⊗1 Y −−−−→ Y ⊗ (X ⊗ Y ) −−−−→YXY (Y ⊗ X ) ⊗ Y −−−−→ Y .

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

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 , denoted Y = X ∗, if there exist evaluation and coevaluation morphisms

ev : Y ⊗ X → 1, coev : 1 → X ⊗ Y ,

such that the following morphisms are the identities:

coev⊗1 α 1⊗ev X −−−−→ (X ⊗ Y ) ⊗ X −−−−→XYX X ⊗ (Y ⊗ X ) −−−−→ X , −1 1⊗coev α ev⊗1 Y −−−−→ Y ⊗ (X ⊗ Y ) −−−−→YXY (Y ⊗ X ) ⊗ Y −−−−→ Y .

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

106 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 , denoted Y = X ∗, if there exist evaluation and coevaluation morphisms

ev : Y ⊗ X → 1, coev : 1 → X ⊗ Y ,

such that the following morphisms are the identities:

coev⊗1 α 1⊗ev X −−−−→ (X ⊗ Y ) ⊗ X −−−−→XYX X ⊗ (Y ⊗ X ) −−−−→ X , −1 1⊗coev α ev⊗1 Y −−−−→ Y ⊗ (X ⊗ Y ) −−−−→YXY (Y ⊗ X ) ⊗ Y −−−−→ Y .

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

107 As the left dual, the right dual is unique up to a unique isomorphism if exists. Definition An object X is rigid if it has both the left and the right dual. A category C is rigid if all its objects are rigid.

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

Duality in monoidal categories, ctd.

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

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

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

Duality in monoidal categories, ctd.

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 unique isomorphism if exists.

109 A category C is rigid if all its objects are rigid.

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

Duality in monoidal categories, ctd.

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 unique isomorphism if exists. Definition An object X is rigid if it has both the left and the right dual.

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

Duality in monoidal categories, ctd.

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 unique isomorphism if exists. Definition An object X is rigid if it has both the left and the right dual. A category C is rigid if all its objects are rigid.

111 Duality in monoidal categories, ctd.

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 unique isomorphism if exists. Definition An object X is rigid if it has both the left and the right dual. A category C is rigid if all its objects are rigid.

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

112 The tensor product is defined by g ⊗ h = gh, and g ∗ = ∗g = g −1. Thus Repf A is a rigid monoidal category. We denote it by Vec(G) (G-graded vector spaces). This category makes sense for any group G (not necessarily finite).

Example The previous example has the following twisted version. Let ∗ αg,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 the group G. If so then this equips C = Vec(G) with another structure of a rigid monoidal category (with the same tensor product functor but different associativity isomorphism).We will denote this category Vec(G, α).

Examples of rigid monoidal categories

Example

Let G be a finite group, A = O(G), then Repf A is spanned by 1-dimensional representations parametrized by g ∈ G.

113 Thus Repf A is a rigid monoidal category. We denote it by Vec(G) (G-graded vector spaces). This category makes sense for any group G (not necessarily finite).

Example The previous example has the following twisted version. Let ∗ αg,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 the group G. If so then this equips C = Vec(G) with another structure of a rigid monoidal category (with the same tensor product functor but different associativity isomorphism).We will denote this category Vec(G, α).

Examples of rigid monoidal categories

Example

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

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

Example The previous example has the following twisted version. Let ∗ αg,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 the group G. If so then this equips C = Vec(G) with another structure of a rigid monoidal category (with the same tensor product functor but different associativity isomorphism).We will denote this category Vec(G, α).

Examples of rigid monoidal categories

Example

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

115 This category makes sense for any group G (not necessarily finite).

Example The previous example has the following twisted version. Let ∗ αg,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 the group G. If so then this equips C = Vec(G) with another structure of a rigid monoidal category (with the same tensor product functor but different associativity isomorphism).We will denote this category Vec(G, α).

Examples of rigid monoidal categories

Example

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

116 Example The previous example has the following twisted version. Let ∗ αg,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 the group G. If so then this equips C = Vec(G) with another structure of a rigid monoidal category (with the same tensor product functor but different associativity isomorphism).We will denote this category Vec(G, α).

Examples of rigid monoidal categories

Example

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

117 Let ∗ αg,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 the group G. If so then this equips C = Vec(G) with another structure of a rigid monoidal category (with the same tensor product functor but different associativity isomorphism).We will denote this category Vec(G, α).

Examples of rigid monoidal categories

Example

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

Example The previous example has the following twisted version.

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

Examples of rigid monoidal categories

Example

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

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

119 If so then this equips C = Vec(G) with another structure of a rigid monoidal category (with the same tensor product functor but different associativity isomorphism).We will denote this category Vec(G, α).

Examples of rigid monoidal categories

Example

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

Example The previous example has the following twisted version. Let ∗ αg,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 the group G.

120 We will denote this category Vec(G, α).

Examples of rigid monoidal categories

Example

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

Example The previous example has the following twisted version. Let ∗ αg,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 the group G. If so then this equips C = Vec(G) with another structure of a rigid monoidal category (with the same tensor product functor but different associativity isomorphism).

121 Examples of rigid monoidal categories

Example

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

Example The previous example has the following twisted version. Let ∗ αg,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 the group G. If so then this equips C = Vec(G) with another structure of a rigid monoidal category (with the same tensor product functor but different associativity isomorphism).We will denote this category Vec(G, α). 122 Definition ∼ A functor F : C → D is a monoidal functor if F (1C) = 1D and F is equipped with a functorial (in X , Y ) isomorphism ∼ JX ,Y : F (X ) ⊗ F (Y ) −→ F (X ⊗ Y ) which makes the diagram

α (F (X ) ⊗ F (Y )) ⊗ F (Z) −−−−→D F (X ) ⊗ (F (Y ) ⊗ F (Z))   J ⊗id id ⊗J y X ,Y Z y X Y ,Z F (X ⊗ Y ) ⊗ F (Z) F (X ) ⊗ F (Y ⊗ Z)   J J y X ⊗Y ,Z y X ,Y ⊗Z F (α ) F ((X ⊗ Y ) ⊗ Z) −−−−→C F (X ⊗ (Y ⊗ Z))

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

Monoidal functors

Let C, D be monoidal categories.

123 which makes the diagram

α (F (X ) ⊗ F (Y )) ⊗ F (Z) −−−−→D F (X ) ⊗ (F (Y ) ⊗ F (Z))   J ⊗id id ⊗J y X ,Y Z y X Y ,Z F (X ⊗ Y ) ⊗ F (Z) F (X ) ⊗ F (Y ⊗ Z)   J J y X ⊗Y ,Z y X ,Y ⊗Z F (α ) F ((X ⊗ Y ) ⊗ Z) −−−−→C F (X ⊗ (Y ⊗ Z))

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

Monoidal functors

Let C, D be monoidal categories. Definition ∼ A functor F : C → D is a monoidal functor if F (1C) = 1D and F is equipped with a functorial (in X , Y ) isomorphism ∼ JX ,Y : F (X ) ⊗ F (Y ) −→ F (X ⊗ Y )

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

Monoidal functors

Let C, D be monoidal categories. Definition ∼ A functor F : C → D is a monoidal functor if F (1C) = 1D and F is equipped with a functorial (in X , Y ) isomorphism ∼ JX ,Y : F (X ) ⊗ F (Y ) −→ F (X ⊗ Y ) which makes the diagram

α (F (X ) ⊗ F (Y )) ⊗ F (Z) −−−−→D F (X ) ⊗ (F (Y ) ⊗ F (Z))   J ⊗id id ⊗J y X ,Y Z y X Y ,Z F (X ⊗ Y ) ⊗ F (Z) F (X ) ⊗ F (Y ⊗ Z)   J J y X ⊗Y ,Z y X ,Y ⊗Z F (α ) F ((X ⊗ Y ) ⊗ Z) −−−−→C F (X ⊗ (Y ⊗ Z))

commutative.

125 Monoidal functors

Let C, D be monoidal categories. Definition ∼ A functor F : C → D is a monoidal functor if F (1C) = 1D and F is equipped with a functorial (in X , Y ) isomorphism ∼ JX ,Y : F (X ) ⊗ F (Y ) −→ F (X ⊗ Y ) which makes the diagram

α (F (X ) ⊗ F (Y )) ⊗ F (Z) −−−−→D F (X ) ⊗ (F (Y ) ⊗ F (Z))   J ⊗id id ⊗J y X ,Y Z y X Y ,Z F (X ⊗ Y ) ⊗ F (Z) F (X ) ⊗ F (Y ⊗ Z)   J J y X ⊗Y ,Z y X ,Y ⊗Z F (α ) F ((X ⊗ Y ) ⊗ Z) −−−−→C F (X ⊗ (Y ⊗ Z))

commutative. A monoidal functor is an equivalence of monoidal categories if it is an equivalence of categories. 126 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 d is the differential in the standard complex of G with coefficients in ∗ C . In particular, F admits a monoidal structure if and only if the cohomology classes of α and β are the same.This shows that 3 ∗ Vec(G, α) is equivalent to Vec(G) iff α is trivial in H (G, C ).

Monoidal functors ctd.

The notion of monoidal equivalence is useful because monoidal categories that are monoidally equivalent are “the same for all practical purposes”.

127 ∗ is monoidal with Jg,h ∈ C : g ⊗ h → g ⊗ h iff dJ = α/β, where d is the differential in the standard complex of G with coefficients in ∗ C . In particular, F admits a monoidal structure if and only if the cohomology classes of α and β are the same.This shows that 3 ∗ Vec(G, α) is equivalent to Vec(G) iff α is trivial in H (G, C ).

Monoidal functors ctd.

The notion of monoidal equivalence is useful because monoidal categories that are monoidally equivalent are “the same for all practical 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

128 In particular, F admits a monoidal structure if and only if the cohomology classes of α and β are the same.This shows that 3 ∗ Vec(G, α) is equivalent to Vec(G) iff α is trivial in H (G, C ).

Monoidal functors ctd.

The notion of monoidal equivalence is useful because monoidal categories that are monoidally equivalent are “the same for all practical 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 d is the differential in the standard complex of G with coefficients in ∗ C .

129 This shows that 3 ∗ Vec(G, α) is equivalent to Vec(G) iff α is trivial in H (G, C ).

Monoidal functors ctd.

The notion of monoidal equivalence is useful because monoidal categories that are monoidally equivalent are “the same for all practical 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 d is the differential in the standard complex of G with coefficients in ∗ C . In particular, F admits a monoidal structure if and only if the cohomology classes of α and β are the same.

130 Monoidal functors ctd.

The notion of monoidal equivalence is useful because monoidal categories that are monoidally equivalent are “the same for all practical 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 d is the differential in the standard complex of G with coefficients in ∗ C . In particular, F admits a monoidal structure if and only if the cohomology classes of α and β are the same.This shows that 3 ∗ Vec(G, α) is equivalent to Vec(G) iff α is trivial in H (G, C ).

131 However, sometimes ∆ =6 ∆op but still X ⊗ Y =∼ Y ⊗ X . Example n For A = Uq(sl2) where q =6 1 define the universal R-matrix ∞ −1 k h⊗h P k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , where[ k]q! = [1]q...[k]q. k=0 [k]q! This is an infinite series, but it makes sense as an operator on X ⊗ Y for any finite dimensional type I representations X , Y , h⊗h λµ because the sum terminates. Here q 2 (x ⊗ y) = q 2 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 representations c : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)R on X ⊗ Y for a ∈ Uq(sl2).

The universal R-matrix of Uq(sl2)

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

132 Example n For A = Uq(sl2) where q =6 1 define the universal R-matrix ∞ −1 k h⊗h P k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , where[ k]q! = [1]q...[k]q. k=0 [k]q! This is an infinite series, but it makes sense as an operator on X ⊗ Y for any finite dimensional type I representations X , Y , h⊗h λµ because the sum terminates. Here q 2 (x ⊗ y) = q 2 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 representations c : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)R on X ⊗ Y for a ∈ Uq(sl2).

The universal R-matrix of Uq(sl2)

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

133 where[ k]q! = [1]q...[k]q. This is an infinite series, but it makes sense as an operator on X ⊗ Y for any finite dimensional type I representations X , Y , h⊗h λµ because the sum terminates. Here q 2 (x ⊗ y) = q 2 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 representations c : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)R on X ⊗ Y for a ∈ Uq(sl2).

The universal R-matrix of Uq(sl2)

If A is a Hopf algebra and X , Y ∈ Rep H then X ⊗ Y  Y ⊗ X op in general, as ∆ =6 ∆ (e.g. g ⊗ h  h ⊗ g for g, h ∈ Vec(G)). However, sometimes ∆ =6 ∆op but still X ⊗ Y =∼ Y ⊗ X . Example n For A = Uq(sl2) where q =6 1 define the universal R-matrix ∞ −1 k h⊗h P k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , k=0 [k]q!

134 This is an infinite series, but it makes sense as an operator on X ⊗ Y for any finite dimensional type I representations X , Y , h⊗h λµ because the sum terminates. Here q 2 (x ⊗ y) = q 2 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 representations c : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)R on X ⊗ Y for a ∈ Uq(sl2).

The universal R-matrix of Uq(sl2)

If A is a Hopf algebra and X , Y ∈ Rep H then X ⊗ Y  Y ⊗ X op in general, as ∆ =6 ∆ (e.g. g ⊗ h  h ⊗ g for g, h ∈ Vec(G)). However, sometimes ∆ =6 ∆op but still X ⊗ Y =∼ Y ⊗ X . Example n For A = Uq(sl2) where q =6 1 define the universal R-matrix ∞ −1 k h⊗h P k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , where[ k]q! = [1]q...[k]q. k=0 [k]q!

135 h⊗h λµ Here q 2 (x ⊗ y) = q 2 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 representations c : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)R on X ⊗ Y for a ∈ Uq(sl2).

The universal R-matrix of Uq(sl2)

If A is a Hopf algebra and X , Y ∈ Rep H then X ⊗ Y  Y ⊗ X op in general, as ∆ =6 ∆ (e.g. g ⊗ h  h ⊗ g for g, h ∈ Vec(G)). However, sometimes ∆ =6 ∆op but still X ⊗ Y =∼ Y ⊗ X . Example n For A = Uq(sl2) where q =6 1 define the universal R-matrix ∞ −1 k h⊗h P k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , where[ k]q! = [1]q...[k]q. k=0 [k]q! This is an infinite series, but it makes sense as an operator on X ⊗ Y for any finite dimensional type I representations X , Y , because the sum terminates.

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

The universal R-matrix of Uq(sl2)

If A is a Hopf algebra and X , Y ∈ Rep H then X ⊗ Y  Y ⊗ X op in general, as ∆ =6 ∆ (e.g. g ⊗ h  h ⊗ g for g, h ∈ Vec(G)). However, sometimes ∆ =6 ∆op but still X ⊗ Y =∼ Y ⊗ X . Example n For A = Uq(sl2) where q =6 1 define the universal R-matrix ∞ −1 k h⊗h P k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , where[ k]q! = [1]q...[k]q. k=0 [k]q! This is an infinite series, but it makes sense as an operator on X ⊗ Y for any finite dimensional type I representations X , Y , h⊗h λµ because the sum terminates. Here q 2 (x ⊗ y) = q 2 x ⊗ y if x, y have weights λ, µ, i.e., Kx = qλx, Ky = qµy.

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

The universal R-matrix of Uq(sl2)

If A is a Hopf algebra and X , Y ∈ Rep H then X ⊗ Y  Y ⊗ X op in general, as ∆ =6 ∆ (e.g. g ⊗ h  h ⊗ g for g, h ∈ Vec(G)). However, sometimes ∆ =6 ∆op but still X ⊗ Y =∼ Y ⊗ X . Example n For A = Uq(sl2) where q =6 1 define the universal R-matrix ∞ −1 k h⊗h P k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , where[ k]q! = [1]q...[k]q. k=0 [k]q! This is an infinite series, but it makes sense as an operator on X ⊗ Y for any finite dimensional type I representations X , Y , h⊗h λµ because the sum terminates. Here q 2 (x ⊗ y) = q 2 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 representations c : X ⊗ Y → Y ⊗ X .

138 The universal R-matrix of Uq(sl2)

If A is a Hopf algebra and X , Y ∈ Rep H then X ⊗ Y  Y ⊗ X op in general, as ∆ =6 ∆ (e.g. g ⊗ h  h ⊗ g for g, h ∈ Vec(G)). However, sometimes ∆ =6 ∆op but still X ⊗ Y =∼ Y ⊗ X . Example n For A = Uq(sl2) where q =6 1 define the universal R-matrix ∞ −1 k h⊗h P k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , where[ k]q! = [1]q...[k]q. k=0 [k]q! This is an infinite series, but it makes sense as an operator on X ⊗ Y for any finite dimensional type I representations X , Y , h⊗h λµ because the sum terminates. Here q 2 (x ⊗ y) = q 2 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 representations c : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)R on X ⊗ Y for a ∈ Uq(sl2). 139 The universal R-matrix of Uq(sl2)

If A is a Hopf algebra and X , Y ∈ Rep H then X ⊗ Y  Y ⊗ X op in general, as ∆ =6 ∆ (e.g. g ⊗ h  h ⊗ g for g, h ∈ Vec(G)). However, sometimes ∆ =6 ∆op but still X ⊗ Y =∼ Y ⊗ X . Example n For A = Uq(sl2) where q =6 1 define the universal R-matrix ∞ −1 k h⊗h P k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , where[ k]q! = [1]q...[k]q. k=0 [k]q! This is an infinite series, but it makes sense as an operator on X ⊗ Y for any finite dimensional type I representations X , Y , h⊗h λµ because the sum terminates. Here q 2 (x ⊗ y) = q 2 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 representations c : X ⊗ Y → Y ⊗ X . In other words, we have R∆(a) = ∆op(a)R on X ⊗ Y for a ∈ Uq(sl2). 140 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 isomorphism ∼ given by ϕX : Y ⊗ X −→ X ⊗ Y ∀ X ∈ C, satisfying the following commutative diagram:

ϕX1⊗X2 Y ⊗ (X1 ⊗ X2) −−−−→ (X1 ⊗ X2) ⊗ Y  x α−1 α−1 y YX1X2  X1X2Y

(Y ⊗ X1) ⊗ X2 X1 ⊗ (X2 ⊗ Y )  x ϕ ⊗id id⊗ϕ y X1  X2

αX1YX2 (X1 ⊗ Y ) ⊗ X2 −−−−→ X1 ⊗ (Y ⊗ X2) Morphisms of such pairs are morphisms in C which preserve ϕ.

The Drinfeld center

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

141 ϕX1⊗X2 Y ⊗ (X1 ⊗ X2) −−−−→ (X1 ⊗ X2) ⊗ Y  x α−1 α−1 y YX1X2  X1X2Y

(Y ⊗ X1) ⊗ X2 X1 ⊗ (X2 ⊗ Y )  x ϕ ⊗id id⊗ϕ y X1  X2

αX1YX2 (X1 ⊗ Y ) ⊗ X2 −−−−→ X1 ⊗ (Y ⊗ X2) Morphisms of such pairs are morphisms in C which preserve ϕ.

The Drinfeld center

A prototypical example of a monoidal category where X ⊗ 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 isomorphism ∼ given by ϕX : Y ⊗ X −→ X ⊗ Y ∀ X ∈ C, satisfying the following commutative diagram:

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

The Drinfeld center

A prototypical example of a monoidal category where X ⊗ 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 isomorphism ∼ given by ϕX : Y ⊗ X −→ X ⊗ Y ∀ X ∈ C, satisfying the following commutative diagram:

ϕX1⊗X2 Y ⊗ (X1 ⊗ X2) −−−−→ (X1 ⊗ X2) ⊗ Y  x α−1 α−1 y YX1X2  X1X2Y

(Y ⊗ X1) ⊗ X2 X1 ⊗ (X2 ⊗ Y )  x ϕ ⊗id id⊗ϕ y X1  X2

αX1YX2 (X1 ⊗ Y ) ⊗ X2 −−−−→ X1 ⊗ (Y ⊗ X2)

143 The Drinfeld center

A prototypical example of a monoidal category where X ⊗ 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 isomorphism ∼ given by ϕX : Y ⊗ X −→ X ⊗ Y ∀ X ∈ C, satisfying the following commutative diagram:

ϕX1⊗X2 Y ⊗ (X1 ⊗ X2) −−−−→ (X1 ⊗ X2) ⊗ Y  x α−1 α−1 y YX1X2  X1X2Y

(Y ⊗ X1) ⊗ X2 X1 ⊗ (X2 ⊗ Y )  x ϕ ⊗id id⊗ϕ y X1  X2

αX1YX2 (X1 ⊗ Y ) ⊗ X2 −−−−→ X1 ⊗ (Y ⊗ X2) Morphisms of such pairs are morphisms in C which preserve ϕ. 144 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 −1 cYZ ,cZY Y ⊗ Z −−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ , ⊗n cZY = ψY . This gives an action of the braid group Bn on V for n V ∈ Z(C). Recall that Bn = π1(C \diagonals/Sn) =

hs1, ..., sn−1 | si sj = sj si if |i − j| ≥ 2, si si+1si = si+1si si+1i

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

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

The Drinfeld center, ctd.

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

145 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 −1 cYZ ,cZY Y ⊗ Z −−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ , ⊗n cZY = ψY . This gives an action of the braid group Bn on V for n V ∈ Z(C). Recall that Bn = π1(C \diagonals/Sn) =

hs1, ..., sn−1 | si sj = sj si if |i − j| ≥ 2, si si+1si = si+1si si+1i

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

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

The Drinfeld center, ctd.

The Drinfeld center Z(C) has a natural monoidal structure defined 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.

146 Moreover, if Y , Z ∈ Z(C) then there are two ways −1 cYZ ,cZY Y ⊗ Z −−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ , ⊗n cZY = ψY . This gives an action of the braid group Bn on V for n V ∈ Z(C). Recall that Bn = π1(C \diagonals/Sn) =

hs1, ..., sn−1 | si sj = sj si if |i − j| ≥ 2, si si+1si = si+1si si+1i

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

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

The Drinfeld center, ctd.

The Drinfeld center Z(C) has a natural monoidal structure defined 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 .

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

hs1, ..., sn−1 | si sj = sj si if |i − j| ≥ 2, si si+1si = si+1si si+1i

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

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

The Drinfeld center, ctd.

The Drinfeld center Z(C) has a natural monoidal structure defined 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 −1 cYZ ,cZY Y ⊗ Z −−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ , cZY = ψY .

148 n Recall that Bn = π1(C \diagonals/Sn) =

hs1, ..., sn−1 | si sj = sj si if |i − j| ≥ 2, si si+1si = si+1si si+1i

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

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

The Drinfeld center, ctd.

The Drinfeld center Z(C) has a natural monoidal structure defined 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 −1 cYZ ,cZY Y ⊗ Z −−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ , ⊗n cZY = ψY . This gives an action of the braid group Bn on V for V ∈ Z(C).

149 hs1, ..., sn−1 | si sj = sj si if |i − j| ≥ 2, si si+1si = si+1si si+1i

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

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

The Drinfeld center, ctd.

The Drinfeld center Z(C) has a natural monoidal structure defined 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 −1 cYZ ,cZY Y ⊗ Z −−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ , ⊗n cZY = ψY . This gives an action of the braid group Bn on V for n V ∈ Z(C). Recall that Bn = π1(C \diagonals/Sn) =

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

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

The Drinfeld center, ctd.

The Drinfeld center Z(C) has a natural monoidal structure defined 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 −1 cYZ ,cZY Y ⊗ Z −−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ , ⊗n cZY = ψY . This gives an action of the braid group Bn on V for n V ∈ Z(C). Recall that Bn = π1(C \diagonals/Sn) =

hs1, ..., sn−1 | si sj = sj si if |i − j| ≥ 2, si si+1si = si+1si si+1i

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

The Drinfeld center, ctd.

The Drinfeld center Z(C) has a natural monoidal structure defined 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 −1 cYZ ,cZY Y ⊗ Z −−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ , ⊗n cZY = ψY . This gives an action of the braid group Bn on V for n V ∈ Z(C). Recall that Bn = π1(C \diagonals/Sn) =

hs1, ..., sn−1 | si sj = sj si if |i − j| ≥ 2, si si+1si = si+1si si+1i

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

152 The Drinfeld center, ctd.

The Drinfeld center Z(C) has a natural monoidal structure defined 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 −1 cYZ ,cZY Y ⊗ Z −−−−−→ Z ⊗ Y to identify Y ⊗ Z and Z ⊗ Y , cYZ = ϕZ , ⊗n cZY = ψY . This gives an action of the braid group Bn on V for n V ∈ Z(C). Recall that Bn = π1(C \diagonals/Sn) =

hs1, ..., sn−1 | si sj = sj si if |i − j| ≥ 2, si si+1si = si+1si si+1i

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

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

153 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 have been hexagons had we not suppressed α. This motivates the definition of a braided monoidal category.

The Drinfeld center, ctd.

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

X ⊗ Y ⊗ Z → Y ⊗ X ⊗ Z & ↓ Y ⊗ Z ⊗ X

154 They are call “hexagon relations” because they would have been hexagons had we not suppressed α. This motivates the definition of a braided monoidal category.

The Drinfeld center, ctd.

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

X ⊗ Y ⊗ Z → Y ⊗ X ⊗ Z X ⊗ Y ⊗ Z → X ⊗ Z ⊗ Y & ↓ & ↓ Y ⊗ Z ⊗ X Z ⊗ X ⊗ Y

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

155 This motivates the definition of a braided monoidal category.

The Drinfeld center, ctd.

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

X ⊗ Y ⊗ Z → Y ⊗ X ⊗ Z X ⊗ Y ⊗ Z → X ⊗ Z ⊗ Y & ↓ & ↓ Y ⊗ Z ⊗ X Z ⊗ X ⊗ Y

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

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

156 The Drinfeld center, ctd.

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

X ⊗ Y ⊗ Z → Y ⊗ X ⊗ Z X ⊗ Y ⊗ Z → X ⊗ Z ⊗ Y & ↓ & ↓ Y ⊗ Z ⊗ X Z ⊗ X ⊗ Y

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

They are call “hexagon relations” because they would have been hexagons had we not suppressed α. This motivates the definition of a braided monoidal category.

157 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 braided subcategory of its Drinfeld center using the inclusion ι : C → Z(C) given by X 7→ (X , cX ?). In this sense the Drinfeld center is a prototypical example of a braided category. Theorem The category of finite dimensional (type I) representations of Uq(sl2) is a braided monoidal category, with c = P ◦ R.

Braided monoidal categories

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

158 Thus we obtain Theorem The Drinfeld center Z(C) is a braided monoidal category.

Moreover, every braided monoidal category C is a braided subcategory of its Drinfeld center using the inclusion ι : C → Z(C) given by X 7→ (X , cX ?). In this sense the Drinfeld center is a prototypical example of a braided category. Theorem The category of finite dimensional (type I) representations of Uq(sl2) is a braided monoidal category, with c = P ◦ R.

Braided monoidal categories

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

159 Moreover, every braided monoidal category C is a braided subcategory of its Drinfeld center using the inclusion ι : C → Z(C) given by X 7→ (X , cX ?). In this sense the Drinfeld center is a prototypical example of a braided category. Theorem The category of finite dimensional (type I) representations of Uq(sl2) is a braided monoidal category, with c = P ◦ R.

Braided monoidal categories

Definition A braided monoidal category is a monoidal category endowed with op a functorial isomorphism c : ⊗ → ⊗ , 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.

160 Braided monoidal categories

Definition A braided monoidal category is a monoidal category endowed with op a functorial isomorphism c : ⊗ → ⊗ , 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 braided subcategory of its Drinfeld center using the inclusion ι : C → Z(C) given by X 7→ (X , cX ?). In this sense the Drinfeld center is a prototypical example of a braided category. Theorem The category of finite dimensional (type I) representations of Uq(sl2) is a braided monoidal category, with c = P ◦ R. 161 Remark

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

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

Braided monoidal categories ctd.

Note that as a consequence R satisfies the Quantum Yang-Baxter equation

R12R13R23 = R23R13R12

(this follows from the relation s1s2s1 = s2s1s2).

162 For example, the categories Rep G and Rep g are symmetric. However, Z(C) is usually not symmetric, and Repf Uq(sl2) isn’t either.

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

Braided monoidal categories ctd.

Note that as a consequence R satisfies the Quantum Yang-Baxter equation

R12R13R23 = R23R13R12

(this follows from the relation s1s2s1 = s2s1s2). Remark

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

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

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

Braided monoidal categories ctd.

Note that as a consequence R satisfies the Quantum Yang-Baxter equation

R12R13R23 = R23R13R12

(this follows from the relation s1s2s1 = s2s1s2). Remark

A braided category is called symmetric if cXY cYX = idX ⊗Y for all X , Y . For example, the categories Rep G and Rep g are symmetric.

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

Braided monoidal categories ctd.

Note that as a consequence R satisfies the Quantum Yang-Baxter equation

R12R13R23 = R23R13R12

(this follows from the relation s1s2s1 = s2s1s2). Remark

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

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

165 Braided monoidal categories ctd.

Note that as a consequence R satisfies the Quantum Yang-Baxter equation

R12R13R23 = R23R13R12

(this follows from the relation s1s2s1 = s2s1s2). Remark

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

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

166 Then the map ϕ A ⊗ → ⊗ A : IndCY = Y A A Y defines a comodule structure τ = ϕA|Y : Y → A ⊗ Y . The compatibility condition between this comodule 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 an A-comodule with τ : Y → A ⊗ Y satisfying the above compatibility condition. The category of such modules is denoted YD(A).

Proposition One has Z(Rep A) =∼ YD(A).

Yetter-Drinfeld modules

Let Y ∈ Z(Rep A).

167 The compatibility condition between this comodule 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 an A-comodule with τ : Y → A ⊗ Y satisfying the above compatibility condition. The category of such modules is denoted YD(A).

Proposition One has Z(Rep A) =∼ YD(A).

Yetter-Drinfeld modules

Let Y ∈ Z(Rep A). Then the map ϕ A ⊗ → ⊗ A : IndCY = Y A A Y defines a comodule structure τ = ϕA|Y : Y → A ⊗ Y .

168 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 an A-comodule with τ : Y → A ⊗ Y satisfying the above compatibility condition. The category of such modules is denoted YD(A).

Proposition One has Z(Rep A) =∼ YD(A).

Yetter-Drinfeld modules

Let Y ∈ Z(Rep A). Then the map ϕ A ⊗ → ⊗ A : IndCY = Y A A Y defines a comodule structure τ = ϕA|Y : Y → A ⊗ Y . The compatibility condition between this comodule structure and the A-module structure on Y is

τ(ay) = a(1)y(1)S(a(2)) ⊗ y(2)a(3).

169 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 an A-comodule with τ : Y → A ⊗ Y satisfying the above compatibility condition. The category of such modules is denoted YD(A).

Proposition One has Z(Rep A) =∼ YD(A).

Yetter-Drinfeld modules

Let Y ∈ Z(Rep A). Then the map ϕ A ⊗ → ⊗ A : IndCY = Y A A Y defines a comodule structure τ = ϕA|Y : Y → A ⊗ Y . The compatibility condition between this comodule 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)

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

Proposition One has Z(Rep A) =∼ YD(A).

Yetter-Drinfeld modules

Let Y ∈ Z(Rep A). Then the map ϕ A ⊗ → ⊗ A : IndCY = Y A A Y defines a comodule structure τ = ϕA|Y : Y → A ⊗ Y . The compatibility condition between this comodule 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).

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

Proposition One has Z(Rep A) =∼ YD(A).

Yetter-Drinfeld modules

Let Y ∈ Z(Rep A). Then the map ϕ A ⊗ → ⊗ A : IndCY = Y A A Y defines a comodule structure τ = ϕA|Y : Y → A ⊗ Y . The compatibility condition between this comodule 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 an A-comodule with τ : Y → A ⊗ Y satisfying the above compatibility condition.

172 Proposition One has Z(Rep A) =∼ YD(A).

Yetter-Drinfeld modules

Let Y ∈ Z(Rep A). Then the map ϕ A ⊗ → ⊗ A : IndCY = Y A A Y defines a comodule structure τ = ϕA|Y : Y → A ⊗ Y . The compatibility condition between this comodule 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 an A-comodule with τ : Y → A ⊗ Y satisfying the above compatibility condition. The category of such modules is denoted YD(A).

173 Yetter-Drinfeld modules

Let Y ∈ Z(Rep A). Then the map ϕ A ⊗ → ⊗ A : IndCY = Y A A Y defines a comodule structure τ = ϕA|Y : Y → A ⊗ Y . The compatibility condition between this comodule 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 an A-comodule with τ : Y → A ⊗ Y satisfying the above compatibility condition. The category of such modules is denoted YD(A).

Proposition One has Z(Rep A) =∼ YD(A). 174 Yetter-Drinfeld modules

Let Y ∈ Z(Rep A). Then the map ϕ A ⊗ → ⊗ A : IndCY = Y A A Y defines a comodule structure τ = ϕA|Y : Y → A ⊗ Y . The compatibility condition between this comodule 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 an A-comodule with τ : Y → A ⊗ Y satisfying the above compatibility condition. The category of such modules is denoted YD(A).

Proposition One has Z(Rep A) =∼ YD(A). 175 so the category YD(A) can be realized as the category of modules over some algebra generated by A and A∗ with some commutation relation between them. Thus we get Proposition For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as a vector space to A ⊗ A∗ such that there is a monoidal equivalence

YD(A) =∼ Rep D(A).

The Hopf algebra D(A) is called the (Drinfeld) quantum double of A. The algebras A and A∗op (with opposite coproduct) are Hopf subalgebras of D(A), and the commutation relations between them −1 have the form ba = (a(1), b(1))(a(3), b(3))a(2)S (b(2)), a ∈ A, b ∈ A∗.

Quantum double

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

176 Thus we get Proposition For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as a vector space to A ⊗ A∗ such that there is a monoidal equivalence

YD(A) =∼ Rep D(A).

The Hopf algebra D(A) is called the (Drinfeld) quantum double of A. The algebras A and A∗op (with opposite coproduct) are Hopf subalgebras of D(A), and the commutation relations between them −1 have the form ba = (a(1), b(1))(a(3), b(3))a(2)S (b(2)), a ∈ A, b ∈ A∗.

Quantum double

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

177 such that there is a monoidal equivalence

YD(A) =∼ Rep D(A).

The Hopf algebra D(A) is called the (Drinfeld) quantum double of A. The algebras A and A∗op (with opposite coproduct) are Hopf subalgebras of D(A), and the commutation relations between them −1 have the form ba = (a(1), b(1))(a(3), b(3))a(2)S (b(2)), a ∈ A, b ∈ A∗.

Quantum double

If A is finite dimensional, an A-comodule is the same as an A∗-module, so the category YD(A) can be realized as the category of modules over some algebra generated by A and A∗ with some commutation relation between them. Thus we get Proposition For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as a vector space to A ⊗ A∗

178 The Hopf algebra D(A) is called the (Drinfeld) quantum double of A. The algebras A and A∗op (with opposite coproduct) are Hopf subalgebras of D(A), and the commutation relations between them −1 have the form ba = (a(1), b(1))(a(3), b(3))a(2)S (b(2)), a ∈ A, b ∈ A∗.

Quantum double

If A is finite dimensional, an A-comodule is the same as an A∗-module, so the category YD(A) can be realized as the category of modules over some algebra generated by A and A∗ with some commutation relation between them. Thus we get Proposition For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as a vector space to A ⊗ A∗ such that there is a monoidal equivalence

YD(A) =∼ Rep D(A).

179 The algebras A and A∗op (with opposite coproduct) are Hopf subalgebras of D(A), and the commutation relations between them −1 have the form ba = (a(1), b(1))(a(3), b(3))a(2)S (b(2)), a ∈ A, b ∈ A∗.

Quantum double

If A is finite dimensional, an A-comodule is the same as an A∗-module, so the category YD(A) can be realized as the category of modules over some algebra generated by A and A∗ with some commutation relation between them. Thus we get Proposition For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as a vector space to A ⊗ A∗ such that there is a monoidal equivalence

YD(A) =∼ Rep D(A).

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

180 Quantum double

If A is finite dimensional, an A-comodule is the same as an A∗-module, so the category YD(A) can be realized as the category of modules over some algebra generated by A and A∗ with some commutation relation between them. Thus we get Proposition For A finite dimensional ∃ a Hopf algebra D(A) isomorphic as a vector space to A ⊗ A∗ such that there is a monoidal equivalence

YD(A) =∼ Rep D(A).

The Hopf algebra D(A) is called the (Drinfeld) quantum double of A. The algebras A and A∗op (with opposite coproduct) are Hopf subalgebras of D(A), and the commutation relations between them −1 have the form ba = (a(1), b(1))(a(3), b(3))a(2)S (b(2)), a ∈ A, b ∈ A∗.

181 This is addressed by the following theorem. Theorem (Drinfeld) The braiding of Rep D(A) = YD(A) is given by the formula c = P ◦ R where R ∈ A ⊗ A∗op ⊂ D(A) ⊗ D(A) is given by the universal R-matrix X ∗ R = ai ⊗ ai , i ∗ ∗ where ai is basis of A and ai the dual basis of A . In particular, R satisfies the quantum Yang-Baxter equation

R12R13R23 = R23R13R12 ∈ D(A)3.

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

Quantum double, ctd.

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

182 Theorem (Drinfeld) The braiding of Rep D(A) = YD(A) is given by the formula c = P ◦ R where R ∈ A ⊗ A∗op ⊂ D(A) ⊗ D(A) is given by the universal R-matrix X ∗ R = ai ⊗ ai , i ∗ ∗ where ai is basis of A and ai the dual basis of A . In particular, R satisfies the quantum Yang-Baxter equation

R12R13R23 = R23R13R12 ∈ D(A)3.

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

Quantum double, ctd.

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

183 where R ∈ A ⊗ A∗op ⊂ D(A) ⊗ D(A) is given by the universal R-matrix X ∗ R = ai ⊗ ai , i ∗ ∗ where ai is basis of A and ai the dual basis of A . In particular, R satisfies the quantum Yang-Baxter equation

R12R13R23 = R23R13R12 ∈ D(A)3.

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

Quantum double, ctd.

It is natural to ask how to express the braided structure of YD(A) in terms of D(A). This is addressed by the following theorem. Theorem (Drinfeld) The braiding of Rep D(A) = YD(A) is given by the formula c = P ◦ R

184 ∗ ∗ where ai is basis of A and ai the dual basis of A . In particular, R satisfies the quantum Yang-Baxter equation

R12R13R23 = R23R13R12 ∈ D(A)3.

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

Quantum double, ctd.

It is natural to ask how to express the braided structure of YD(A) in terms of D(A). This is addressed by the following theorem. Theorem (Drinfeld) The braiding of Rep D(A) = YD(A) is given by the formula c = P ◦ R where R ∈ A ⊗ A∗op ⊂ D(A) ⊗ D(A) is given by the universal R-matrix X ∗ R = ai ⊗ ai , i

185 In particular, R satisfies the quantum Yang-Baxter equation

R12R13R23 = R23R13R12 ∈ D(A)3.

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

Quantum double, ctd.

It is natural to ask how to express the braided structure of YD(A) in terms of D(A). This is addressed by the following theorem. Theorem (Drinfeld) The braiding of Rep D(A) = YD(A) is given by the formula c = P ◦ R where R ∈ A ⊗ A∗op ⊂ D(A) ⊗ D(A) is given by the universal R-matrix X ∗ R = ai ⊗ ai , i ∗ ∗ where ai is basis of A and ai the dual basis of A .

186 Moreover, the commutation relation between A and A∗ is uniquely determined by this equation.

Quantum double, ctd.

It is natural to ask how to express the braided structure of YD(A) in terms of D(A). This is addressed by the following theorem. Theorem (Drinfeld) The braiding of Rep D(A) = YD(A) is given by the formula c = P ◦ R where R ∈ A ⊗ A∗op ⊂ D(A) ⊗ D(A) is given by the universal R-matrix X ∗ R = ai ⊗ ai , i ∗ ∗ where ai is basis of A and ai the dual basis of A . In particular, R satisfies the quantum Yang-Baxter equation

R12R13R23 = R23R13R12 ∈ D(A)3.

187 Quantum double, ctd.

It is natural to ask how to express the braided structure of YD(A) in terms of D(A). This is addressed by the following theorem. Theorem (Drinfeld) The braiding of Rep D(A) = YD(A) is given by the formula c = P ◦ R where R ∈ A ⊗ A∗op ⊂ D(A) ⊗ D(A) is given by the universal R-matrix X ∗ R = ai ⊗ ai , i ∗ ∗ where ai is basis of A and ai the dual basis of A . In particular, R satisfies the quantum Yang-Baxter equation

R12R13R23 = R23R13R12 ∈ D(A)3.

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

188 Then D(A) = CG n O(G), where G acts on itself by conjugation (i.e., the commutation relation for the double gives the conjugation action). We have P R = 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 by e, K ±1 with the relations Ke = q2eK, e` = 0, K ` = 1, and coproduct defined by ∆(e) = e ⊗ K + 1 ⊗ e, ∆(K) = K ⊗ K. Then D(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is a Hopf algebra of dimension `3 generated by e, f , K ±1 with relations −2 K−K −1 as above and Kf = q fK, ef − fe = q−q−1 and coproduct given by the above formulas and ∆(f ) = f ⊗ 1 + K −1 ⊗ f .

Examples

Example Let A = CG where G is a finite group.

189 We have P R = 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 by e, K ±1 with the relations Ke = q2eK, e` = 0, K ` = 1, and coproduct defined by ∆(e) = e ⊗ K + 1 ⊗ e, ∆(K) = K ⊗ K. Then D(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is a Hopf algebra of dimension `3 generated by e, f , K ±1 with relations −2 K−K −1 as above and Kf = q fK, ef − fe = q−q−1 and coproduct given by the above formulas and ∆(f ) = f ⊗ 1 + K −1 ⊗ f .

Examples

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

190 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 by e, K ±1 with the relations Ke = q2eK, e` = 0, K ` = 1, and coproduct defined by ∆(e) = e ⊗ K + 1 ⊗ e, ∆(K) = K ⊗ K. Then D(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is a Hopf algebra of dimension `3 generated by e, f , K ±1 with relations −2 K−K −1 as above and Kf = q fK, ef − fe = q−q−1 and coproduct given by the above formulas and ∆(f ) = f ⊗ 1 + K −1 ⊗ f .

Examples

Example Let A = CG where G is a finite group. Then D(A) = CG n O(G), where G acts on itself by conjugation (i.e., the commutation relation for the double gives the conjugation action). We have P R = g∈G g ⊗ δg ,

191 Let A be the Taft Hopf algebra (of dimension `2), generated by e, K ±1 with the relations Ke = q2eK, e` = 0, K ` = 1, and coproduct defined by ∆(e) = e ⊗ K + 1 ⊗ e, ∆(K) = K ⊗ K. Then D(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is a Hopf algebra of dimension `3 generated by e, f , K ±1 with relations −2 K−K −1 as above and Kf = q fK, ef − fe = q−q−1 and coproduct given by the above formulas and ∆(f ) = f ⊗ 1 + K −1 ⊗ f .

Examples

Example Let A = CG where G is a finite group. Then D(A) = CG n O(G), where G acts on itself by conjugation (i.e., the commutation relation for the double gives the conjugation action). We have P R = 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 `.

192 and coproduct defined by ∆(e) = e ⊗ K + 1 ⊗ e, ∆(K) = K ⊗ K. Then D(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is a Hopf algebra of dimension `3 generated by e, f , K ±1 with relations −2 K−K −1 as above and Kf = q fK, ef − fe = q−q−1 and coproduct given by the above formulas and ∆(f ) = f ⊗ 1 + K −1 ⊗ f .

Examples

Example Let A = CG where G is a finite group. Then D(A) = CG n O(G), where G acts on itself by conjugation (i.e., the commutation relation for the double gives the conjugation action). We have P R = 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 by e, K ±1 with the relations Ke = q2eK, e` = 0, K ` = 1,

193 Then D(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is a Hopf algebra of dimension `3 generated by e, f , K ±1 with relations −2 K−K −1 as above and Kf = q fK, ef − fe = q−q−1 and coproduct given by the above formulas and ∆(f ) = f ⊗ 1 + K −1 ⊗ f .

Examples

Example Let A = CG where G is a finite group. Then D(A) = CG n O(G), where G acts on itself by conjugation (i.e., the commutation relation for the double gives the conjugation action). We have P R = 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 by e, K ±1 with the relations Ke = q2eK, e` = 0, K ` = 1, and coproduct defined by ∆(e) = e ⊗ K + 1 ⊗ e, ∆(K) = K ⊗ K.

194 D0 is a Hopf algebra of dimension `3 generated by e, f , K ±1 with relations −2 K−K −1 as above and Kf = q fK, ef − fe = q−q−1 and coproduct given by the above formulas and ∆(f ) = f ⊗ 1 + K −1 ⊗ f .

Examples

Example Let A = CG where G is a finite group. Then D(A) = CG n O(G), where G acts on itself by conjugation (i.e., the commutation relation for the double gives the conjugation action). We have P R = 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 by e, K ±1 with the relations Ke = q2eK, e` = 0, K ` = 1, and coproduct defined by ∆(e) = e ⊗ K + 1 ⊗ e, ∆(K) = K ⊗ K. Then D(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where

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

Examples

Example Let A = CG where G is a finite group. Then D(A) = CG n O(G), where G acts on itself by conjugation (i.e., the commutation relation for the double gives the conjugation action). We have P R = 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 by e, K ±1 with the relations Ke = q2eK, e` = 0, K ` = 1, and coproduct defined by ∆(e) = e ⊗ K + 1 ⊗ e, ∆(K) = K ⊗ K. Then D(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is a Hopf algebra of dimension `3 generated by e, f , K ±1 with relations −2 K−K −1 as above and Kf = q fK, ef − fe = q−q−1

196 Examples

Example Let A = CG where G is a finite group. Then D(A) = CG n O(G), where G acts on itself by conjugation (i.e., the commutation relation for the double gives the conjugation action). We have P R = 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 by e, K ±1 with the relations Ke = q2eK, e` = 0, K ` = 1, and coproduct defined by ∆(e) = e ⊗ K + 1 ⊗ e, ∆(K) = K ⊗ K. Then D(A) is the Hopf algebra of the form D0 ⊗ C[Z/`], where D0 is a Hopf algebra of dimension `3 generated by e, f , K ±1 with relations −2 K−K −1 as above and Kf = q fK, ef − fe = q−q−1 and coproduct given by the above formulas and ∆(f ) = f ⊗ 1 + K −1 ⊗ f .

197 It is called the small quantum group and was introduced by G. Lusztig (for any simple Lie algebra), and is denoted by uq(sl2). Note that by virtue of the construction D0 has a universal R-matrix, which can be computed to be `−1 −1 k h⊗h X k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , [k]q! k=0

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

Small quantum group

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

198 Note that by virtue of the construction D0 has a universal R-matrix, which can be computed to be `−1 −1 k h⊗h X k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , [k]q! k=0

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

Small quantum group

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

199 `−1 −1 k h⊗h X k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , [k]q! k=0

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

Small quantum group

Note that D0 is nothing but the quotient of Uq(sl2) by the relations E ` = F ` = K ` − 1 = 0. It is called the small quantum group and was introduced by G. Lusztig (for any simple Lie algebra), and is denoted by uq(sl2). Note that by virtue of the construction D0 has a universal R-matrix, which can be computed to be

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

Small quantum group

Note that D0 is nothing but the quotient of Uq(sl2) by the relations E ` = F ` = K ` − 1 = 0. It is called the small quantum group and was introduced by G. Lusztig (for any simple Lie algebra), and is denoted by uq(sl2). Note that by virtue of the construction D0 has a universal R-matrix, which can be computed to be `−1 −1 k h⊗h X k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , [k]q! k=0

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

Small quantum group

Note that D0 is nothing but the quotient of Uq(sl2) by the relations E ` = F ` = K ` − 1 = 0. It is called the small quantum group and was introduced by G. Lusztig (for any simple Lie algebra), and is denoted by uq(sl2). Note that by virtue of the construction D0 has a universal R-matrix, which can be computed to be `−1 −1 k h⊗h X k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , [k]q! k=0

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

202 Small quantum group

Note that D0 is nothing but the quotient of Uq(sl2) by the relations E ` = F ` = K ` − 1 = 0. It is called the small quantum group and was introduced by G. Lusztig (for any simple Lie algebra), and is denoted by uq(sl2). Note that by virtue of the construction D0 has a universal R-matrix, which can be computed to be `−1 −1 k h⊗h X k(k−1) (q − q ) k k R = q 2 q 2 e ⊗ f , [k]q! k=0

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

203 Given a Cartan matrix( aij ) with symmetrizing numbers di ∈ Z+ (i.e., di aij is symmetric), we start with the Hopf algebra Uq(b+) (quantum Borel) generated by ±1 ei , Ki with relations di aij Ki ej = q ej Ki , [Ki , Kj ] = 0,

with coproduct defined by ∆(Ki ) = Ki ⊗ Ki , ∆(ei ) = ei ⊗ Ki + 1 ⊗ ei and also quantum Serre relations, which are the most constraining relations you can impose preserving the Hopf algebra structure without killing any of the ei . Then the quantum group Uq(g) is the quantum double of Uq(b+) (understood appropriately) modded out by the “redundant copy of the maximal torus” created by the quantum double construction. This algebra has additional generators fi also satisfying quantum −1 Serre relations and ∆(fi ) = fi ⊗ 1 + Ki ⊗ fi with commutation K −K −1 relations[ e , f ] = δ i i . i j ij qdi −q−di

Higher rank quantum groups

Quantum groups attached to any simple Lie algebra can be constructed similarly.

204 di aij Ki ej = q ej Ki , [Ki , Kj ] = 0,

with coproduct defined by ∆(Ki ) = Ki ⊗ Ki , ∆(ei ) = ei ⊗ Ki + 1 ⊗ ei and also quantum Serre relations, which are the most constraining relations you can impose preserving the Hopf algebra structure without killing any of the ei . Then the quantum group Uq(g) is the quantum double of Uq(b+) (understood appropriately) modded out by the “redundant copy of the maximal torus” created by the quantum double construction. This algebra has additional generators fi also satisfying quantum −1 Serre relations and ∆(fi ) = fi ⊗ 1 + Ki ⊗ fi with commutation K −K −1 relations[ e , f ] = δ i i . i j ij qdi −q−di

Higher rank quantum groups

Quantum groups attached to any simple Lie algebra can be constructed similarly. Given a Cartan matrix( aij ) with symmetrizing numbers di ∈ Z+ (i.e., di aij is symmetric), we start with the Hopf algebra Uq(b+) (quantum Borel) generated by ±1 ei , Ki with relations

205 with coproduct defined by ∆(Ki ) = Ki ⊗ Ki , ∆(ei ) = ei ⊗ Ki + 1 ⊗ ei and also quantum Serre relations, which are the most constraining relations you can impose preserving the Hopf algebra structure without killing any of the ei . Then the quantum group Uq(g) is the quantum double of Uq(b+) (understood appropriately) modded out by the “redundant copy of the maximal torus” created by the quantum double construction. This algebra has additional generators fi also satisfying quantum −1 Serre relations and ∆(fi ) = fi ⊗ 1 + Ki ⊗ fi with commutation K −K −1 relations[ e , f ] = δ i i . i j ij qdi −q−di

Higher rank quantum groups

Quantum groups attached to any simple Lie algebra can be constructed similarly. Given a Cartan matrix( aij ) with symmetrizing numbers di ∈ Z+ (i.e., di aij is symmetric), we start with the Hopf algebra Uq(b+) (quantum Borel) generated by ±1 ei , Ki with relations di aij Ki ej = q ej Ki , [Ki , Kj ] = 0,

206 and also quantum Serre relations, which are the most constraining relations you can impose preserving the Hopf algebra structure without killing any of the ei . Then the quantum group Uq(g) is the quantum double of Uq(b+) (understood appropriately) modded out by the “redundant copy of the maximal torus” created by the quantum double construction. This algebra has additional generators fi also satisfying quantum −1 Serre relations and ∆(fi ) = fi ⊗ 1 + Ki ⊗ fi with commutation K −K −1 relations[ e , f ] = δ i i . i j ij qdi −q−di

Higher rank quantum groups

Quantum groups attached to any simple Lie algebra can be constructed similarly. Given a Cartan matrix( aij ) with symmetrizing numbers di ∈ Z+ (i.e., di aij is symmetric), we start with the Hopf algebra Uq(b+) (quantum Borel) generated by ±1 ei , Ki with relations di aij Ki ej = q ej Ki , [Ki , Kj ] = 0,

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

207 Then the quantum group Uq(g) is the quantum double of Uq(b+) (understood appropriately) modded out by the “redundant copy of the maximal torus” created by the quantum double construction. This algebra has additional generators fi also satisfying quantum −1 Serre relations and ∆(fi ) = fi ⊗ 1 + Ki ⊗ fi with commutation K −K −1 relations[ e , f ] = δ i i . i j ij qdi −q−di

Higher rank quantum groups

Quantum groups attached to any simple Lie algebra can be constructed similarly. Given a Cartan matrix( aij ) with symmetrizing numbers di ∈ Z+ (i.e., di aij is symmetric), we start with the Hopf algebra Uq(b+) (quantum Borel) generated by ±1 ei , Ki with relations di aij Ki ej = q ej Ki , [Ki , Kj ] = 0,

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

208 This algebra has additional generators fi also satisfying quantum −1 Serre relations and ∆(fi ) = fi ⊗ 1 + Ki ⊗ fi with commutation K −K −1 relations[ e , f ] = δ i i . i j ij qdi −q−di

Higher rank quantum groups

Quantum groups attached to any simple Lie algebra can be constructed similarly. Given a Cartan matrix( aij ) with symmetrizing numbers di ∈ Z+ (i.e., di aij is symmetric), we start with the Hopf algebra Uq(b+) (quantum Borel) generated by ±1 ei , Ki with relations di aij Ki ej = q ej Ki , [Ki , Kj ] = 0,

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

209 with commutation K −K −1 relations[ e , f ] = δ i i . i j ij qdi −q−di

Higher rank quantum groups

Quantum groups attached to any simple Lie algebra can be constructed similarly. Given a Cartan matrix( aij ) with symmetrizing numbers di ∈ Z+ (i.e., di aij is symmetric), we start with the Hopf algebra Uq(b+) (quantum Borel) generated by ±1 ei , Ki with relations di aij Ki ej = q ej Ki , [Ki , Kj ] = 0,

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

210 Higher rank quantum groups

Quantum groups attached to any simple Lie algebra can be constructed similarly. Given a Cartan matrix( aij ) with symmetrizing numbers di ∈ Z+ (i.e., di aij is symmetric), we start with the Hopf algebra Uq(b+) (quantum Borel) generated by ±1 ei , Ki with relations di aij Ki ej = q ej Ki , [Ki , Kj ] = 0,

with coproduct defined by ∆(Ki ) = Ki ⊗ Ki , ∆(ei ) = ei ⊗ Ki + 1 ⊗ ei and also quantum Serre relations, which are the most constraining relations you can impose preserving the Hopf algebra structure without killing any of the ei . Then the quantum group Uq(g) is the quantum double of Uq(b+) (understood appropriately) modded out by the “redundant copy of the maximal torus” created by the quantum double construction. This algebra has additional generators fi also satisfying quantum −1 Serre relations and ∆(fi ) = fi ⊗ 1 + Ki ⊗ fi with commutation K −K −1 relations[ e , f ] = δ i i . i j ij qdi −q−di 211 It is well known that any knot K can be obtained by closing up a braid b. On the other hand, as we have learned, b ⊗n acts as an operator on the space V , where V = V1 is the 2-dimensional representation of the quantum sl2. Theorem The trace of b · K in V ⊗n (called the quantum trace of b) is the Jones polynomial of K (up to normalization).

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

Jones polynomial

These commutation relations, as well as the R-matrix (which is now much more complicated) are produced automatically by the double construction. In fact, this works more generally, for any symmetrizable Kac-Moody algebra. Example In conclusion let us point out a connection to the Jones polynomial.

212 On the other hand, as we have learned, b ⊗n acts as an operator on the space V , where V = V1 is the 2-dimensional representation of the quantum sl2. Theorem The trace of b · K in V ⊗n (called the quantum trace of b) is the Jones polynomial of K (up to normalization).

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

Jones polynomial

These commutation relations, as well as the R-matrix (which is now much more complicated) are produced automatically by the double construction. In fact, this works more generally, for any symmetrizable Kac-Moody algebra. Example In conclusion let us point out a connection to the Jones polynomial. It is well known that any knot K can be obtained by closing up a braid b.

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

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

Jones polynomial

These commutation relations, as well as the R-matrix (which is now much more complicated) are produced automatically by the double construction. In fact, this works more generally, for any symmetrizable Kac-Moody algebra. Example In conclusion let us point out a connection to the Jones polynomial. It is well known that any knot K can be obtained by closing up a braid b. On the other hand, as we have learned, b ⊗n acts as an operator on the space V , where V = V1 is the 2-dimensional representation of the quantum sl2.

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

Jones polynomial

These commutation relations, as well as the R-matrix (which is now much more complicated) are produced automatically by the double construction. In fact, this works more generally, for any symmetrizable Kac-Moody algebra. Example In conclusion let us point out a connection to the Jones polynomial. It is well known that any knot K can be obtained by closing up a braid b. On the other hand, as we have learned, b ⊗n acts as an operator on the space V , where V = V1 is the 2-dimensional representation of the quantum sl2. Theorem The trace of b · K in V ⊗n (called the quantum trace of b) is the Jones polynomial of K (up to normalization).

215 Jones polynomial

These commutation relations, as well as the R-matrix (which is now much more complicated) are produced automatically by the double construction. In fact, this works more generally, for any symmetrizable Kac-Moody algebra. Example In conclusion let us point out a connection to the Jones polynomial. It is well known that any knot K can be obtained by closing up a braid b. On the other hand, as we have learned, b ⊗n acts as an operator on the space V , where V = V1 is the 2-dimensional representation of the quantum sl2. Theorem The trace of b · K in V ⊗n (called the quantum trace of b) is the Jones polynomial of K (up to normalization).

For Vi for i > 1 one gets the colored Jones polynomial, and for other Lie algebras – more complex invariants of knots called the Reshetikhin-Turaev invariants. 216 Thank you!

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