U.U.D.M. Project Report 2018:3

Examensarbete i matematik, 30 hpHandledare: Volodymyr MazorchukExaminator: Denis GaidashevApril 2018

Department of MathematicsUppsala University

Drinfeld centers

Markus Thuresson

Drinfeld centersMarkus Thuresson

Abstract

In the first part of this paper, we describe the structure of the center Z(R-Mod) of thecategory of left R-modules. Its natural structure as a ring is shown to be isomorphic to thesubring Z(R).

In the sections that follow, we present the basics of monoidal categories by regarding themas single-object bicategories. The Drinfeld center Z(C ) of a monoidal category C is definedand its basic properties presented.

The second half of the paper is devoted to describing the structure of the Drinfeld centerof the monoidal categories VectC and Z2-mod.

Contents

1 Basics of cateogries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Categories of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 The center of R-mod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Tensor product of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Tensor product of bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 2-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Bicategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Monoidal categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1 The Drinfeld center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 The Drinfeld center of VectC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.1 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7 Categories of group representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.1 Drinfeld centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8 The Drinfeld center of Z2-mod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Markus Thuresson 3

1 Basics of cateogries

In order to make this paper as self-contained as possible, we present the basics of categorytheory. This secion, and those similar to it, are of course highly skippable.

Definition 1.1. A category C consists of the following:

- a class Ob (C) of objects.

- for every pair of objectsX,Y ∈ Ob (C), a class of morphisms or arrows, denoted by Hom (X,Y ).In particular, for the pair (X,X), we require the existence of an identity morphismidX : X → X.

- for every three objects X,Y, Z ∈ Ob (C) and morphisms ϕ ∈ Hom (X,Y ) and ψ ∈ Hom (Y, Z),a binary operation : Hom (X,Y ) × Hom (X,Z) → Hom (Y,Z) with (ϕ,ψ) 7→ ψ ϕ, calledthe composition.

We require that the composition satisfies the following axioms:

i) if ϕ ∈ Hom (X,Y ) , ψ ∈ Hom (Y,Z) and ξ ∈ Hom (Z,W ) then ξ (ψ ϕ) = (ξ ψ) ϕ.

ii) if ϕ ∈ Hom (X,Y ) and ψ ∈ Hom (Y,Z) then ϕ idX = ϕ and idZ ψ = ψ.

Remark 1.2. The identity morphism idX is unique for every object X ∈ Ob (C). If id′X wereanother identity morphism, we would have

idX = idX id′X = id′X .

Definition 1.3. A category C is said to be small if both the class of objects and the morphismclasses are sets, and not proper classes.

Definition 1.4. The terminal category, denoted by 111, is the category having a single objectand a single morphism(the identity).

Definition 1.5. Let C be a category. A morphism ϕ ∈ Hom (X,Y ) is called an isomorphism ifif there exists ψ ∈ Hom (Y,X) such that ψ ϕ = idX and ϕ ψ = idY .

Definition 1.6. Let C be a category. Two objects X,Y ∈ Ob (C) are said to be isomorphic ifthere exists ϕ ∈ Hom (X,Y ) which is an isomorphism.

Definition 1.7. Let C and D be categories. Then a functor F : C → D consists of the following:

- a map F : Ob (C)→ Ob (D).

- for every pair of objects X,Y ∈ Ob (C), a map F : HomC (X,Y )→ HomD (F (X), F (Y )) suchthat

i) for every X ∈ Ob (C), we have F (idX) = idF (X).

ii) for composable morphisms ϕ and ψ we have F (ψ ϕ) = F (ψ) F (ϕ).

Drinfeld centers 4

Definition 1.8. Let C,D be categories and let F : C → D be a functor. Then F is full if eachmap

FX,Y : HomC (X,Y )→ HomD (F (X), F (Y ))

is surjective.

Definition 1.9. Let C,A be categories and let F : C → D be a functor. Then F is faithful ifeach map

FX,Y : HomC (X,Y )→ HomD (F (X), F (Y ))

is injective.

Definition 1.10. Let C,D be categories and let F : C → D be a functor. Then F is dense ifeach Y ∈ Ob (D) is isomorphic to an object F (X) for some X ∈ Ob (C).

Definition 1.11. Let C and D be categories and let F,G : C → D be functors. A naturaltransformation η : F ⇒ G is a map Ob (C)→ HomD (F (X), G(X)) such that the diagram

F (X)F (ϕ)

//

ηX

F (Y )

ηY

G(X)G(ϕ)

// G(Y )

commutes for every morphism ϕ ∈ Hom (X,Y ).

Definition 1.12. A natural transformation η such that each component ηX is an isomorphismis called a natural isomorphism.

Definition 1.13. Let C,D be categories. An equivalence of C and D is a functor F : C → Dsuch that there exists another functor G : D → C and two natural isomorphisms η : F G→ idDand µ : G F → idC .

Theorem 1.14. Let C, D be categories and F : C → D a functor. Then F is an equivalence ifand only if F is full, faithful and dense.

Example 1.15. Let the categories C and D be defined as follows:Fix a positive integer n. The category C has a single object and its morphisms are n × n

complex matrices, with composition given by matrix multiplication.The category D has as objects complex vector spaces of dimension n, and its morphisms are

linear maps. Composition is given by the usual composition of maps.Fixing the standard basis of the vector space Cn, we define a functor F : C → D by mapping

the object of C to the vector space Cn and mapping each matrix to the linear map which in thestandard basis is given by that matrix.

Then it is clear by linear algebra that F is a functor and that F is full and faithful. More-over, F is dense since all objects of D, being vector spaces of the same finite dimension, areisomorphic.

Since the functor F is full, faithful and dense it is an equivalence.

Markus Thuresson 5

Definition 1.16. Let C, D be categories. Then the product category C × D consists of thefollowing:

- pairs (X,Y ) of objects, where X ∈ Ob (C) and Y ∈ Ob (D).

- pairs (ϕ,ψ) : (X,Y ) → (Z,W ) of morphisms, where for X,Z ∈ Ob (C) and Y,W ∈ Ob (D),we have ϕ ∈ HomC (X,Z) and ψ ∈ HomD (Y,W ).

- componentwise composition (ϕ2, ψ2) C×D (ϕ1, ψ1) = (ϕ2 C ϕ1, ψ2 D ψ1).

- identity morphisms of the form id(X,Y ) = (idX , idY ).

Proposition 1.17. Let C be a small category and 111 the terminal category. Then we haveC × 111 ∼= C (as objects in the category of small categories).

Proof. Consider the map F : C → C × 111 given by

X 7→ (X, 1)

f 7→ (f, id).

Then we have F (idX) = (idX , id) = id(X,1) = idF (X) and

F (g f) = (g f, id) = (g, id) (f, id) = F (g) F (f)

so F is a functor. It is obvious that we can construct the inverse functor F−1 : C × 111→ C by

(X, 1) 7→ X

(f, id) 7→ f.

Definition 1.18. Let C be a category. We define the center of C, denoted by Z(C) to be theclass of natural transformations from the identity functor idC to itself.

Example 1.19. We recall that a monoid is a set with a binary associative operation withidentity. Any monoid M may be regarded as a category. Define the category M as follows:

- M has one object, •.

- Morphisms are all elements of M .

- Composition is given by the multiplication of M . The identity of M then acts as identity forthe composition, and composition is associative since the multiplication of M is associative.

Drinfeld centers 6

It is clear that given such a category we can reconstruct our monoid, so the above is an equivalentdefinition.

Now consider the center Z(M) of M. Since M has only one object and the morphismsare elements of M , a natural transformation from the identity functor to itself consists of anelement z ∈M such that the diagram

Mx //

z

M

z

M x//M

commutes for all x ∈ M . Since the composition is just multiplication in M , this is equivalentto zx = xz for all x ∈M . Now it is clear that Z(M) = Z(M) = z ∈M : zx = xz ∀x ∈M.

2 Categories of modules

In the following section, let R be a unital ring. Recall that the center of a ring R is the subringZ(R) = z ∈ R : zr = rz ∀r ∈ R.

Definition 2.1. A left R-module is an abelian group (M,+) together with a binary operation· : R×M →M such that

i) r · (x+ y) = r · x+ r · y

ii) (r + s) · x = r · x+ s · x

iii) (rs) · x = r · (s · x)

iv) 1R · x = x

for all x, y ∈M and r, s ∈ R. When necessary, we refer to the action of R on M as ·M .

Definition 2.2. Let M,N be left R-modules. A homomorphism of R-modules is a mapϕ : M → N such that

ϕ(r ·M x+ s ·M y) = r ·N ϕ(x) + s ·N ϕ(y)

for all x, y ∈M and r, s ∈ R.

For a fixed ring R the left R-modules together with module homomorphisms form a category,R-Mod.

2.1 The center of R-mod

Proposition 2.3. Z(R-Mod) is a ring under componentwise addition and composition of ho-momorphisms.

Markus Thuresson 7

Proof. Z(R-Mod) consists of natural transformations from idR-Mod to itself, so η ∈ Z(R-Mod)maps every module M to an endomorphism ηM of M .

For any module, its endomorphisms carry the natural structure of a ring with pointwiseaddition and composition. This implies that the componenwise addition and composition ofa family of endomorphisms yields another family. We check that these operations preservenaturality. Let ϕ : M → N be a homomorphism and let x ∈M .

ϕ((ηM + µM )(x)) = ϕ(ηM (x) + µM (x))

= ϕ(ηM (x)) + ϕ(µM (x))

= ηM (ϕ(x)) + µM (ϕ(x))

= (ηM + µM )(ϕ(x))

(ϕ (ηM µM ))(x) = ((ϕ ηM ) µM )(x)

= ((ηM ϕ) µM )(x)

= (ηM (ϕ µM ))(x)

= (ηM (µM ϕ))(x)

= ((ηM µM ) ϕ)(x)

Definition 2.4. For any z ∈ Z(R), define a natural transformation ηz : idR-Mod → idR-Mod

by ηzM (x) = z · x, x ∈M ∈ R-Mod.

Remark 2.5. The natural transformation ηz is indeed in the center of R-Mod. Since ηzM isgiven by left multiplication with z, we get:

ηzN ϕ(x) = ηzN (ϕ(x))

= z ·N ϕ(x)

= ϕ(z ·M x)

= ϕ(ηzM (x))

= ϕ ηzM (x)

which is equivalent to commutativity of the diagram

Mϕ//

ηzM

N

ηzN

Mϕ// N

Definition 2.6. For any z ∈ Z(R), define the endomorphism ϕz of R by ϕz(x) = z · x.

Proposition 2.7. The homomorphisms ϕz induced by Z(R) form a subring EndZR−(R) ofEndR−(R).

Drinfeld centers 8

Proof. The identity endomorphism is ϕ1. Let z, c ∈ Z(R). We observe that

i) (ϕz + ϕc)(x) = (z + c) · x

ii) (ϕz + ϕ−z)(x) = (z + (−z)) · x = 0

iii) (ϕz(ϕc))(x) = ϕz(c · x) = z · c · x = (zc) · x

and we have z + c,−z, zc ∈ Z(R) since Z(R) is a subring.

Proposition 2.8. The rings Z(R) and EndZR−(R) are isomorphic.

Proof. Consider the map F : Z(R) → EndZR−(R) defined by F (z) = ϕz. Let z, c ∈ Z(R). Wehave:

F (z + c)(x) = ϕz+c(x)

= (z + c)(x)

= (ϕz + ϕc)(x)

= (F (z) + F (c))(x)

F (zc)(x) = ϕzc(x)

= (zc) · x= (ϕz ϕc)(x)

= (F (z) F (c))(x).

We clearly have F (1) = ϕ1 = idR so F is a homomorphism. For injectivity, note thatF (z) = F (c) =⇒ ϕz = ϕc which implies z = ϕz(1) = ϕc(1) = c. For any ϕz we have F (z) = ϕz

so F is surjective, hence an isomorphism.

Theorem 2.9. The evaluation map ε : Z(R-Mod) → EndR−(R) defined by η 7→ ηR inducesan isomorphism between Z(R-Mod) and Z(R).

Lemma 2.10. Let M be an R-module and let R be the regular module. Then, for any x ∈M ,there exists a unique homomorphism ξ : R→M such that ξ(1) = x.

Proof of lemma 2.10. Consider the map ξ(r) = r ·M x. Then ξ is a homomorphism:

ξ(r + s) = (r + s) ·M x

= r ·M x+ s ·M x

= ξ(r) + ξ(s)

ξ(rs) = (rs) ·M x

= r ·M (s ·M x)

= r ·M ξ(s)

Markus Thuresson 9

Suppose ψ : R→M is another homomorphism with ψ(1) = x. Then

ψ(r) = r ·M ψ(1) = r ·M ξ(1) = ξ(r), ∀r ∈ R

so ξ is unique.

Corollary 2.11. The rings EndR−(R) and Rop, are isomorphic.

Proof. For every endomorphism ϕ in EndR−(R) we have ϕ(r) = r ·ϕ(1) and hence ϕ is given byright multiplication with ϕ(1). But from lemma 2.10 it follows that every element of Rop(wichis the same as an element of R) is given as the image of 1 under a unique endomorphism,so the map f : EndR−(R) → Rop with ϕ 7→ ϕ(1) gives a clear bijection. Moreover, f is ahomomorphism since

f(ϕ+ ψ) = (ϕ+ ψ)(1) = ϕ(1) + ψ(1)

and

f(ϕ ψ) = ϕ(ψ(1)) = ψ(1)ϕ(1) = f(ψ)f(ϕ).

Corollary 2.12. The ring EndR−R(R), where R is the regular R-R-bimodule, is isomorphic toZ(R).

Proof. For every endomorphism ϕ , we have r · ϕ(1) = ϕ(r) = ϕ(1) · r, so ϕ is given bymultiplication with the central element ϕ(1). Again, by lemma 2.10, this induces the bijectionf : EndR−R(R) → Z(R) defined ϕ 7→ ϕ(1). That f is a homomorphism can be checked in thesame way as in the previous corollary.

Proof of theorem 2.9. Consider ε : Z(R-Mod) → EndR−(R) defined by η 7→ ηR. For everyx ∈ R there exists a unique module homomorphism ψ : R → R with ψ(1) = x by lemma 2.10.Then we have:

ηR(1) · x = ηR(1) · ψ(1)

= ψ(ηR(1))

= ηR(ψ(1))

= ψ(1) · ηR(1)

= x · ηR(1).

so z = ηR(1) ∈ Z(R). Now, for any r ∈ R, we have ηR(r) = r · ηR(1) = ηR(1) · r. So we haveηR = ϕz.

By definition of ε, ηz is mapped to ηzR. But by the above, ηR = ϕz for any η ∈ Z(R-Mod).So in particular, ε(ηz) = ηzR = ϕz. Thus, the map ε is surjective onto EndZR−(R), sinceηz ∈ Z(R-Mod) exists for every z ∈ Z(R).

If ε(η) = ε(µ), then ηR = µR. By naturality of η and µ we get the following commutativediagrams:

Drinfeld centers 10

Rϕ//

ηR=µR

M

ηM

Rϕ//M

Rϕ//

ηR=µR

M

µM

Rϕ//M

which imply

ηM ϕ = ϕ ηR = ϕ µR = µM ϕ

for every homomorphism ϕ : R → M . By lemma 2.10 there is a unique homomorphismψ : R→M with ψ(1) = x for every x ∈M . From this, we get:

ηM (x) = ηM (ψ(1))

= µM (ψ(1))

= µM (x)

so ηM = µM for any M and hence η = µ, so ε is injective.Since addition and composition are performed componentwise in Z(R-Mod) we get:

ε(η + µ) = (η + µ)R

= ηR + µR

= ε(η) + ε(µ)

ε(η µ) = (η µ)R

= ηR µR= ε(ηR) ε(µR).

If we let id denote the identity natural transformation and let idR denote the identity homo-morphism, then ε(id) = idR so ε is a homomorphism of unital rings.

Lemma 2.13. Every natural transformation η ∈ Z(R-Mod) is of the form η = ηz for somez ∈ Z(R).

Proof of lemma 2.13. Let x ∈M . By lemma 2.10 there is a unique homomorphism ψ : R→Mwith ψ(1) = x. We also know that ηR = ϕz for some z ∈ Z(R). Then, using naturality of η, weget:

ηM (x) = ηM (ψ(1))

= ψ(ηR(1))

= ψ(z)

= z · ψ(1)

= z · x.

so we see that the map ηM is actually ηzM for every module M , so we have η = ηz.

Markus Thuresson 11

Using lemma 2.13 we now see that the map ε : Z(R-Mod)→ EndZR−(R) is an isomorphism.By proposition 2.8 EndZR−(R) is isomorphic to Z(R), so in conclusion we get

Z(R-Mod) ∼= EndZR−(R) ∼= Z(R) =⇒ Z(R-Mod) ∼= Z(R).

2.2 Tensor product of modules

Definition 2.14. Let R be a ring, let M be a right R-module, let N be a left R-module andlet G be an abelian group. Then an R-balanced product is a map ϕ : M × N → G satisfyingthe following:

i) ϕ(m,n+ n′) = ϕ(m,n) + ϕ(m,n′)

ii) ϕ(m+m′, n) = ϕ(m,n) + ϕ(m′, n)

iii) ϕ(m ·M r, n) = ϕ(m, r ·N n)

for all m,m′ ∈M , n, n′ ∈ N and r ∈ R.

Definition 2.15. Let R be a ring, let M be a right R-module, and let N be a left R-module.Then the tensor product over R, denoted by M ⊗R N is an abelian group together with abalanced product ⊗ : M ×N →M ⊗R N satisfying the following:

For every abelian group G and every balanced product ϕ : M ×N → G there exists a uniquegroup homomorphism ϕ : M ⊗R N → G such that ϕ ⊗ = ϕ. In a commutative diagram:

M ×N

ϕ&&

⊗//M ⊗R N

∃!ϕ

G

This condition is known as the universal property of the tensor product.

Proposition 2.16. Elements of the form x⊗R y with x ∈M,y ∈ N generate M ⊗R N .

Proof. Consider the subgroup S ⊂ M ⊗R N generated by elements of the form x ⊗R y. Letπ be the projection onto the quotient group M ⊗R N

/S . Note that the zero map M × N →

M ⊗R N/S is a balanced product. By the universal property, there exists a unique group

homomorphism ϕ which makes the diagram

M ×N

0 &&

⊗//M ⊗R N

ϕ

M ⊗R N/S

commute. We clearly have 0 = 0 ⊗, but we also have (π ⊗)(x, y) = π(x ⊗ y) = 0 since(x, y) ∈ S. By the universal property, we have π = 0 which implies S = M ⊗R N .

Drinfeld centers 12

Corollary 2.17. Let M be a left R-module. Then the tensor product R ⊗R M is itself a leftR-module with module structure given by r · (x⊗ y) = rx⊗ y.

Proof. Note that R is an R−R−bimodule. First we have

r ·∑

xi ⊗ yi = r ·∑

1⊗ xiyi

= r · (1⊗∑

xiyi)

= r ⊗∑

xiyi

=∑

r ⊗ xiyi

=∑

rxi ⊗ yi

so our scalar multiplication extends nicely to sums of elements of the form x⊗y. Next we checkthe module axioms:

i)

r ·

n∑i=1

xi ⊗ yi +m∑j=1

xj ⊗ yj

=n∑i=1

rxi ⊗ yi +m∑j=1

rxj ⊗ yj

= r ·n∑i=1

xi ⊗ yi + r ·m∑j=1

xj ⊗ yj .

ii)

(r + s) ·n∑i=1

xi ⊗ yi = (r + s) ·n∑i=1

1⊗ xiyi

= (r + s) ·

(1⊗

n∑i=1

xiyi

)

= (r + s)⊗n∑i=1

xiyi

= r ⊗n∑i=1

xiyi + s⊗n∑i=1

xiyi

=

n∑i=1

rxi ⊗ yi +

n∑i=1

sxi ⊗ yi

= r ·n∑i=1

xi ⊗ yi + s ·n∑i=1

xi ⊗ yi.

Markus Thuresson 13

iii)

(rs) ·n∑i=1

xi ⊗ yi =n∑i=1

(rs)xi ⊗ yi

=n∑i=1

r(sxi)⊗ yi

= r ·n∑i=1

sxi ⊗ yi

= r · (s ·n∑i=1

xi ⊗ yi).

iv)

1 ·n∑i=1

xi ⊗ yi =n∑i=1

1xi ⊗ yi =n∑i=1

xi ⊗ yi.

Proposition 2.18. Let M be a left R-module. Then the left R-modules R ⊗R M and M areisomorphic.

Proof. Consider the map ϕ : R⊗RM →M defined byn∑i=1

xi ⊗ yi 7→n∑i=1

xiyi.

ϕ

n∑i=1

xi ⊗ yi +m∑j=1

xj ⊗ yj

= ϕ

n∑i=1

1⊗ xiyi +m∑j=1

1⊗ xjyj

= ϕ

1⊗n∑i=1

xiyi + 1⊗m∑j=1

xjyj

= ϕ

1⊗

n∑i=1

xiyi +m∑j=1

xjyj

= 1 ·

n∑i=1

xiyi +m∑j=1

xjyj

=

n∑i=1

xiyi +

m∑j=1

xjyj

= ϕ

(n∑i=1

xi ⊗ yi

)+ ϕ

m∑j=1

xj ⊗ yj

.

Drinfeld centers 14

ϕ

(r ·

n∑i=1

xi ⊗ yi

)= ϕ

(n∑i=1

rxi ⊗ yi

)

=

n∑i=1

rxiyi

= r ·n∑i=1

xiyi

= r · ϕ

(n∑i=1

xi ⊗ yi

)

so ϕ is a homomorphism. Suppose

ϕ

(n∑i=1

xi ⊗ yi

)=

n∑i=1

xiyi =

m∑j=1

x′jy′j = ϕ

m∑j=1

x′j ⊗ y′j

.

Then we get:

n∑i=1

xi ⊗ yi =

n∑i=1

1⊗ xiyi

= 1⊗n∑i=1

xiyi

= 1⊗m∑j=1

x′jy′j

=

m∑j=1

1⊗ x′jy′j

=m∑j=1

x′j ⊗ y′j

so ϕ is injective. For every m ∈ M , we have ϕ(1 ⊗ m) = m, so ϕ is surjective, hence anisomorphism.

Proposition 2.19. Let M,N be left R-modules and let ϕ : M → N be a homomorphism. Thenthe map ϕ⊗ : R⊗RM → R⊗R N defined by x⊗ y 7→ x⊗ ϕ(y) is a homomorphism.

Markus Thuresson 15

Proof.

ϕ⊗

(n∑i=1

xi ⊗ yi

)= ϕ⊗

(n∑i=1

1⊗ xiyi

)

= ϕ⊗

(1⊗

n∑i=1

xiyi

)

= 1⊗ ϕ

(n∑i=1

xiyi

)

= 1⊗n∑i=1

ϕ(xiyi)

= 1⊗n∑i=1

xiϕ(yi)

=n∑i=1

1⊗ xiϕ(yi)

=n∑i=1

xi ⊗ ϕ(yi)

ϕ⊗

n∑i=1

xi ⊗ yi +m∑j=1

xj ⊗ yj

= ϕ⊗

1⊗

n∑i=1

xiyi +m∑j=1

xjyj

= 1⊗ ϕ

n∑i=1

xiyi +m∑j=1

xjyj

= 1⊗

n∑i=1

xiϕ(yi) +

m∑j=1

xjϕ(yj)

=

n∑i=1

1⊗ xiϕ(yi) +

m∑j=1

1⊗ xjϕ(yj)

=

n∑i=1

xi ⊗ ϕ(yi) +

m∑j=1

xj ⊗ ϕ(yj)

= ϕ⊗

(n∑i=1

xi ⊗ yi

)+ ϕ⊗

m∑j=1

xj ⊗ yj

ϕ⊗

(r

n∑i=1

xi ⊗ yi

)= ϕ⊗

(n∑i=1

rxi ⊗ yi

)

=

n∑i=1

rxi ⊗ ϕ(yi)

= r ·n∑i=1

xi ⊗ ϕ(yi)

= r · ϕ⊗

(n∑i=1

xi ⊗ yi

).

Drinfeld centers 16

Proposition 2.20. Define F on R-Mod by

i) M 7→ R⊗RM for modules.

ii) ϕ : M → N 7→ ϕ⊗ : R⊗RM → R⊗R N for homomorphisms.

Then F is an endofunctor of R-Mod.

Proof. For every module M , we have:

F (idM ) = idM⊗ = idR⊗RM .

For homomorphisms ϕ : M → N and ψ : N → L we have:

F (ψ ϕ)

(n∑i=1

xi ⊗ yi

)= (ψ ϕ)⊗

(n∑i=1

xi ⊗ yi

)

=

n∑i=1

xi ⊗ (ψ ϕ)(yi)

=

n∑i=1

xi ⊗ ψ(ϕ(yi))

= ψ⊗

(n∑i=1

xi ⊗ ϕ(yi)

)

= ψ⊗

(ϕ⊗

(n∑i=1

xi ⊗ yi

))

= F (ψ) F (ϕ)

(n∑i=1

xi ⊗ yi

).

Theorem 2.21. Let F be the functor defined in proposition 2.20. Then F ∼= idR-Mod.

Proof. Consider η : F → idR-Mod with components ηM : R⊗RM →M given by

ηM

(n∑i=1

xi ⊗ yi

)=

n∑i=1

xiyi.

This is an isomorphism of modules by proposition 2.18. The diagram

R⊗RMηM

ϕ⊗// R⊗R N

ηN

M ϕ// N

Markus Thuresson 17

commutes since

ϕ

(ηM

(n∑i=1

xi ⊗ yi

))= ϕ

(n∑i=1

xiyi

)

=n∑i=1

xiϕ(yi)

= ηN

(n∑i=1

xi ⊗ ϕ(yi)

)

= ηN

(ϕ⊗

(n∑i=1

xi ⊗ yi

))

so η is a natural isomorphism.

Theorem 2.22. Bimodule endomorphisms of R are exactly the components at R of the naturaltransformations in Z(R-Mod).

Proof. Let ϕ ∈ EndR−R(R) be an endomorphism of bimodules. Then, for every r ∈ R, we have

ϕ(r) = r · ϕ(1) and ϕ(r) = ϕ(1) · r.

This implies that ϕ is given by multiplication with the element z = ϕ(1) ∈ Z(R). We knowthat such endomorphisms are components of natural transformations from idR-Mod itself at R.Moreover, by lemma 2.13, Z(R-Mod) consists only of such natural transformations.

2.2.1 Tensor product of bimodules

Proposition 2.23. Let R,S, T be (unital) rings. Let RMS be an R-S-bimodule and let SNT bean S-T -bimodule. Then M ⊗S N is an R-T -bimodule, with scalar multiplication given by

r ·

n∑i=1

mi ⊗ ni =n∑i=1

rmi ⊗ nim∑j=1

mj ⊗ nj · t =m∑j=1

mj ⊗ nit∀r ∈ R, t ∈ T

Drinfeld centers 18

Proof.

r ·

n∑i=1

mi ⊗ ni +

m∑j=1

mj ⊗ nj

=

n∑i=1

rmi ⊗ ni +

m∑j=1

rmj ⊗ nj

= r ·n∑i=1

mi ⊗ ni + r ·m∑j=1

mj ⊗ nj

(r + r′)n∑i=1

mi ⊗ ni =n∑i=1

(r + s)mi ⊗ ni

=n∑i=1

(rmi + smi)⊗ ni

=n∑i=1

rmi ⊗ ni + r′mi ⊗ ni

= r ·n∑i=1

mi ⊗ ni + r′ ·n∑i=1

mi ⊗ ni

(rr′) ·n∑i=1

mi ⊗ ni =n∑i=1

(rr′)mi ⊗ ni

=n∑i=1

r(r′mi)⊗ ni

= r ·n∑i=1

r′mi ⊗ ni

1R ·n∑i=1

mi ⊗ ni =

n∑i=1

1Rmi ⊗ ni

=

n∑i=1

mi ⊗ ni

so we have left R-module structure by using the left R-module strucutre of M . The right

Markus Thuresson 19

T -module structure is checked similarly. Moreover,(r ·

n∑i=1

mi ⊗ ni

)· t =

(n∑i=1

rmi ⊗ ni

)· t

=n∑i=1

rmi ⊗ nit

= r ·

(n∑i=1

mi ⊗ nit

).

Proposition 2.24. If R,S, T, U are (unital) rings and we have bimodules RMS ,S NT ,T LU , then

(M ⊗S N)⊗T L ∼= M ⊗S (N ⊗T L) .

Lemma 2.25. Elements of the form (m⊗S n)⊗T l generate (M ⊗S N)⊗T L and elements ofthe form m⊗S (n⊗T l) generate M ⊗S (N ⊗T L).

Proof of lemma 2.25. Let S be the subgroup of (M ⊗S N) ⊗T L generated by the elements

(m ⊗S n) ⊗T l and let π be the projection onto the quotient (M ⊗S N)⊗T L/S . By the

universal property of the tensor product, there exists a unique homomorphism ϕ which makesthe following diagram commute:

(M ⊗S N)× L ⊗T //

0 ))

(M ⊗S N)⊗T L

ϕ

(M ⊗S N)⊗T L/S

Clearly, we have 0 = 0 ⊗T . But since ((m⊗S n)⊗T l) ∈ S we have

π ⊗T ((m⊗ n), l) = π((m⊗S n)⊗T l) = 0 =⇒ π = 0 =⇒ S = (M ⊗S N)⊗T L.

The second statement can be proved in the same way.

Proof of proposition 2.24. Consider the map

f : (M ⊗S N)⊗T L→M ⊗S (N ⊗T L)

defined by

n∑i=1

(mi ⊗S ni)⊗T li 7→n∑i=1

mi ⊗S (ni ⊗T li).

Drinfeld centers 20

f

n∑i=1

(mi ⊗S ni)⊗T li +m∑j=1

(mj ⊗S nj)⊗T lj

=n∑i=1

mi ⊗S (ni ⊗T li) +m∑j=1

mj ⊗S (nj ⊗T lj)

= f

(n∑i=1

(mi ⊗S ni)⊗T li

)+ f

m∑j=1

(mj ⊗S nj)⊗T lj

f

(r ·

n∑i=1

(mi ⊗S ni)⊗T li · u

)= f

(n∑i=1

r(mi ⊗S ni)⊗T liu

)

= f

(n∑i=1

(rmi ⊗S ni)⊗T liu

)

=n∑i=1

rmi ⊗S (ni ⊗T liu)

= r ·n∑i=1

mi ⊗S (ni ⊗T li) · u

= r · f

(n∑i=1

(mi ⊗S ni)⊗T li

)· u

for any r ∈ R, u ∈ U so f is a homomorphism of bimodules. Moreover, f is clearly invertiblewith inverse given by

n∑i=1

mi ⊗S (ni ⊗T li) 7→n∑i=1

(mi ⊗S ni)⊗ li

so f is an isomorphism.

Proposition 2.26. Let f :R MS →R M ′S and g : SNT → SN′T be bimodule homomorphisms.

Then the map f ⊗ g :R (M ⊗S N)T →R (M ′ ⊗S N ′)T defined by

n∑i=1

mi ⊗ ni 7→n∑i=1

f(mi)⊗ g(ni)

is a homomorphism of R-T -bimodules.

Markus Thuresson 21

Proof.

f ⊗ g

n∑i=1

mi ⊗ ni +m∑j=1

mj ⊗ nj

=n∑i=1

f(mi)⊗ g(ni) +m∑j=1

f(mj)⊗ g(nj)

= f ⊗ g

(n∑i=1

mi ⊗ ni

)+ f ⊗ g

m∑j=1

mj ⊗ nj

f ⊗ g

(r ·

n∑i=1

mi ⊗ ni · t

)= f ⊗ g

(n∑i=1

rmi ⊗ nit

)

=

n∑i=1

f(rmi)⊗ g(nit)

=

n∑i=1

rf(mi)⊗ g(ni)t

= r ·n∑i=1

f(mi)⊗ g(ni) · t

= r · f ⊗ g

(n∑i=1

mi ⊗ ni

)· t.

3 2-categories

If we consider a category where the morphism classes are themselves equipped with the structureof a category, we arrive at the notion of a 2-category. Much of this section follows from [1], withmore details spelled out.

Definition 3.1. A 2-category C consists of the following:

- a class Ob (C) of objects.

- for every pair of objects X,Y , a small category C(X,Y ), also called the hom-category. Theobjects f, g : X → Y of this category are the morphisms from X to Y , called 1-morphisms.Its morphisms α : f ⇒ g are called 2-morphisms. The composition of this category is denotedby • and is also called vertical composition.

- for every object X, a functor IX from the terminal category 111 to C(X,X). This functor mapsthe object of 111 to the identity 1-morphism idX : X → X and the morphism of 111 to the identity2-morphism idf : f ⇒ f .

Drinfeld centers 22

- for all objects X,Y, Z a bifunctor : C(Y,Z)×C(X,Y )→ C(X,Z). This functor is also calledhorizontal composition.

Example 3.2 (Vertical composition). For objects X,Y , 1-morphisms f, g, h and 2-morphismsα, β as in

f

α

X g

β

// Y

h

GG

we get the 2-morphism β • α : f ⇒ h as in

f

β•α

X Y

h

FF

Example 3.3 (Horizontal composition). For f, f ′, g, g′, α, β as in

f

α

g

β

X Y Z

f ′

FF

g′

FF

we get the 2-morphism β α : g f ⇒ g′ f ′ as in

g f

βα

X Z

g′ f ′

DD

Remark 3.4 (Interchange law). Since is a functor, it commutes with the (vertical) compositionof the hom-categories, so we have, for composable 2-morphisms as in

f

α

f ′

α′

X g //

β

Y g′ //

β′

Z

h

GG

h′

GG

Markus Thuresson 23

we have

(β′ β) • (α′ α) = (β′ • α′) (β • α).

Or, in terms of diagrams:

f ′ f

α′α

X g′ g //

β′β

Z

h′ h

DD=

f

β•α

f ′

β′•α′

X Y Z

h

GG

h′

FF

The left diagram corresponds to the left hand side of the above equation and the right diagramcorresponds to the right hand side of the above equation.

Example 3.5. An intuitive and motivating example of a 2-category is Cat, where the objectsare categories, the 1-morphisms are functors and the 2-morphisms are natural transformations.

Remark 3.6. [1] This structure also enables us to horizontally compose 2-morphisms with 1-morphisms, by composing the 2-morphism with the with the identity 2-morphism on the 1-morphism. We denote the horizontal composition of 1- and 2-morphisms by juxtaposition. For1-morphisms f, g, h and a 2-morphism α we get the following diagram for hα:

f

α

h

idh

X Y Z

g

GG

h

GG =

h f

idh •α

X Z

h g

CC

It is clear that we also can compose in the opposite direction.

Next we define the 2-categorical notions corresponding to functors and natural transforma-tions.

Definition 3.7. Let C and D be 2-categories. Then a 2-functor F : C → D consists of a tripleof maps; a map of objects, a map of 1-morphisms and a map of 2-morphisms, satisfying thefollowing:

i) For every object X we have F (idX) = idF (X) and for every 1-morphism f we have F (idf ) =idF (f) .

ii) For composable 1-morphisms f, g we have F (g f) = F (g) F (f).

iii) For horizontally composable 2-morphisms α, α′, we have F (α′ α) = F (α′) F (α).

iv) For vertically composable 2-morphisms β, β′ we have, F (β′ • β) = F (β′) • F (β).

Drinfeld centers 24

Remark 3.8. Note that a 2-functor F : C → D, when applied to the objects and 1-morphisms isan ordinary functor between the categories formed by the objects and 1-morphisms of C and D,so we can think of a 2-functor as an extension of an ordinary functor, respecting the additionalstructure of a 2-category.

Definition 3.9. Let C and D be 2-categories and let F,G : C → D be 2-functors. Then a2-natural transformation η : F → G is a map sending every object X of C to a 1-morphismηX : F (X)→ G(X) such that for 1-morphisms f, g : X → Y and every 2-morphism α : f ⇒ Gthe following holds

F (f)

F (α)

F (X) F (Y )ηY // G(Y )

F (g)

AA=

G(f)

G(α)

F (X)ηX // G(X) G(Y )

G(g)

AA

Remark 3.10. If we consider the identity 2-morphism on f the above diagram becomes

F (X)F (f)

// F (Y )ηY // G(Y ) = F (X)

ηX // G(X)G(f)

// G(Y )

which is just the usual naturality square

F (X)

ηX

F (f)// F (Y )

ηY

G(X)G(f)

// G(Y )

for a natural transformations between the ordinary functors of F and G, so in the same wayas with 2-functors, 2-natural transformations can be seen as extensions of ordinary naturaltransformations to the 2-categorical framework.

4 Bicategories

If we weaken the requirements on 2-categories, by instead of requiring associativity of thehorizontal composition, require associativity up to a natural isomorphism, we arrive at thenotion of a bicategory. This section essentially follows from [2], but with more details spelledout.

4.1 Basics

Definition 4.1. A bicategory B consists of the following:

Markus Thuresson 25

- a class Ob (B) of objects.

- for every pair X,Y of objects, a small hom-category B(X,Y ). We denote its (vertical) com-position by •.

- for every object X, a functor IX from 111 to B(X,X) as in the definition of a 2-category.

- for ordered triples of objects X,Y, Z, a bifunctor ? : B(Y, Z) × B(X,Y ) → B(X,Z). For no-tational convenience, we denote the horizontal composition of 1-morphisms by juxtaposition,so for 1-morphisms f, g and 2-morphisms α, β we get

? : B(Y, Z)× B(X,Y )→ B(X,Z)

(g, f) 7→ gf

(β, α) 7→ β ? α.

We might denote this specific bifunctor by ?XY Z . Here we differ from the definition of a 2-category. We do not require associativity of ?, we only require it up to a natural isomorphism.

This is made precise in the following way:

- for objects X,Y, Z,W , a natural isomorphism αXY ZW as given in

B(Z,W )× B(Y, Z)× B(X,Y )

?Y ZW×idB(X,Y )

idB(Z,W )×?XY Z// B(Z,W )× B(X,Z)

?XZW

B(Y,W )× B(X,Y )

αXY ZW

55

?XY W

// B(X,W )

called the associator. For 1- and 2-morphisms as given in

f

ϕ

g

ψ

h

ξ

X Y Z W

f ′

FF

g′

FF

h′

FF

the naturality of α yields the commutative diagram

(hg)f(ξ?ψ)?ϕ

//

αhgf

(h′g′)f ′

αh′g′f ′

h(gf)ξ?(ψ?ϕ)

// h′(g′f ′)

Drinfeld centers 26

which means that for composable 1-morphisms h, g, f we have an invertible 2-morphism

αhgf : (hg)f ⇒ h(gf).

- for each pair X,Y of objects, natural isomorphisms λXY and ρXY as given in

111× B(X,Y )

IY ×idB(X,Y )

∼

&&

B(Y, Y )× B(X,Y ) ?XY Y

//

λXY

88

B(X,Y )

andB(X,Y )× 111

idB(X,Y )×IX

∼

&&

B(X,Y )× B(X,X) ?XY Y

//

ρXY

88

B(X,Y )

called left and right unitors, respectively. So for a 1-morphism f ∈ B(X,Y ), we have invertible2-morphisms

λf : idY f ⇒ f

ρf : f idX ⇒ f.

Finally, we require the two following diagrams commute for composable 1-morphisms f, g, h, k.

((kh)g)f

α

~~

α?id // (k(hg))f

α

(kh)(gf)

α''

k((hg)f)

id ?αww

k(h(gf))

(gI)f

ρ?id""

α // g(If)

id ?λ||

gf

Remark 4.2. It is clear that if the natural isomorphisms α, λ, ρ are all identities, in which casethe composition is strictly associative, then the definition of bicategory coincides with that ofa 2-category.

Markus Thuresson 27

Definition 4.3. Let B be a bicategory. An internal equivalence in B consists if a pair of1-morphisms as given in

Xf))Y

gii

together with an isomorphism idX∼=⇒ gf in the hom-category B(X,X) and an isomorphism

fg∼=⇒ idY in the hom-category B(Y, Y ). We say that X and Y are equivalent inside B.

Definition 4.4. Let B and C be bicategories. A lax functor (F,ϕ) from B to C consists of thefollowing:

- a map F : Ob (B)→ Ob (C) of objects

- for objects X,Y ∈ Ob (B), a functor of hom-categories

FXY : B(X,Y )→ C(F (X), F (Y ))

- for objects X,Y, Z ∈ Ob (B), a natural transformation ϕXY Z as given in

B(Y, Z)× B(X,Y )?B //

FY Z×FXY

B(X,Z)

FXZ

C(F (Y ), F (Z))× C(F (X), F (Y )) ?C//

ϕXY Z

55

C(F (X), F (Z))

which, for composable 1-morphisms f, g gives the 2-morphism

ϕgf : F (g)F (f)⇒ F (gf)

and a natural transformation ϕX as given in

B(X,X)

FXX

111

IX

;;

IF (X)

//

ϕX

55

C(F (X), F (X))

which gives the 2-morphism ϕX : idF (X) ⇒ F (idX).

We require that the following diagrams commute for composable 1-morphisms f, g, h, denotingthe associators in the categories B, C by αB and αC respectively:

(F (h)F (g))F (f)ϕ?id

//

αC

F (hg)F (f)ϕ// F ((hg)f)

F (αB)

F (h)(F (g)F (f))id ?ϕ

// F (h)F (gf) ϕ// F (h(gf))

Drinfeld centers 28

F (f) idF (X)id ?ϕ

//

ρF (f)''

F (f)F (idX)ϕ// F (f idX)

F (ρf )ww

F (f)

idF (Y ) F (f)ϕ?id//

λF (f) ''

F (idY )F (f)ϕ// F (idY f)

F (λf )ww

F (f)

Definition 4.5. If for some property of functors every functor FXY has this property, we saythat the lax functor F locally has this property. For example, a lax functor might be locallyfull.

Definition 4.6. If (F,ϕ) is a lax functor such that all the natural transformations ϕXY Z andϕX are natural isomorphisms, then (F,ϕ) is called a pseudofunctor.

Definition 4.7. If (F,ϕ) is a lax functor such that all the natural transformations ϕXY Z andϕX are identities, then (F,ϕ) is called a strict 2-functor.

Definition 4.8. Let (F,ϕ) and (G,ψ) be lax functors from B to C. Then a lax natural trans-formation η consists of the following:

- for each X ∈ Ob (B), a 1-morphism ηX : F (X)→ G(X).

- natural transformations as given in:

B(X,Y )FXY //

GXY

C(F (X), F (Y ))

ηY

C(G(X), G(Y ))

ηXY

55

ηX// C(F (X), G(Y ))

so we have a 2-morphism

ηf : G(f)ηX ⇒ ηY F (f).

Additionally, we require that the following diagrams commute for composable 1-morphisms f, g:

Markus Thuresson 29

(G(g)G(f)) ηXαC //

ψ?id

G(g) (G(f)ηX)id ?ηf

// G(g) (ηY F (f))α−1C // (G(g)ηY )F (f)

ηg?id

(ηZF (g))F (f)

αC

ηZ (F (g)F (f))

id ?ϕ

G(gf)ηX ηgf// ηZF (gf)

idG(X) ηX

ψ?id

λC // ηXρ−1C // ηX idF (X)

id ?ϕ

G(idX)ηX ηidX// ηXF (idX)

Definition 4.9. If η is a lax natural transformation such that the natural transformations ηXYare all natural isomorphisms, then η is called a pseudonatural transformation.

Definition 4.10. Let η and µ be lax natural transformations between the lax functors (F,ϕ), (G,ψ)from B to C. Then a modification Γ : η → µ consists of 2-morphisms ΓX : ηX ⇒ µX such thatthe following diagram commutes:

G(f)ηXid ?ΓX //

ηf

G(f)µX

µf

ηY F (f)ΓY ?id

// µY F (f)

Example 4.11. There is a bicategory Bimod whose objects are rings, 1-morphisms are bimod-ules and 2-morphisms are bimodule homomorphisms. Then a typical structure in Bimod wouldlook like this:

RMS

ϕ

SM′T

ϕ′

R RNS

//

ψ

S SN′T

//

ψ′

T

RLS

EE

SL′T

EE

Composition of 1-morphisms is given by the bimodule tensor product and composition of 2-morphisms is just composition of bimodule homomorphisms. By proposition 2.23, the tensorproduct behaves nicely with respect to the bimodule structure, so the composite would look like:

Drinfeld centers 30

R(M ⊗S M ′)T

!!

ϕ⊗ϕ′

R R(N ⊗S N ′)T //

ψ⊗ψ′

T

R(L⊗S L′)T

==

For larger composites, we have the required associativity up to isomorphism by proposition 2.24,that is

(M ⊗S N)⊗T L ∼= M ⊗S (N ⊗T L) .

Example 4.12. If B is a bicategory, we may form a new bicategory Bop by reversing the1-morphisms. So the diagram

f

α

X Y

g

FF

in B becomes the diagram

f

α

X Y

g

XX

in Bop.

Example 4.13. For bicategories B and C, there is a functor bicategory Lax(B, C). Its ob-jects are lax functors, the 1-morphisms are lax natural transformations and the 2-morphismsare modifications. It has a sub-bicategory [B, C] consisting of pseudofunctors, pseudonaturaltransformations, and modifications.

Proposition 4.14. Let B be a bicategory and C a 2-category. Then Lax(B, C) is a 2-category.

Markus Thuresson 31

Proof. Suppose we have lax functors and natural transformations as given in

F

αG

β

B C

H

DD

γL

JJ

Then the composition of 1-morphisms in Lax(B, C) is given by the composition of the transfor-mations α, β, γ. But since these are transformations, this is just the componentwise composition.So for an object X ∈ Ob (B), we have 1-morphisms

F (X)αX // G(X)

βX // H(X)γX // L(X)

but these components are 1-morphisms of the 2-category C, and this composition is associative,so we get (γβ)α = γ(βα). By the same argument we have α id = α = idα for any transformationα.

Similarily, if we have lax functors, natural transformations and modifications as given in

B

α

Γ

β

(Σ

G

""

F

||

γ

6>

Ω

δ

>F

C

we get 2-morphisms

αXΓX +3 βX

ΣX +3 γXΩX +3 δX

which now are 2-morphisms of C and again this yields associativity. It is clear that we by thesame argument have Γ ? id = Γ = id ?Γ for any modification Γ.

Drinfeld centers 32

4.2 Coherence

Definition 4.15. Let B and C be bicategories. A biequivalence of B and C consists of a pair ofpsuedofunctors

BF(( C

G

ii

together with an internal equivalence idB → GF in [B,B] and an internal equivalence FG→ idCin [C, C]. It can be shown that a pseudofunctor F : B → C admits a biequivalence if and only ifF is a local equivalence and if for every Y ∈ Ob (B) there exists an X ∈ Ob (C) such that F (X)is internally equivalent to Y .

Example 4.16. Let C be a category.We define the bicategory X as follows: it has only one object and only one 1-morphism, idC.

Its 2-morphisms are natural transformations.We define the bicategory Y as follows: it has only one object. Its 1-morphisms are functors

isomorphic to idC. Its 2-morphisms are natural transformations.Then we have a clear embedding F : X → Y. Clearly, F is a pseudofunctor which is surjective

on objects.The induced functor of the hom-categories is clearly dense, since it sends idC to itself and

every 1-morphism in Y is isomorphic to idC by construction. Moreover, it is faithful since it isan inclusion on 1-morphisms, and it is full since the 2-morphisms of X and Y are the same.

So F is a local equivalence and hence a biequivalence.

The following result is a version of the Yoneda lemma for bicategories, which we state withoutproof.

Theorem 4.17 (Yoneda lemma for bicategories). Let B be a bicategory and let F : Bop → Catbe a pseudofunctor. Then, for any X ∈ Ob (B), there is an equivalence of categories

[Bop,Cat] (B( , X), F ) ' F (X)

which is pseudonatural in X and in F .

From the Yoneda lemma it follows that there is an analogue of the usual Yoneda embedding.This means that we have a pseudofunctor

Y : B → [Bop,Cat]

which is locally full, faithful and dense. In other words, Y is a local equivalence.

Theorem 4.18. Let B be a bicategory. Then B is biequivalent to a 2-category.

Proof. Let Y be the Yoneda pseudofunctor and let C be the image of Y in [Bop,Cat]. By thiswe mean that C is the sub-2-category of [Bop,Cat] whose objects are the objects in the imageof Y, with all 1- and 2-morphisms of [Bop,Cat]. Then, seen as a psuedofunctor Y : B → C,we have that Y is surjective on objects by construction and a local equivalence, so it is abiequivalence.

Markus Thuresson 33

5 Monoidal categories

A monoidal category is usually defined as a category equipped with a tensor product. So for acategory C we would define the tensor product as a bifunctor

⊗ : C × C → C

obeying certain axioms.

However, thanks to the previous section, we can simply define a monoidal category as thehom-category of a bicategory with one single object.

Then, taking the tensor product as horizontal composition and the identity 1-morphism asthe tensor unit, the associator and unitor isomorphisms together with their coherence axiomsyield exactly the standard definition of monoidal category.

In the same way, we effortlessly get the definitions of a monoidal functor and a monoidal trans-formation from the definitions of lax functors and lax natural transformations in the previoussection.

A monoidal category where the associator and unitors are all identities, is, unsurprisingly,called a strict monoidal category. This section essentially follows from [3], with more detailsspelled out.

Definition 5.1. Let C be a monoidal category. Then a braiding β of C is a natural isomorphismwith components

βX,Y : X ⊗ Y → Y ⊗X.

We require that the braiding satisfies the hexagon identities, given by the following commutativediagrams:

X ⊗ (Y ⊗ Z)β// (Y ⊗ Z)⊗X

α

))

(X ⊗ Y )⊗ Z

α55

β⊗id ))

Y ⊗ (Z ⊗X)

(Y ⊗X)⊗ Z α// Y ⊗ (X ⊗ Z)

id⊗β

55

(X ⊗ Y )⊗ Z β// Z ⊗ (X ⊗ Y )

α−1

))

X ⊗ (Y ⊗ Z)

α−155

id⊗β ))

(Z ⊗X)⊗ Y

X ⊗ (Z ⊗ Y )α−1// (X ⊗ Z)⊗ Y

β⊗id

55

A monoidal category C together with chosen braiding is called a braided monoidal category.

Drinfeld centers 34

Definition 5.2. A braided monoidal category is called symmetric if the braiding satisfies

βY,X βX,Y = idX⊗Y

for every pair of objects.

5.1 The Drinfeld center

Definition 5.3. Let C be a monoidal category. Then we define its Drinfeld center Z(C ) asthe following monoidal category:

- objects are pairs (X, ηX, ) where X is an object of C and ηX, is a natural isomorphism

ηX, : X ⊗ → ⊗X

such that

ηX,Y⊗Z = (idY ⊗ηX,Z)(ηX,Y ⊗ idZ).

The naturality of ηX, yields commutativity of the square

X ⊗ Y idX ⊗g //

ηX,Y

X ⊗ ZηX,Z

Y ⊗Xg⊗idX

// Z ⊗X

for any morphism g : Y → Z.

- a morphism f : (X, ηX, )→ (Y, ηY, ) is a morphism f : X → Y in C such that

(idZ ⊗f)ηX,Z = ηY,Z(f ⊗ idZ)

for every Z ∈ C . This is equivalent to the square

X ⊗ ZηX,Z

f⊗idZ // Y ⊗ ZηY,Z

Z ⊗XidZ ⊗f

// Z ⊗ Y

commuting for every Z ∈ C .

- the tensor product of Z(C ) is given by

(X, ηX, )⊗ (Y, ηY, ) = (X ⊗ Y, ηX⊗Y, )

where ηX⊗Y,Z : (X ⊗ Y )⊗ Z → Z ⊗ (X ⊗ Y ) is given by

ηX⊗Y,Z = (ηX,Z ⊗ idY )(idX ⊗ηY,Z).

Markus Thuresson 35

Remark 5.4. The conditions

ηX,Y⊗Z = (idY ⊗ηX,Z)(ηX,Y ⊗ idZ)

ηX⊗Y,Z = (ηX,Z ⊗ idY )(idX ⊗ηY,Z)

are is not quite correct. In the above definition, we have left out some associators necessaryto make sense of the equations. With the associators spelled out, the conditions amount tocommutativity of the following diagrams:

(Y ⊗X)⊗ Z α // Y ⊗ (X ⊗ Z)idY ⊗ηX,Z

((

(X ⊗ Y )⊗ Z

ηX,Y ⊗idZ

66

Y ⊗ (Z ⊗X)

α−1

X ⊗ (Y ⊗ Z)

α−1

OO

ηX,Y⊗Z

// (Y ⊗ Z)⊗X

X ⊗ (Z ⊗ Y )α−1// (X ⊗ Z)⊗ Y

ηX,Z⊗idY

((

X ⊗ (Y ⊗ Z)

idX ⊗ηY,Z

66

(Z ⊗X)⊗ Y

α

(X ⊗ Y )⊗ Z

α

OO

ηX⊗Y,Z

// Z ⊗ (X ⊗ Y )

which in turn yield the accurate equations:

ηX,Y⊗Z = α−1Y ZX(idY ⊗ηX,Z)αY XZ(ηX,Y ⊗ idZ)α−1

XY Z

ηX⊗Y,Z = αZXY (ηX,Z ⊗ idY )α−1XZY (idX ⊗ηY,Z)αXY Z .

If, however, C is a strict monoidal category, then the previously stated conditions are just fine.

Example 5.5. Recall the category defined in example 1.19, the categorical equivalent of amonoid. We can impose a tensor product on M by putting • ⊗ • = • for the object of Mand x⊗ y = xy for the morphisms to get a strict monoidal category M .

Now we want to consider possible objects in the Drinfeld center Z (M ). They must be of theform (•, η•, ). Since • is the only object in M , η•, has only the component η•,•. Denoting thiscomponent by z, we require that the diagram

• ⊗ • id•⊗x//

z

• ⊗ •z

• ⊗ •x⊗id•

// • ⊗ •

Drinfeld centers 36

commutes for any x ∈M . But this diagram is actually just the diagram

• x //

z

•z

• x

// •

which we know commutes if and only if z ∈ Z(M). Since z must also be an isomorphism, werequire that z be invertible. Writing out the condition for a morphism x : • → • to be in Z(M )yields the same diagram as above, so we conclude that

HomM (•, •) = HomZ(M )(•, •).

So the Drinfeld center Z(M ) consists of objects of the form (•, z) where z ∈ M is invertibleand central. Morphisms (•, z)→ (•, c) are elements of M .

Proposition 5.6. Let C be a strict monoidal category. Then its Drinfeld center Z(C ) is astrict braided monoidal category, with braiding given by

ηX,Y : (X, ηX, )⊗ (Y,⊗ηY, )→ (Y,⊗ηY, )⊗ (X, ηX, ).

Proof. We check that ηX,Y is indeed a morphism in Z(C ). We have

(X, ηX, )⊗ (Y, ηY, ) = (X ⊗ Y, ηX⊗Y, )

(Y, ηY, )⊗ (X, ηX, ) = (Y ⊗X, ηY⊗X, )

so ηX,Y is a morphism between the correct objects of C . The criterion for ηX,Y being a morphismin Z(C ) is

(idZ ⊗ηX,Y )ηX⊗Y,Z = ηY⊗X,Z(ηX,Y ⊗ idZ).

We have

(idZ ⊗ηX,Y )ηX⊗Y,Z = (idZ ⊗ηX,Y )(ηX,Z ⊗ idY )(idX ⊗ηY,Z)

= ηX,Z⊗Y (idX ⊗ηY,Z)

= (ηY,Z ⊗ idX)ηX,Y⊗Z

= (ηY,Z ⊗ idX)(idY ⊗ηX,Z)(ηX,Y ⊗ idZ)

= ηY⊗X,Z(ηX,Y ⊗ idX).

Equivalently, we can show that the diagram

X ⊗ Y ⊗ ZidX ⊗ηY,Z

//

ηX,Y⊗Z

))

ηX,Y ⊗idZ

ηX⊗Y,Z

))

X ⊗ Z ⊗ YηX,Z⊗idY

//

ηX,Z⊗Y

))

Z ⊗X ⊗ YidZ ⊗ηX,Y

Y ⊗X ⊗ ZidY ⊗ηX,Z

//

ηY⊗X,Z

55Y ⊗ Z ⊗X

ηY,Z⊗idX

// Z ⊗ Y ⊗X

Markus Thuresson 37

commutes. It does, since the triangles commute by our conditions and the center parallelogramis just a naturality square of ηX, . Left to check are the hexagon identities. In a strict monoidalcategory, these are equivalent to the diagrams

X ⊗ Y ⊗ ZηX,Y⊗Z

//

ηX,Y ⊗idZ

Y ⊗ Z ⊗X

Y ⊗X ⊗ ZidY ⊗ηX,Z

66X ⊗ Y ⊗ Z

ηX⊗Y,Z//

idX ⊗ηY,Z

Z ⊗X ⊗ Y

X ⊗ Z ⊗ YηX,Z⊗idY

66

commuting, which they do by our definition of Z(C ).

Remark 5.7. This result holds even for non-strict monoidal categories. That is, for any monoidalcategory C , its Drinfeld center Z(C ) is a braided monoidal category. The proof, however, isnot given here.

6 The Drinfeld center of VectC

Now we consider the category VectC, consisting of finite-dimensional vector spaces over C andlinear maps between them. We impose the structure of a monoidal category on VectC usingthe usual tensor product of vector spaces.

Note that for vector spaces V and W , there is a canonical isomorphism V ⊗W → W ⊗ V ,defined by v ⊗ w 7→ w ⊗ v. For any pair of vector spaces in, let ΦV,W denote this isomorphism.

Proposition 6.1. Let V be a finite-dimensional complex vector space and let

ΦV, : V ⊗ → ⊗ V

have components given by the canonical isomorphism ΦV,W . Then the pair (V,ΦV, ) is inZ(VectC).

Proof. We need the square

V ⊗W idV ⊗F //

ΦV,W

V ⊗XΦV,X

W ⊗ VF⊗idV

// X ⊗ V

to commute for any linear map F : W → X.

(F ⊗ idV ) (ΦV,W (v ⊗ w)) = (F ⊗ idV )(w ⊗ v)

= F (w)⊗ v= ΦV,X (v ⊗ F (w))

= ΦV,X ((idV ⊗F )(v ⊗ w))

Drinfeld centers 38

so we get a natural family of isomorphisms. Left to check is the condition

ΦV,W⊗X = α−1WXV (idW ⊗ΦV,X)αWVX(ΦV,W ⊗ idX)α−1

VWX .

α−1(idW ⊗ΦV,X)α(ΦV,W ⊗ idX)α−1(v ⊗ (w ⊗ x)) = α−1(idW ⊗ΦV,X)α(ΦV,W ⊗ idX)((v ⊗ w)⊗ x)

= α−1(idW ⊗ΦV,X)α((w ⊗ v)⊗ x)

= α−1(idW ⊗ΦV,X)((w ⊗ (v ⊗ x))

= α−1(w ⊗ (x⊗ v))

= (w ⊗ x)⊗ v= ΦV,W⊗X(v ⊗ (w ⊗ x)).

This condition is also easily checked by chasing the element (v⊗(w⊗x)) through the diagrambelow.

(W ⊗ V )⊗X α //W ⊗ (V ⊗X)idW ⊗ΦV,X

((

(V ⊗W )⊗X

ΦV,W⊗idX

66

W ⊗ (X ⊗ V )

α−1

V ⊗ (W ⊗X)

α−1

OO

ΦV,W⊗X

// (W ⊗X)⊗ V

To avoid notational clutter, from this point forward, we drop the indices of morphisms whendomain and codomain are clear from context.

Proposition 6.2.

HomZ(VectC) ((V,Φ), (W,Φ)) = HomVectC (V,W ) .

Proof. By definition, HomZ(VectC) ((V,Φ), (W,Φ)) consists of linear maps F : V →W such thatthe diagram

V ⊗X F⊗id//

Φ

W ⊗X

Φ

X ⊗ Vid⊗F

// X ⊗W

for every X. We see that

Φ(F ⊗ id)(v ⊗ x) = Φ(F (v)⊗ x)

= x⊗ F (v)

= (id⊗F )(x⊗ v)

= (id⊗F )Φ(v ⊗ x)

holds for any linear map F .

Markus Thuresson 39

Corollary 6.3. If V and W are isomorphic as vector spaces, then (V,Φ) and (W,Φ) are iso-morphic as objects of Z(VectC).

Proof. Any invertible linear map G : V → W is in HomZ(VectC) ((V,Φ), (W,Φ)) by proposition6.2 and, similarly, its inverse is in HomZ(VectC) ((W,Φ), (V,Φ)), so G is an isomorphism inZ(VectC).

Proposition 6.4. If (C,Ψ) is in Z(VectC), then Ψ = Φ.

Proof. We consider the component of Ψ at the vector space V . Fix the basis 1 of C and thebasis vi of V . Then 1 ⊗ vi is a basis of C ⊗ V and vi ⊗ 1 is a basis of V ⊗ C. Putn = dimV . Let F : V → V be some linear map. Then, the square

C⊗ V id⊗F//

Ψ

C⊗ V

Ψ

V ⊗ CF⊗id

// V ⊗ C

commutes so we have (F ⊗ id)Ψ = (id⊗F )Ψ. Note that with respect to our bases, we have

[id⊗F ] = [F ⊗ id] = [F ].

This means that, in terms of matrices, we have the equation [F ][Ψ] = [Ψ][F ]. Since this musthold for any linear map F , we see that [Ψ] is a matrix that commutes with every other matrix.From linear algebra, we know that such a matrix must be a scalar multiple of the identity matrix.Now it follows that Ψ is a (non-zero) scalar multiple of the canonical isomorphism C⊗V → V⊗C,say Ψ = λV Φ for some complex number λV . So now we know that Ψ : C⊗ V → V ⊗C is givenby

1⊗ v 7→ λV v ⊗ 1.

We note that the diagram

C⊗ V id⊗F//

Ψ

C⊗WΨ

V ⊗ CF⊗id

//W ⊗ C

must commute for any vector space W and any linear map F : V →W . So we must have

λV F (v)⊗ 1 = (F ⊗ id)(λV v ⊗ 1)

= (F ⊗ id)Ψ(1⊗ v)

= Ψ(id⊗F )(1⊗ v)

= Ψ(1⊗ F (v))

= λWF (v)⊗ 1

Drinfeld centers 40

which implies λV = λW . So the constant λV is the same across every vector space. To reflectthis, we drop the index and put λ := λV . Moreover, since (C,Ψ) is in Z(VectC), the diagram

(V ⊗ C)⊗W α // V ⊗ (C⊗W )

id⊗Ψ

((

(C⊗ V )⊗W

Ψ⊗id66

V ⊗ (W ⊗ C)

α−1

C⊗ (V ⊗W )

α−1

OO

Ψ// (V ⊗W )⊗ C

commutes. This is equivalent to

λ(v ⊗ w)⊗ 1 = Ψ(1⊗ (v ⊗ w))

= α−1(id⊗Ψ)α(Ψ⊗ id)α−1(1⊗ (v ⊗ w))

= α−1(id⊗Ψ)α(Ψ⊗ id)((1⊗ v)⊗ w)

= α−1(id⊗Ψ)α((λv ⊗ 1)⊗ w)

= α−1(id⊗Ψ)(λv ⊗ (1⊗ w))

= α−1(λ(n)v ⊗ (λw ⊗ 1))

= (λv ⊗ λw)⊗ 1

= λ2(v ⊗ w)⊗ 1

which implies λ2 = λ and since λ 6= 0, we have λ = 1.

Corollary 6.5. If (X,Ψ) is in Z(VectC) and dimX = 1, then Ψ = Φ.

Proof. Follows immediately from the proof of the case X = C.

Now that we’ve established the behavior of objects of Z(VectC) of the form (C,Ψ), we seekto generalize the previous arguments higher dimensions. Our first objects of study are elementsof the form

(C2,Ψ

). Throughout this section, we fix the standard basis of Ck. If V is some

vector space with a basis v1, . . . , vn, we fix the basis

e1 ⊗ v1, . . . , e1 ⊗ vn, . . . , ek ⊗ v1, . . . , ek ⊗ vn

of Ck ⊗ V and the basis

v1 ⊗ e1, . . . , vn ⊗ e1, . . . , v1 ⊗ ek, . . . , vn ⊗ ek

of V ⊗ Ck. To help us classify objects of the form(C2,Ψ

)in Z(VectC), we have the following

lemma.

Markus Thuresson 41

Lemma 6.6. If(C2,Ψ

)∈ Z(VectC), then the component of Ψ

ΨX : C2 ⊗X → X ⊗ C2

is given by the (invertible) matrix

[ΨX ] =

[aXIdimX bXIdimX

cXIdimX dXIdimX

]with respect to the bases chosen above, for any vector space X. In particular, the component of

Ψ at the vector space C is given by an invertible 2× 2 matrix

[aC bCcC dC

].

Proof. Consider the component of Ψ at X. Let n = dimX and let F : X → X be a linear mapgiven by the n× n matrix [F ]. Note that C2 ⊗X and X ⊗C2 have dimension 2n, so the linearisomorphism Ψ is given by a 2n × 2n matrix [Ψ]. We write the matrix of Ψ as a 2 × 2 block

matrix with blocks of size n× n, so we have [Ψ] =

[A BC D

]. By naturality of Ψ, the square

C2 ⊗ V id⊗F//

Ψ

C2 ⊗ VΨ

V ⊗ C2F⊗id

// V ⊗ C2

commutes. With respect to the chosen bases, we have

[id⊗F ] = [F ⊗ id] =

[[F ] 00 [F ]

].

Naturality then amounts to the matrix equation[A BC D

] [[F ] 00 [F ]

]=

[[F ] 00 [F ]

] [A BC D

]which yields the following relations among the blocks:

A[F ] = [F ]A, B[F ] = [F ]B, C[F ] = [F ]C, D[F ] = [F ]D.

By the same argument as in the proof of proposition 6.4, we have

A = aXIn, B = bXIn, C = cXIn, D = dXIn.

Now it is clear that

[Ψ] =

[A BC D

]=

[aXIn bXIncXIn dXIn

].

Drinfeld centers 42

More explicitly, Ψ has the matrix

[Ψ] =

aX 0 . . . bX 0 . . .0 aX 0 bX...

. . ....

. . .

cX 0 . . . dX 0 . . .0 cX 0 dX...

. . ....

. . .

.

Proposition 6.7. If(C2,Ψ

)is in Z(VectC) then Ψ = Φ.

Proof. By naturality, the square

C2 ⊗ V id⊗F//

Ψ

C2 ⊗WΨ

V ⊗ C2F⊗id

//W ⊗ C2

commutes for all vector spaces V,W and every linear map F : V → W . Let n = dimV andm = dimW . Fixing bases v1, . . . , vn and w1, . . . , wm of V and W respectively, we obtainbases for all involved vector spaces. Denote the component at V by ΨV and the component atW by ΨW . By lemma 6.6, we have

[ΨV ] =

[aV In bV IncV In dV In

]and [ΨW ] =

[aW Im bW ImcW Im dW Im

].

Now we note the following:

(i) the linear map F is given by an m× n matrix [F ],

(ii) the linear maps id⊗F and F ⊗ id are given by 2m× 2n matrices,

(iii) with respect to our bases we have

[id⊗F ] = [F ⊗ id] =

[[F ] 0m×n

0m×n [F ]

].

Here 0m×n denotes a m× n block of zeroes.

Commutativity of the square now yields:

(F ⊗ id)ΨV = ΨW (id⊗F ) ⇐⇒ [F ⊗ id][ΨV ] = [ΨW ][id⊗F ]

⇐⇒[

[F ] 0m×n0m×n [F ]

] [aV In bV IncV In dV In

]=

[aW Im bW ImcW Im dW Im

] [[F ] 0m×n

0m×n [F ]

]⇐⇒

[aV [F ] bV [F ]cV [F ] dV [F ]

]=

[aW [F ] bW [F ]cW [F ] dW [F ]

].

Markus Thuresson 43

Since this must hold for every matrix [F ] this implies

aV = aW , bV = bW , cV = cW , dV = dW .

Since V and W are arbitrary vector spaces this shows that these numbers are invariant acrossall vector spaces, so we may drop the indices. This is an improvement upon the result of lemma6.6, in which the numbers may depend on the vector space.

Since(C2,Ψ

)∈ Z(VectC), the diagram

(V ⊗ C2)⊗W α // V ⊗ (C2 ⊗W )

id⊗Ψ

((

(C2 ⊗ V )⊗W

Ψ⊗id66

V ⊗ (W ⊗ C2)

α−1

C2 ⊗ (V ⊗W )

α−1

OO

Ψ// (V ⊗W )⊗ C2

commutes. Putting V = W = C, this diagram becomes

(C⊗ C2)⊗ C α // C⊗ (C2 ⊗ C)

id⊗Ψ

((

(C2 ⊗ C)⊗ C

Ψ⊗id66

C⊗ (C⊗ C2)

α−1

C2 ⊗ (C⊗ C)

α−1

OO

Ψ// (C⊗ C)⊗ C2

and now we observe that, with respect to the chosen bases, we have

[α] = [α−1] = I2 and [Ψ⊗ id] = [id⊗Ψ] = [Ψ] =

[a bc d

].

This means that the commutativity of the above diagram is equivalent to the matrix equation[Ψ]2 = [Ψ]. Since [Ψ] is invertible we get

[Ψ]2 = [Ψ] ⇐⇒ [Ψ]2[Ψ]−1 = [Ψ][Ψ]−1

⇐⇒ [Ψ] = I2

⇐⇒ a = d = 1 and b = c = 0.

This means that for any vector space, the matrix of Ψ : C2 ⊗ → ⊗ C2 with respect to thechosen basis is the identity matrix. This shows that Ψ = Φ.

Finally, we want to mimic the approach taken in the two-dimensional case in order to establishthe following proposition.

Drinfeld centers 44

Proposition 6.8. If(Ck,Ψ

)is in Z(VectC), then Ψ = Φ.

Proof. First, we consider the component of Ψ at some vector space X. Let n = dimX. Bynaturality, the square

Ck ⊗X id⊗F//

Ψ

Ck ⊗XΨ

X ⊗ CkF⊗id

// X ⊗ Ck

commutes. Writing [Ψ] as a k × k block matrix with block size n× n, we have

[Ψ] =

A11 . . . A1k...

. . ....

Ak1 . . . Akk

and with respect to our bases we have

[id⊗F ] = [F ⊗ id] =

[F ] . . . 0...

. . ....

0 . . . [F ]

.Naturality now yields Aij [F ] = [F ]Aij for all i, j ∈ 1, . . . , k. This implies Aij = aXij In as inthe case k = 2. The square

Ck ⊗ V id⊗F//

Ψ

Ck ⊗WΨ

V ⊗ CkF⊗id

//W ⊗ Ck

commutes for all vector spaces V,W and every linear map F : V → W . The exact sametechnique as in the case k = 2 can be used to show that aVij = aWij for all V,W .

Putting V = W = C and using the commutativity of the diagram

(C⊗ Ck)⊗ C α // C⊗ (Ck ⊗ C)

id⊗Ψ

((

(Ck ⊗ C)⊗ C

Ψ⊗id66

C⊗ (C⊗ Ck)

α−1

Ck ⊗ (C⊗ C)

α−1

OO

Ψ// (C⊗ C)⊗ Ck

we get that the component at a one-dimensional vector space is given by [Ψ] = Ik which implies

aij =

0 if i 6= j1 if i = j

so Ψ acts via the identity matrix and hence Ψ = Φ.

Markus Thuresson 45

Corollary 6.9. If (V,Ψ) is in Z(VectC), then Ψ = Φ.

Proof. The proof of proposition 6.8 applies without modification.

6.1 Equivalence

Thanks to the results of the previous section, we can now completely describe the Drinfeldcenter of VectC.

- objects are pairs (V,Φ)

- morphisms from (V,Φ) to (W,Φ) are linear maps from V to W

- the tensor product is given by (V,Φ)⊗ (W,Φ) = (V ⊗W,Φ) and the tensor unit is (C,Φ).

Theorem 6.10. VectC and Z(VectC) are isomorphic as categories.

Proof. Define a map F : VectC → Z(VectC) as follows:

- V 7→ (V,Φ) for objects,

- HomVectC (V,W ) 3 f 7→ f ∈ HomZ(VectC) ((V,Φ) , (W,Φ)) for morphisms.

It is clear from the results of the previous sections that F is a bijection on objects and onmorphisms, hence an isomorphism of categories.

Denoting the tensor product of Z(VectC) by ⊗Z to distinguish it from the usual tensor productin VectC, we note the following properties of F :

F (V )⊗Z F (W ) = (V,Φ)⊗Z (W,Φ) = (V ⊗W,Φ) = F (V ⊗W )

F (C) = (C,Φ)

so F preserves the tensor product and the tensor unit.Now it can be easily checked that the pair (F, id) satisfies the axioms of a monoidal functor.

This fact together with theorem 6.10 shows that (F, id) is an isomorphism of monoidal categories.

7 Categories of group representations

Our next objects of study are suitable categories of group representations. The basics followfrom [4].

Definition 7.1. Let K be a field, V a vector space over K and let G be a group. Then arepresentation of V is a homomorphism of groups

ρ : G→ GL(V ).

Definition 7.2. Let K be a field, V a vector space over K and let G be a group. Then V is aG-module if there exists a linear action of G on V , that is, a map ϕ : G× V → V such that

Drinfeld centers 46

i) ϕ(g, (v + w)) = ϕ(g, v) + ϕ(g, w)

ii) ρ(g, λv) = λϕ(g, v)

iii) ϕ(gh, v) = ϕ(g, ϕ(h, v))

iv) ϕ(e, v) = v.

Proposition 7.3. The definitions 7.1 and 7.2 are equivalent.

Proof. Let ρ : G→ GL(V ) be a representation. Then, for any g ∈ G, ρ(g) is an invertible linearmap so defining the map ϕ : G× V → V by ϕ(g, v) = ρ(g)(v) defines a G-module structure onV . Let ϕ : G× V → V be a G-module structure. Define the map ρ : G→ GL(V ) by

g 7→ ϕ(g, ).

Then ρ is a homomorphism of groups:

ρ(gh) = ϕ(gh, )

= ϕ(g, ϕ(h, ))

= ρ(g)ρ(h).

Remark 7.4. Any field K viewed as a vector space over itself becomes a G-module for any groupG with action given by

g(x) = x,∀g ∈ G, x ∈ K.

This is called the trivial G-module.

Definition 7.5. Let V and W be G-modules. Then a G-homomorphism is a linear mapF : V →W such that

F (g(v)) = g(F (v))

for every g ∈ G and v ∈ V . In other words, the diagram

VF //

g·

W

g·

VF//W

commutes.

Definition 7.6. Let V be a G-module and let W ⊂ V be a linear subspace. Then W is asubmodule of V if w ∈W =⇒ gw ∈W for all g ∈ G.

Every G-module has two submodules: 0 and itself. These are called trivial submodules. Asubmodule which is not trivial is called proper.

Markus Thuresson 47

Proposition 7.7. Let V,W be G-modules and let F : V → W be a G-homomorphism. Thenker(F ) and im(F ) are submodules of V and W , respectively.

Proof. By linear algebra, we know that ker(F ) and im(F ) are subspaces. Left to check is thatthey are invariant under the action of G. Suppose v ∈ ker(F ). Then

F (gv) = g(F (v))

= g(0)

= 0.

Suppose w ∈ im(F ). Then there exists v0 ∈ V such that F (v0) = w. Then

F (gv0) = gF (v0)

= gw.

Definition 7.8. A G-module which has no proper submodules is said to be simple. Note thatany one-dimensional module is automatically simple.

Proposition 7.9. Let V and W be G-modules. Then the tensor product V ⊗W is a G-modulewith action given by

g(v ⊗ w) = g(v)⊗ g(w).

Proof. Clearly we can extend the action of g to sums of elements of the form v⊗w, which yieldsthe condition

g(v ⊗ w + v′ ⊗ w′) = g(v ⊗ w) + g(v′ ⊗ w′).

We then have

i)

g(λ(v ⊗ w)) = g(λv ⊗ w)

= g(λv)⊗ g(w)

= λg(v)⊗ g(w)

= λ(g(v)⊗ g(w))

ii)

(gh)(v ⊗ w)) = (gh)(v)⊗ (gh)(w)

= g(h(v))⊗ g(h(w))

= g(h(v)⊗ h(w))

Drinfeld centers 48

iii)

e(v ⊗ w) = e(v)⊗ e(w)

= v ⊗ w.

Proposition 7.10. Let V be a G-module and K the trivial G-module. Then there is an iso-morphism of G-modules

F : V ⊗K ∼−→ V.

Proof. Fix a basis vi and some k ∈ K. Then vi ⊗ k is a basis of V ⊗K and we know thatdefining F by vi ⊗ k 7→ vi yields a linear isomorphism. But we also have

g(F (v ⊗ k)) = g(v)

= F (g(v)⊗ k))

= F (g(v)⊗ g(k))

= F (g(v ⊗ k))

so F is an isomorphism of G-modules.

Remark 7.11. In a similar manner, it may be checked that the canonical isomorphism

F : (U ⊗ V )⊗W → U ⊗ (V ⊗W )

is in fact a G-isomorphism, when the action of G on the tensor product is defined as in propo-sition 7.9.

Proposition 7.12 (Maschke’s theorem). Let G be a finite group and K a field such thatchar(K) - |G|. Then any finite-dimensional G-module can be written as a direct sum of simplemodules.

Proof. If V is simple, we are done. If not, we use induction on the dimension of V . Sinceone-dimensional modules are automatically simple, it suffices to show that any submodule of Vhas a submodule complement.

Let Y be a submodule of V . Let X be a complement of Y , that is, a subspace such thatV = X ⊕Y . Note that X is not necessarily a submodule.Let P0 : V → V be a linear projectiononto Y .

Now, define the linear endomorphism P = 1|G|∑g∈G

g−1P0g.

Markus Thuresson 49

Then, for any y ∈ Y , we have

P (y) =1

|G|∑g∈G

g−1P0g(y)

=1

|G|∑g∈G

g−1P0(g(y))

=1

|G|∑g∈G

g−1g(y)

=1

|G|∑g∈G

y

= y

which follows from the fact that Y is a submodule and P0|Y = idY . Moreover, we have im(P ) =Y , so P is also a linear projection. It follows that we have a decomposition V = Z ⊕ Y , wherenow Z = ker(P ) and Y = im(P ). Next, we note that P is a G-homomorphism. Indeed,

P (h(v)) =1

|G|∑g∈G

g−1P0g(h(v))

=1

|G|∑g∈G

hh−1g−1Pgh(v)

=1

|G|h∑g∈G

(gh)−1P (gh)(v)

= hP (v).

Now Z is a submodule since it is the kernel of a G-homomorphism, so we have a submodulecomplement Z of Y , and we are done.

In what follows, we assume that K = C.

Proposition 7.13 (Schur’s lemma). Let V and W be simple G-modules. Then the followinghold:

i) Every G-homomorphism F : V →W is either zero or an isomorphism.

ii) If V = W , the only non-zero G-homomorphisms are scalar multiples of the identity.

Proof. i) Suppose F : V → W is non-zero. Since ker(F ) is a submodule, we have eitherker(F ) = 0 or ker(F ) = V since V is simple. Since F is non-zero, we have ker(F ) = 0which implies that F is injective.

Similarily, since im(F ) is a submodule, we have either im(F ) = 0 or im(F ) = W since Wis simple. Since F -is non-zero we have im(F ) = W , so F is surjective. So F is bijective, soit is an isomorphism.

Drinfeld centers 50

ii) Suppose F : V → V is non-zero. Since our base field is C, the map F has an eigenvalue,λ, with corresponding eigenvector v. Put F ′ = F − λ idV . Then F ′(v) = 0.

Since ker(F ) is either 0 or V , it must be V since it contains v, so F ′ = 0 which impliesF = λ idV .

Corollary 7.14. Let V and W be simple G-modules. Then

dim HomG-mod(V,W ) =

1 if V ∼= W0 if V 6∼= W

Proof. If V 6∼= W , then the only G-homomorphism between them is the zero map by Schur’slemma. Suppose V ∼= W and let F,G : V → W be G-isomorphisms. Then G−1F : V → V isan endomorphism of V , so we have G−1F = λ idV . Then

G−1F = λ idV =⇒ G(G−1F ) = G(λ idV )

=⇒(GG−1

)F = λG idV

=⇒ idW F = λG idV

=⇒ F = λG.

7.1 Drinfeld centers

For a fixed group G and a field K, the finite-dimensional G-modules over K and their G-homomorphisms form a category, G-mod.

The basic properties of G-modules established thus far make it clear that we can impose thestructure of a monoidal category on G-mod, by taking the tensor product as the usual tensorproduct of K-vector spaces, with the addition of defining action of G on this tensor productas in proposition 7.9. The tensor unit will be the trivial G-module K and the associator andunitor isomorphisms well be the canonical isomorphisms we are familiar with from the monoidalcategory VectC.

Definition 7.15. Let V and W be finite dimensional G-modules. For any z ∈ Z(G), define

Φz : V ⊗W →W ⊗ V

by

v ⊗ w 7→ z(w)⊗ v.

Since action of z is an invertible linear operator, this yields a linear isomorphism.

Proposition 7.16. Let V and W be finite dimensional G-modules. Then any isomorphism

Φz : V ⊗W →W ⊗ V

is a G-isomorphism.

Markus Thuresson 51

Proof. We immediately have

Φz (g(v ⊗ w)) = Φz(g(v)⊗ g(w))

= (zg)(w)⊗ g(v)

= (gz)(w)⊗ g(v)

= g (Φz(v ⊗ w))

Remark 7.17. Since for any group we have e ∈ Z(G), the above definition contains the canonicalisomorphism we are familiar with as a special case, namely Φ = Φe.

Proposition 7.18. Let V be a finite dimensional G-module and z ∈ Z(G). Then the pair(V,Φz) is in the Drinfeld center Z (G-mod).

Proof. Let W and X be G-modules and F : W → X some G-homomorphism. We need thediagram

V ⊗W id⊗F//

Φz

V ⊗XΦz

W ⊗ VF⊗id

// X ⊗ V

to commute, which is verified by

(F ⊗ id)(Φz(v ⊗ w)) = (F ⊗ id)(zw ⊗ v)

= F (zw)⊗ v= zF (w)⊗ v= Φz(v ⊗ F (w))

= Φz(id⊗F )(v ⊗ w).

Moreover, chasing the element v ⊗ (w ⊗ x) through the diagram

(W ⊗ V )⊗X α //W ⊗ (V ⊗X)

id⊗Φz

((

(V ⊗W )⊗X

Φz⊗id66

W ⊗ (X ⊗ V )

α−1

V ⊗ (W ⊗X)

α−1

OO

Φz

// (W ⊗X)⊗ V

shows that it is commutative.

Proposition 7.19. Let V,W be finite-dimensional G-modules. Then

HomZ(G-mod) ((V,Φz), (W,Φz)) = HomG-mod (V,W ) .

Drinfeld centers 52

Proof. Let F : V →W be a G-homomorphism. We need the diagram

V ⊗X F⊗id//

Φz

W ⊗XΦz

X ⊗ Vid⊗F

// X ⊗W

to commute for any G-module X. We have

(id⊗F )(Φz(v ⊗ x)) = (id⊗F )(zx⊗ v)

= zx⊗ F (v)

= Φz(F (v)⊗ x)

= Φz(F ⊗ id)(v ⊗ x).

Definition 7.20. Let V,W,X be G-modules. Denote by

ε : (V ⊕W )⊗X → (V ⊗X)⊕ (W ⊗X)

the canonical isomorphism defined by

(v, w)⊗ x 7→ (v ⊗ x,w ⊗ x).

Definition 7.21. Let V,W,X be G-modules. Denote by

δ : X ⊗ (V ⊕W )→ (X ⊗ V )⊕ (X ⊗W )

the canonical isomorphism defined by

x⊗ (v, w) 7→ (x⊗ v, x⊗ w).

Definition 7.22. Let (V,Ψ) and (W,Θ) be in Z(VectC) so that we for any module X haveisomorphisms

Ψ : V ⊗X → X ⊗ V and W ⊗X → X ⊗W.

Define Ψ Θ by

Ψ Θ := δ−1 (Ψ⊕Θ) ε : (V ⊕W )⊗X → X ⊗ (V ⊕W ).

Proposition 7.23. If (V,Ψ) and (W,Θ) are in Z(G-mod), then

(V,Ψ)⊕ (W,Θ) := (V ⊕W,Ψ Θ)

is in Z(G-mod).

Markus Thuresson 53

Proof. Let F : X → Y be a G-homomorphism and cosider the following diagram:

(V ⊕W )⊗X id⊗F//

ε

(V ⊕W )⊗ Y

ε

(V ⊗X)⊕ (W ⊗X)

Ψ⊕Θ

(id⊗F )⊕(id⊗F )// (V ⊗ Y )⊕ (W ⊗ Y )

Ψ⊕Θ

(X ⊗ V )⊕ (X ⊗W )(F⊗id)⊕(F⊗id)

//

δ−1

(Y ⊗ V )⊕ (Y ⊗W )

δ−1

X ⊗ (V ⊕W )F⊗id

// Y ⊗ (V ⊕W )

It is clear that the perimeter is a naturality square of Ψ Θ. Moreover, the middle squareis just the sum of naturality squares of Ψ and Θ, which commute by assumption. Hence, themiddle square commutes.

In the top square, we have

ε(id⊗F )((v, w)⊗ x) = ε((v, w)⊗ F (x))

= (v ⊗ F (x), w ⊗ F (x))

= ((id⊗F )⊕ (id⊗F )) (v ⊗ x,w ⊗ x)

= ((id⊗F )⊕ (id⊗F )) ε((v, w)⊗ x)

so the top square commutes.In the bottom square, we have

δ−1 ((F ⊗ id)⊕ (F ⊗ id)) ((x⊗ v, x⊗ w)) = δ−1((F (x)⊗ v, F (x)⊗ w))

= F (x)⊗ (v, w)

= (F ⊗ id)(x⊗ (v, w))

= (F ⊗ id)δ−1((x⊗ v, x⊗ w))

so the bottom square commutes. This shows that the perimeter commutes so Ψ Θ is anatural family of isomorphisms.

Next, we consider the diagram

Drinfeld centers 54

(V ⊕W )⊗ (X ⊗ Y )α−1

//

ε

((V ⊕W )⊗X)⊗ Y

ε⊗id

((V ⊗X)⊕ (W ⊗X))⊗ Y

ε

(Ψ⊕Θ)⊗id

ss

(V ⊗ (X ⊗ Y ))⊕ (W ⊗ (X ⊗ Y ))

Ψ⊕Θ

α−1⊕α−1

%%

((X ⊗ V )⊕ (X ⊗W ))⊗ Y

δ−1⊗id

ε

++

((V ⊗X)⊗ Y )⊕ ((W ⊗X)⊗ Y )

(Ψ⊗id)⊕(Θ⊗id)

(X ⊗ (V ⊕W ))⊗ Y

α

((X ⊗ V )⊗ Y )⊕ ((X ⊗W )⊗ Y )

α⊕α

X ⊗ ((V ⊕W )⊗ Y )

id⊗ε

(X ⊗ (V ⊗ Y ))⊕ (X ⊗ (W ⊗ Y ))

(id⊗Ψ)⊕(id⊗Θ)

δ−1

ss

((X ⊗ Y )⊗ V )⊕ ((X ⊗ Y )⊗W )

δ−1

X ⊗ ((V ⊗ Y )⊕ (W ⊗ Y ))

id⊗(Ψ⊕Θ)

++

(X ⊗ (Y ⊗ V ))⊕ (X ⊗ (Y ⊗W ))

δ−1

α−1⊕α−1

ee

X ⊗ ((Y ⊗ V )⊕ (Y ⊗W ))

id⊗δ−1

(X ⊗ Y )⊗ (V ⊕W ) X ⊗ (Y ⊗ (V ⊕W ))α−1

oo

Now the diagram given by the curved arrows together with the first and third columns is just thesum of diagrams which commute by the assumption that (V,Ψ) and (W,Θ) are in Z(G-mod),so it commutes.

Next, consider the diagram given by the top arrow, the first and third columns and the topcurved arrow α−1 ⊕ α−1.

ε(ε⊗ id)α−1((v, w)⊗ (x⊗ y)) = ε(ε⊗ id)(((v, w)⊗ x)⊗ y)

= ε((v ⊗ x,w ⊗ x), y)

= ((v ⊗ x)⊗ y, (w ⊗ x)⊗ y)

= (α−1 ⊕ α−1)((v ⊗ (x⊗ y), w ⊗ (x⊗ y)))

= (α−1 ⊕ α−1)ε((v, w)⊗ (x⊗ y))

so this diagram commutes. The corresponding diagram on the bottom is shown to be commu-tative in the same way. These observations together show that the outer perimeter commutes.

Markus Thuresson 55

Next, we consider the diagram where the third column branches out into the second. Thisdiagram is divided into three parts.

In the top part, we have

((Ψ⊗ id)⊕ (Θ⊗ id))ε((v ⊗ x,w ⊗ x)⊗ y) = ((Ψ⊗ id)⊕ (Θ⊗ id))((v ⊗ x)⊗ y, (w ⊗ x)⊗ y)

= (Ψ(v ⊗ x)⊗ y,Θ(w ⊗ x)⊗ y)

= ε((Ψ(v ⊗ x),Θ(w ⊗ x))⊗ y)

= ε((Ψ⊕Θ)⊗ id)((v ⊗ x,w ⊗ x)⊗ y)

so the top part commutes. The bottom part is shown to be commutative in the same way.In the middle part, we have

δ−1(α⊕ α)ε((x⊗ v, x⊗ w)⊗ y) = δ−1(α⊕ α)((x⊗ v)⊗ y, (x⊗ w)⊗ y)

= δ−1(x⊗ (v ⊗ y), x⊗ (w ⊗ y))

= x⊗ (v ⊗ y, w ⊗ y)

= (id⊗ε)(x⊗ ((v, w)⊗ y))

= (id⊗ε)α((x⊗ (v, w))⊗ y)

= (id⊗ε)α(δ−1 ⊗ id)((x⊗ v, x⊗ w)⊗ y)

so the middle part commutes. Now it follows that the whole branch commutes with the thirdcolumn.

This implies, that if we follow the perimeter but instead follow the branch into the innerperimeter, the diagram still commutes. This diagram commuting verifies that (V ⊕W,Ψ Θ)is in Z(G-mod).

8 The Drinfeld center of Z2-mod

We write G = Z2 = e, s with the relation s2 = e. It can be shown that Z2 has two simplemodules. The first one is the trivial module Ctriv. The second one is the sign module Csign

where s acts as multiplication with −1. By Maschke’s theorem, any module V is isomorphic toa direct sum of simples, that is V ∼= C⊕mtriv ⊕ C⊕nsign. Fixing the standard basis, we have a basis

e1, . . . , em, em+1, . . . , em+n

of C⊕mtriv ⊕ C⊕nsign.

Proposition 8.1. Any G-homomorphism F : C⊕m1triv ⊕C⊕n1

triv → C⊕m2triv ⊕C⊕n2

triv is given by a blockmatrix of the form [

A 0m2×n1

0n2×m1 B

]where A is an arbitrary m2 ×m1 matrix and B is an arbitrary n2 × n1 matrix.

Drinfeld centers 56

Proof. Consider a linear map

F : C⊕m1triv ⊕ C⊕n1

triv → C⊕m2triv ⊕ C⊕n2

triv

It is given by an (m2 + n2)× (m1 + n1) matrix. We write the matrix of F as a block matrix

[F ] =

[[F11]m2×m1 [F12]m2×n1

[F21]n2×m1 [F22]n2×n1

]where Fij denotes the component of F mapping the j:th summand to the i:th summand. Inother words, a linear map F as above corresponds to four linear maps

F11 : C⊕m1triv → C⊕m2

triv

F12 : C⊕n1sign → C⊕m2

triv

F21 : C⊕m1triv → C⊕n2

sign

F22 : C⊕n1sign → C⊕n2

sign.

It is clear that F is a G-homomorphism if and only if all of its components are. By additivityof the Hom-functor and Schur’s lemma, we have

dim HomG-mod

(C⊕mtriv,C

⊕nsign

)= mdim HomG-mod

(Ctriv,C⊕nsign

)= mndim HomG-mod (Ctriv,Csign)

= 0

for any m,n and

dim Hom(C⊕m

′

sign,C⊕n′triv

)= m′ dim Hom

(Csign,C⊕n

′

triv

)= m′n′ dim Hom (Csign,Ctriv)

= 0

for any m′, n′. This shows that Fij = 0 whenever i 6= j, so a G-homomorphism

F : C⊕m1triv ⊕ C⊕n1

triv → C⊕m2triv ⊕ C⊕n2

triv

is given by a matrix

[F ] =

[[F11] 0

0 [F22]

].

The element s ∈ Z2 acts as the identity on Ctriv, and as multiplication with −1 on Csign. Thismeans that on C⊕mtriv ⊕ C⊕nsign, the action of s is the linear extension of the map defined on thebasis by

s(ei) =

ei if 1 ≤ i ≤ m−ei if m+ 1 ≤ i ≤ m+ n

Markus Thuresson 57

so we have

[s] =

[Im 00 −In

].

Denoting the matrix of the action of s on the different modules by [s]1 and [s]2, the condition ofF being a G-homomorphism now amounts to the equation [s]2[F ] = [F ][s]1. Using our previousobservation we see that

[s]2[F ] = [F ][s]1 ⇐⇒[Im2 00 −In2

] [[F11] 0

0 [F22]

]=

[[F11] 0

0 [F22]

] [Im1 00 −In1

]⇐⇒

[Im2 [F11] 0

0 −In2 [F22]

]=

[[F11]Im1 0

0 −[F22]In1

].

This holds trivially for any matrices [Fii].

Definition 8.2. Let V and W be finite-dimensional G-modules. Define the mapΦe : V ⊗W →W ⊗ V by

v ⊗ w 7→ e(w)⊗ v.

This is a G-isomorphism by proposition 7.16 and (V,Φe) is in Z(G-mod) by proposition 7.18.

Definition 8.3. Let V,W be finite-dimensional G-modules. Define the mapΦs : V ⊗W →W ⊗ V by

v ⊗ w 7→ s(w)⊗ v.

This is a G-isomorphism by proposition 7.16 and (V,Φe) is in Z(G-mod) by proposition 7.18.

Lemma 8.4. Let (Ctriv,Ψ) be in Z(G-mod) and let X be a one-dimensional module. Thenthe component

ΨX : Ctriv ⊗X → X ⊗ Ctriv

has the form ΨX = λXΦeX for some nonzero λX ∈ C. Moreover,

λX =

λCtriv

if X ∼= Ctriv

λCsignif X ∼= Csign

.

Proof. Fix the basis 1 of C and the basis x of X, so that we have bases 1⊗ x and x⊗ 1 of therespective tensor products.

Since ΨX : Ctriv ⊗X → X ⊗ Ctriv is a linear isomorphism by assumption, we have

ΨX(1⊗ x) = λX(x⊗ 1) = λXΦeX(1⊗ x)

Drinfeld centers 58

so the first statement follows by linearity. Suppose X ∼= Ctriv. By naturality the square

Ctriv ⊗ Ctrivid⊗ϕ

//

ΨCtriv

Ctriv ⊗X

ΨX

Ctriv ⊗ Ctrivϕ⊗id

// X ⊗ Ctriv

commutes for any G-homomorphism. In particular, it commutes for any isomorphism ϕ. Wethen have

λCtrivϕ(1)⊗ 1 = (ϕ⊗ id) (λCtriv

(1⊗ 1))

= (ϕ⊗ id)(λCtrivΦeCtriv

(1⊗ 1))

= (ϕ⊗ id)ΨCtriv(1⊗ 1)

= ΨX(id⊗ϕ)(1⊗ 1)

= ΨX(1⊗ ϕ(1))

= λXΦeX(1⊗ ϕ(1))

= λXϕ(1)⊗ 1.

This implies λCtriv= λX . The case X ∼= Csign can be proved in the same way.

Proposition 8.5. If (Ctriv,Ψ) is in Z(G-mod), then Ψ = Φe or Ψ = Φs.

Proof. The component of Ψ at some module V = C⊕mtriv ⊕ C⊕nsign is an isomorphism

Ψ : Ctriv ⊗(C⊕mtriv ⊕ C⊕nsign

)→(C⊕mtriv ⊕ C⊕nsign

)⊗ Ctriv.

Fix bases 1 ⊗ ei and ei ⊗ 1 of the respective tensor products. We write the matrix of Ψwith respect to these bases as a block matrix

[Ψ] =

[Am×m Bm×nCn×m Dn×n

].

The action of the element s ∈ Z2 on the chosen bases is given by

s(1⊗ ei) = s(1)⊗ s(ei) = 1⊗ s(ei)s(ei ⊗ 1) = s(ei)⊗ s(1) = s(ei)⊗ 1

since s acts as the identity in the trivial module. This means that in our bases we have

[s] =

[Im 00 −In

].

Markus Thuresson 59

Now, Ψ being aG-homomorphism amounts to the matrix equation [s][Ψ] = [Ψ][s]. This equationyields

[s][Ψ] = [Ψ][s] ⇐⇒[Im 00 −In

] [A BC D

]=

[A BC D

] [Im 00 −In

]⇐⇒

[A B−C −D

]=

[A −BC −D

]so we must have B = −B and C = −C which implies B = C = 0. This shows that

[Ψ] =

[A 00 D

]. By proposition 8.1, any endomorphism of the module C⊕mtriv ⊕C⊕nsign is given by

a matrix [F ] =

[F11 00 F22

]where F11 is an arbitrary m × m matrix and F22 is an arbitrary

n× n matrix. By naturality of Ψ the square

Ctriv ⊗(C⊕mtriv ⊕ C⊕nsign

)Ψ

id⊗F// Ctriv ⊗

(C⊕mtriv ⊕ C⊕nsign

)Ψ(

C⊕mtriv ⊕ C⊕nsign

)⊗ Ctriv

F⊗id//

(C⊕mtriv ⊕ C⊕nsign

)⊗ Ctriv

commutes for any endomorphism F . With respect to our bases we have [id⊗F ] = [F⊗id] = [F ].Commutativity of the diagram amounts to the equation [Ψ][F ] = [F ][Ψ]. We have

[Ψ][F ] = [F ][Ψ] ⇐⇒[A 00 D

] [F11 00 F22

]=

[F11 00 F22

] [A 00 D

]⇐⇒

[AF11 0

0 DF22

]=

[F11A 0

0 F22D

]⇐⇒ AF11 = F11A and DF22 = F22D

and since Fii is arbitrary we have A = aV Im and D = dV In.Now let V1 = C⊕m1

triv ⊕C⊕n1sign and V2 = C⊕m2

triv ⊕C⊕n2sign. By proposition 8.1 a G-homomorphism

F : V1 → V2 is given by a matrix of the form

[F ] =

F11m2×m10m2×n1

0n2×m1 F22n2×n1

.The commutativity of the naturality square

Ctriv ⊗ V1

ΨV1

id⊗F// Ctriv ⊗ V2

ΨV2

V1 ⊗ CtrivF⊗id

// V2 ⊗ Ctriv

Drinfeld centers 60

is equivalent to [ΨV2 ][F ] = [F ][ΨV1 ] since we have [id⊗F ] = [F ⊗ id] = [F ] with respect toour chosen bases. Then we have

[ΨV2 ][F ] = [F ][ΨV1 ] ⇐⇒[aV2Im2 0

0 dV2In2

] [F11m2×m1

00 F22n2×n1

]=

[F 11m2×m1

00 F 22

n2×n1

] [aV1Im1 0

0 dV1In1

]⇐⇒

[aV2F11 0

0 dV2F22

]=

[aV1F11 0

0 dV1F22

]which imples aV1 = aV2 and dV1 = dV2 so we may drop the indices. Consider the commutativediagram

Csign ⊗ (Ctriv ⊗ Ctriv)

Ψ

α−1// (Csign ⊗ Ctriv)⊗ Ctriv

Ψ⊗id

(Ctriv ⊗ Csign)⊗ Ctriv

α

Ctriv ⊗ (Csign ⊗ Ctriv)

id⊗Ψ

(Ctriv ⊗ Ctriv)⊗ Ctriv Ctriv ⊗ (Ctriv ⊗ Csign)α−1

oo

Since [ΨCtriv] = [a] and (Ctriv ⊗ Ctriv) ∼= Ctriv, the left arrow is given by a by lemma 8.4. With

respect to our bases, we have [α] = [α−1] = [1], so commutativity is equivalent to a2 = a. Thisimplies a = 1 since a = 0 would contradict Ψ being an isomorphism.

Now we consider the diagram

Ctriv ⊗ (Csign ⊗ Csign)

Ψ

α−1// (Ctriv ⊗ Csign)⊗ Csign

Ψ⊗id

(Csign ⊗ Ctriv)⊗ Csign

α

Csign ⊗ (Ctriv ⊗ Csign)

id⊗Ψ

(Csign ⊗ Csign)⊗ Ctriv Csign ⊗ (Csign ⊗ Ctriv)α−1

oo

We observe that Csign ⊗ Csign∼= Ctriv, so the left arrow is given by a = 1 by lemma 8.4.

Commutativity is then equivalent to d2 = 1 =⇒ d = ±1. Clearly, d = 1 corresponds to theisomorphism Φe, since then we have

[Ψ] =

[Im 00 In

]= Im+n.

Markus Thuresson 61

If d = −1, then we have

[Ψ] =

[Im 00 −In

]= [s]

which implies

Ψ(1⊗ ei) =

ei ⊗ 1 if 1 ≤ i ≤ m−ei ⊗ 1 if m+ 1 ≤ i ≤ m+ n

= Ψs(1⊗ ei)

so Ψ = Φs.

Proposition 8.6. If (Csign,Ψ) is in Z(G-mod), then Ψ = Φe or Ψ = Φs.

Proof. We aim to mimic the proof of the previous proposition, so consider the component of Ψat some direct sum of simples:

Ψ : Csign ⊗(C⊕mtriv ⊕ C⊕nsign

)→(C⊕mtriv ⊕ C⊕nsign

)⊗ Csign.

Fix bases 1⊗ ei and ei ⊗ 1 of the respective tensor products. Again we write

[Ψ] =

[A BC D

].

Now, since we have s(1) = −1 in Csign, we differ from the previous case and get

[s] =

[−Im 0

0 In

].

The isomorphism Ψ commuting with the group action again yields [Ψ] =

[A 00 D

]. Using

proposition 8.1 now gives A = aV Im and D = dV In just like before and naturality then impliesaV1 = aV2 and dV1 = dV2 for any modules V1 and V2. Now we consider the commutative diagram

Csign ⊗ (Ctriv ⊗ Ctriv)

Ψ

α−1// (Csign ⊗ Ctriv)⊗ Ctriv

Ψ⊗id

(Ctriv ⊗ Csign)⊗ Ctriv

α

Ctriv ⊗ (Csign ⊗ Ctriv)

id⊗Ψ

(Ctriv ⊗ Ctriv)⊗ Ctriv Ctriv ⊗ (Ctriv ⊗ Csign)α−1

oo

Drinfeld centers 62

which shows a = 1 and the diagram

Csign ⊗ (Csign ⊗ Csign)

Ψ

α−1// (Csign ⊗ Csign)⊗ Csign

Ψ⊗id

(Csign ⊗ Csign)⊗ Csign

α

Csign ⊗ (Csign ⊗ Csign)

id⊗Ψ

(Csign ⊗ Csign)⊗ Csign Csign ⊗ (Csign ⊗ Csign)α−1

oo

then shows that d = ±1. If d = 1 then Ψ = Φe and if d = −1 then Ψ = Φs.

Proposition 8.7. Let (V1,Ψ1), . . . , (Vk,Ψk) be objects in Z(G-mod) where all Vi are simplemodules, so that Vi ∈ Ctriv,Csign and Ψi ∈ Φe,Φs for all i. Let Vi be the linear span of thevector vi. Then

k⊕i=1

(Vi,Ψi) = (V,Ψ)

where V =⊕k

i=1 Vi and Ψ : V ⊗X → X ⊗ V is defined by

(v1, . . . , vk)⊗ x 7→ x⊗ (c1v1, . . . , ckvk)

where

ci =

1 if Ψi = Φe

−1 if Ψi = Φs

Proof. The fact that V =⊕k

i=1 Vi is clear. What this proposition aims to prove is that theaddition of natural isomorphisms used in the Drinfeld center addition of proposition 7.23 iswell-behaved. We proceed by induction. If k = 2 then

(V1,Ψ1)⊕ (V2,Ψ2) = (V1 ⊕ V2, δ−1(Ψ1 ⊕Ψ2)ε)

by definition. We have

δ−1(Ψ1 ⊕Ψ2)ε((v1, v2)⊗ x) = δ−1(Ψ1 ⊕Ψ2)((v1 ⊗ x, v2 ⊗ x))

= δ−1((Ψ1(v1 ⊗ x),Ψ2(v2 ⊗ x)))

= δ−1((x⊗ c1v1, x⊗ c2v2))

= x⊗ ((c1v1, c2v2))

= Ψ((v1, v2)⊗ x).

Markus Thuresson 63

Next, we have (V,Ψ) ⊕ (Vk+1,Ψk+1) = (V ⊕ Vk+1, δ−1(Ψ ⊕ Ψk+1)ε). Let Ψ′ denote the

isomorphism in the formulation.

δ−1(Ψ⊕Ψk+1)ε(((v1, . . . , vk), vk+1)⊗ x) = δ−1(Ψ⊕Ψk+1)(((v1, . . . , vk)⊗ x, vk+1 ⊗ x))

= δ−1((x⊗ (c1v1, . . . , ckvk), x⊗ ck+1vk+1))

= x⊗ ((c1v1, . . . , ckvk), ck+1vk+1)

= x⊗ (c1v1, . . . , ckvk, ck+1vk+1)

= Ψ′((v1, . . . , vk+1)⊗ x).

Lemma 8.8. Let (V,Ψ) be in Z(G-mod) and let W ∼= C⊕mtriv ⊕ C⊕nsign be some module. Then,for any fixed basis of V , there exists a basis of W such that with respect to these bases, we have

[ΨC⊕mtriv ⊕C

⊕nsign

] = [ΨW ].

Proof. Let viki=1 be a basis of V and let ϕ : C⊕mtriv ⊕C⊕nsign →W be a G-isomorphism and . Inparticular, it is a linear isomorphism so the set ϕ(ei) forms a basis of W . Fixing this basis ofW , we have [ϕ] = Im+n. By naturality, the square

V ⊗(C⊕mtriv ⊕ C⊕nsign

)id⊗ϕ

//

ΨC⊕mtriv⊕C⊕n

sign

V ⊗W

ΨW

(C⊕mtriv ⊕ C⊕nsign

)⊗ V

ϕ⊗id//W ⊗ V

commutes and since in our bases, we have

[id⊗ϕ] = [ϕ⊗ id] =

[ϕ] . . . 0...

. . ....

0 . . . [ϕ]

= Ik(m+n)

the result follows.

Proposition 8.9. If (C⊕ktriv,Ψ) is in Z(G-mod), then (C⊕ktriv,Ψ) is isomorphic to

(Ctriv,Φe)⊕k1 ⊕ (Ctriv,Φs)

⊕k2

for some k1, k2 such that k1 + k2 = k.

Proof. Fix the standard basis ei of C⊕ktriv and the standard basis vi of C⊕mtriv ⊕ C⊕nsign. Thenfix bases

e1 ⊗ v1, . . . , e1 ⊗ vm+n, . . . , ek ⊗ v1, . . . , ek ⊗ vm+nv1 ⊗ e1, . . . , v1 ⊗ ek, . . . , vm+n ⊗ e1, . . . , vm+n ⊗ ek

Drinfeld centers 64

of the respective tensor products. We write [Ψ] as a k × k block matrix with block size(m+ n)× (m+ n), so that

[Ψ] =

A11 . . . A1k...

. . ....

Ak1 . . . Akk

.In the module V , we have

s(vi) =

vi if 1 ≤ i ≤ m−vi if m+ 1 ≤ i ≤ m+ n

so that we have

[s]V =

[Im 0m×n0n×m −In

].

In other words, we have

s(ei ⊗ vj) = ei ⊗ s(vj) and s(vj ⊗ ei) = s(vj)⊗ ei

since s acts as the identity in the trivial module. Then it is clear that the matrix of s in thetensor products is the k×k block matrix with blocks [s]V on the diagonal and zeroes everywhereelse. Writing this out we have

[s]C⊕ktriv⊗V

= [s]V⊗C⊕ktriv

=

[s]V . . . 0...

. . ....

0 . . . [s]V

.Since Ψ is a G-isomorphism by assumption, it commutes with the action of the group, so we haveΨ(s(x⊗ y)) = sΨ(x⊗ y). In terms of matrices, this is equivalent to the equation [Ψ][s] = [s][Ψ].Then A11 . . . A1k

.... . .

...Ak1 . . . Akk

[s]V . . . 0

.... . .

...0 . . . [s]V

=

[s]V . . . 0...

. . ....

0 . . . [s]V

A11 . . . A1k

.... . .

...Ak1 . . . Akk

which holds if and only ifA11[s]V . . . A1k[s]V

.... . .

...Ak1[s]V . . . Akk[s]V

=

[s]VA11 . . . [s]VA1k...

. . ....

[s]VAk1 . . . [s]VAkk

.This shows that every matrix Aij must commute with [s]V . We write Aij as a block matrix

Aij =

[Bm×m Cm×nDn×m En×n

].

Markus Thuresson 65

Writing out the condition Aij [s]V = [s]VAij we get[Bm×m Cm×nDn×m En×n

] [Im 0m×n0n×m −In

]=

[Im 0m×n0n×m −In

] [Bm×m Cm×nDn×m En×n

]which holds if and only if [

B −CD −E

]=

[B C−D −E

].

This implies C = −C and D = −D so we must have C = D = 0. Consider the component ofΨ at the module V = C⊕mtriv ⊕ C⊕nsign. By assumption, the diagram

C⊕ktriv ⊗(C⊕mtriv ⊕ C⊕nsign

)id⊗F

//

Ψ

C⊕ktriv ⊗(C⊕mtriv ⊕ C⊕nsign

)Ψ(

C⊕mtriv ⊕ C⊕nsign

)⊗ C⊕ktriv F⊗id

//

(C⊕mtriv ⊕ C⊕nsign

)⊗ C⊕ktriv

commutes for any G-homomorphism F : V → V . With respect to our bases, we know that thematrices of the homomorphisms id⊗F and F ⊗ id are given by

[id⊗F ] = [F ⊗ id] =

[F ] . . . 0...

. . ....

0 . . . [F ]

.It is now clear that the diagram commutes if and only if this matrix commutes with the matrixof Ψ. This condition is equivalent to Aij [F ] = [F ]Aij for all i, j. By our previous observationthe matrices Aij have the form

Aij =

[Bij 00 B′ij

]and by proposition 8.1 the matrix [F ] has the form

[F ] =

[F11 00 F22

].

This means that our condition is equivalent to[Bij 00 B′ij

] [F11 00 F22

]=

[F11 00 F22

] [Bij 00 B′ij

]which holds if and only if

BijF11 = F11Bij and B′ijF22 = F22B′ij .

Drinfeld centers 66

Since the matrices Fii are arbitrary, this implies that Bij = λV Im and B′ij = µV In, where

λV , µV ∈ C. Let W = C⊕xtriv ⊕ C⊕ysign be another module. By assumption, the diagram

C⊕ktriv ⊗(C⊕mtriv ⊕ C⊕nsign

)id⊗F

//

Ψ

C⊕ktriv ⊗(C⊕xtriv ⊕ C⊕ysign

)Ψ(

C⊕mtriv ⊕ C⊕nsign

)⊗ C⊕ktriv F⊗id

//

(C⊕xtriv ⊕ C⊕ysign

)⊗ C⊕ktriv

commutes for any G-homomorphism F : V →W . A similar argument now shows that we musthave AWij [F ] = [F ]AVij for all i, j. This is equivalent to[

λWij Ix 0

0 µWij Iy

] [F11 00 F22

]=

[F11 00 F22

] [λVijIm 0

0 µVijIn

]which holds if and only if λWF11 = λV F11 and µWF22 = µV F22. This implies λVij = λWij and

µVij = µWij . Now, we consider the commutative diagram

C⊕ktriv ⊗ (Ctriv ⊗ Ctriv)

Ψ

α−1//

(C⊕ktriv ⊗ Ctriv

)⊗ Ctriv

Ψ⊗id(

Ctriv ⊗ C⊕ktriv

)⊗ Ctriv

α

Ctriv ⊗(C⊕ktriv ⊗ Ctriv

)id⊗Ψ

(Ctriv ⊗ Ctriv)⊗ C⊕ktriv Ctriv ⊗(Ctriv ⊗ C⊕ktriv

)α−1

oo

Since Ctriv ⊗ Ctriv∼= Ctriv, there exists a suitable basis of Ctriv ⊗ Ctriv such that

[ΨCtriv⊗Ctriv] = [ΨCtriv

] by lemma 8.8. We also note that

[idCtriv⊗ΨCtriv

] = [ΨCtriv⊗ idCtriv

] = [ΨCtriv].

Since the associators are given by identity matrices, commutativity of the diagram is equivalentto [ΨCtriv

]2 = [ΨCtriv] and since [ΨCtriv

] is invertible it follows that [ΨCtriv] = Ik. This is

equivalent to

λij =

1 if i = j0 if i 6= j

.

Markus Thuresson 67

If we instead consider the commutative diagram

C⊕ktriv ⊗ (Csign ⊗ Csign)

Ψ

α−1//

(C⊕ktriv ⊗ Csign

)⊗ Csign

Ψ⊗id(

Csign ⊗ C⊕ktriv

)⊗ Csign

α

Csign ⊗(C⊕ktriv ⊗ Csign

)id⊗Ψ

(Csign ⊗ Csign)⊗ C⊕ktriv Csign ⊗(Csign ⊗ C⊕ktriv

)α−1

oo

Since Csign ⊗ Csign∼= Ctriv we may assume that the left arrow is given by Ik, by lemma 8.8.

Commutativity of the diagram is then equivalent to [ΨCsign]2 = Ik.

Such a matrix must have eigenvalues ±1 and be diagonalizable. This means that there is abasis of C⊕ksign such that

[ΨCsign] =

[Ip 00 −Iq

]with respect to this basis, where p + q = k. At first sight, it seems like this change of basiscould affect the arguments made thus far, but we are saved by the following observations:

- The matrices representing the associators, their inverses and [ΨCtriv] are identity matrices.

The identity matrix is invariant under any change of basis.

- The form of the module C⊕ktriv is not affected by a change of basis.

So, we can safely make this change of basis, which shows that we may assume

µij =

±1 if i = j0 otherwise

and moreover,

µii =

1 if 1 ≤ i ≤ p−1 if p+ 1 ≤ i ≤ k .

This shows that Ψ : C⊕ktriv ⊗ (C⊕mtriv ⊕ C⊕nsign)→ (C⊕mtriv ⊕ C⊕nsign)C⊕ktriv is defined by

ei ⊗ vj 7→ vj ⊗ ei if 1 ≤ i ≤ pei ⊗ vj 7→ −vj ⊗ ei if p+ 1 ≤ i ≤ k = p+ q

which corresponds to p copies of (Ctriv,Φe) and q copies of (Ctriv,Φs).

Drinfeld centers 68

Proposition 8.10. If (C⊕ksign,Ψ) is in Z(G-mod), then (C⊕ksign,Ψ) is isomorphic to

(Ctriv,Φe)⊕k1 ⊕ (Ctriv,Φs)

⊕k2

for some k1, k2 such that k1 + k2 = k.

Proof. We use the same setup as in the proof of proposition 8.9. We start by considering thecomponent of Ψ at the module V = C⊕mtriv ⊕ C⊕nsign. The first point where we differ is the actionof the group element s on the tensor products. We have

s(ei ⊗ vj) = −ei ⊗ s(vj) and s(vj ⊗ ei) = −s(vj)⊗ ei

since s acts as −1 in the sign module. Like before, we have

[s]V =

[Im 0m×n0n×m −In

]but now we get a different matrix representing the action of s in the tensor products, namely:

[s]C⊕ksign⊗V

= [s]V⊗C⊕ksign

=

−[s]V . . . 0...

. . ....

0 . . . −[s]V

.If we write

[Ψ] =

A11 . . . A1k...

. . ....

Ak1 . . . Akk

.The assumption that Ψ be a G-homomorphism again yields [s]VAij = Aij [s]V for all i, j. Justlike in the previous proposition, this implies that the matrices Aij have the form

Aij =

[Bij 00 B′ij

].

Moreover, it follows in the same exact way that

Bij = λVijIm and B′ij = µVijIn, λV , µV ∈ C.

Next, naturality shows that if W is some other module, then λV = λW and µV = µW . Thismeans that so far, we know that the matrix representing the component of Ψ at V is given by

[Ψ] =

A11 . . . A1k...

. . ....

Ak1 . . . Akk

Markus Thuresson 69

where the matrices Aij have the form

Aij =

[λijIm 0

0 µijIn

].

Now, we consider the commutative diagram

C⊕ksign ⊗ (Ctriv ⊗ Ctriv)

Ψ

α−1//

(C⊕ksign ⊗ Ctriv

)⊗ Ctriv

Ψ⊗id(

Ctriv ⊗ C⊕ksign

)⊗ Ctriv

α

Ctriv ⊗(C⊕ksign ⊗ Ctriv

)id⊗Ψ

(Ctriv ⊗ Ctriv)⊗ C⊕ksign Ctriv ⊗(Ctriv ⊗ C⊕ksign

)α−1

oo

By the same arguments as in the proof of proposition 8.9, commutativity yields [ΨCtriv]2 =

[ΨCtriv] which implies [ΨCtriv

] = Ik. This is equivalent to

λij =

1 if i = j0 if i 6= j

.

Now, we consider the commutative diagram

C⊕ksign ⊗ (Csign ⊗ Csign)

Ψ

α−1//

(C⊕ksign ⊗ Csign

)⊗ Csign

Ψ⊗id(

Csign ⊗ C⊕ksign

)⊗ Csign

α

Csign ⊗(C⊕ksign ⊗ Csign

)id⊗Ψ

(Csign ⊗ Csign)⊗ C⊕ksign Csign ⊗(Csign ⊗ C⊕ksign

)α−1

oo

Since Csign ⊗ Csign∼= Ctriv we may assume that the left arrow is given by Ik, by lemma 8.8.

Commutativity of the diagram is then equivalent to [ΨCsign]2 = Ik. Now we use the same

argument as in the proof of proposition 8.9 to arrive at the conclusion.

Drinfeld centers 70

Proposition 8.11. If (C⊕ktriv⊕C⊕lsign,Ψ) is in Z(G-mod), then (C⊕ktriv⊕C⊕lsign,Ψ) is isomorphic

to (C⊕ktriv,Ψ1)⊕ (C⊕lsign,Ψ2).

Proof. Considering the component of Ψ at V = C⊕mtriv ⊕C⊕nsign we get the commutative diagram(C⊕ktriv ⊕ C⊕lsign

)⊗(C⊕mtriv ⊕ C⊕nsign

)id⊗F

//

Ψ

(C⊕ktriv ⊕ C⊕lsign

)⊗(C⊕mtriv ⊕ C⊕nsign

)Ψ(

C⊕mtriv ⊕ C⊕nsign

)⊗(C⊕ktriv ⊕ C⊕lsign

)F⊗id

//

(C⊕mtriv ⊕ C⊕nsign

)⊗(C⊕ktriv ⊕ C⊕lsign

)We write [Ψ] as a (k + l)× (k + l) block matrix with block size (m+ n)× (m+ n). So we have

[Ψ] =

A11 . . . A1(k+l)...

. . ....

A(k+l)1 . . . A(k+l)(k+l)

Let ui denote the standard basis of C⊕ktriv⊕C⊕lsign and vi the standard basis of C⊕mtriv⊕C⊕nsign.Fix the basis

u1 ⊗ v1, . . . , u1 ⊗ vm+n, . . . , uk+l ⊗ v1, . . . , uk+l ⊗ vm+n

of the module(C⊕ktriv ⊕ C⊕lsign

)⊗(C⊕mtriv ⊕ C⊕nsign

)and the basis

v1 ⊗ u1, . . . , vm+n ⊗ u1, . . . , vm+n ⊗ u1, . . . , vm+n ⊗ uk+l

of the module(C⊕mtriv ⊕ C⊕nsign

)⊗(C⊕ktriv ⊕ C⊕lsign

). Now we note that

s(ui) =

ui if 1 ≤ i ≤ k−ui if k + 1 ≤ i ≤ k + l

, s(vi) =

vi if 1 ≤ i ≤ m−vi if m+ 1 ≤ i ≤ m+ n

.

Letting E =

[Im 00 −In

]we get the matrices for the action of s in U ⊗ V and in V ⊗ U as

(k + l) × (k + l) block matrices with blocks of the form E or −E on the diagonal and zeroeseverywhere else. Clearly, there are k + l such blocks. Then, the first k blocks are E and thelast l blocks are −E.

[s]U⊗V =

E . . . 0...

. . ....

0 . . . −E

= [s]V⊗U .

Now Ψ being a G-homomorphism is equivalent to [Ψ][s]U⊗V = [s]V⊗U [Ψ]. Writing the conditionout we have

Markus Thuresson 71

A11 . . . A1(k+l)...

. . ....

A(k+l)1 . . . A(k+l)(k+l)

E . . . 0

.... . .

...0 . . . −E

=

E . . . 0...

. . ....

0 . . . −E

A11 . . . A1(k+l)

.... . .

...A(k+l)1 . . . A(k+l)(k+l)

which holds if and only if

A11E . . . A1kE −A1(k+1)E . . . −A1(k+l)E...

......

...Ak+l1E . . . Ak+lkE −A(k+l)(k+1)E . . . −A(k+l)(k+l)E

=

EA11 . . . EA1(k+l)...

...EAk1 . . . EAk(k+l)

−EA(k+1)1 . . . −EA(k+1)(k+l)...

...−EA(k+l)1 . . . −EA(k+l)(k+l)

.

This means that we have four possibly different conditions on the matrices Aij .

1) If i ≤ k, j ≤ k, we require EAij = AijE. Writing Aij =

[B11 B12

B21 B22

]we get

[Im 00 −In

] [B11 B12

B21 B22

]=

[B11 B12

B21 B22

] [Im 00 −In

]⇐⇒

[B11 B12

−B21 −B22

]=

[B11 −B12

B21 −B22

]which implies B12 = B21 = 0. The same holds if i > k, j > k.

2) If i ≤ k, j > k, we require EAij = −AijE. The condition becomes[B11 B12

−B21 −B22

]=

[−B11 B12

−B21 B22

]which implies B11 = B22 = 0. The same holds if i > k, j ≤ k.

So Ψ is given by a matrix of the form

[Ψ] =

A11 . . . A1(k+l)...

. . ....

A(k+l)1 . . . A(k+l)(k+l)

where the blocks Aij have either the form

[B11 00 B22

]or

[0 B12

B21 0

], according to the cases

above. We return to the naturality square(C⊕ktriv ⊕ C⊕lsign

)⊗(C⊕mtriv ⊕ C⊕nsign

)id⊗F

//

Ψ

(C⊕ktriv ⊕ C⊕lsign

)⊗(C⊕mtriv ⊕ C⊕nsign

)Ψ(

C⊕mtriv ⊕ C⊕nsign

)⊗(C⊕ktriv ⊕ C⊕lsign

)F⊗id

//

(C⊕mtriv ⊕ C⊕nsign

)⊗(C⊕ktriv ⊕ C⊕lsign

)

Drinfeld centers 72

and just like before

[id⊗F ] = [F ⊗ id] =

[F ] . . . 0...

. . ....

0 . . . [F ]

.By the commutativity of the diagram, we get Aij [F ] = [F ]Aij for all i, j. Note that

[F ] =

[F11 00 F22

]by proposition 8.1. If Aij has the form Aij =

[B11 00 B22

]then we have

Aij [F ] = [F ]Aij ⇐⇒[B11 00 B22

] [F11 00 F22

]=

[F11 00 F22

] [B11 00 B22

]⇐⇒

[B11F11 0

0 B22F22

]=

[F11B11 0

0 F22B22

]which implies B11 = aVijIm and B22 = dVijIn since the matrices Fii are arbitrary. If Aij has the

form Aij =

[0 B12

B21 0

]then we have

Aij [F ] = [F ]Aij ⇐⇒[

0 B12

B21 0

] [F11 00 F22

]=

[F11 00 F22

] [0 B12

B21 0

]⇐⇒

[0 B12F22

B21F11 0

]=

[0 F11B12

F22B21 0

]which holds if and only if

B12F22 = F11B12 and B21F11 = F22B21.

Since the matrices Fii are arbitrary, this implies B12 = B21 = 0. Using all of this, we may write

Ψ as a 2 × 2 block matrix [Ψ] =

[M 00 N

]where all entries are block matrices with block size

(m + n) × (m + n), where M is a k × k block matrix and N is an l × l block matrix. We seethat M corresponds to a map C⊕ktriv ⊗ V → V ⊗C⊕ktriv and N to a map C⊕lsign ⊗ V → V ⊗C⊕lsign,so we are done.

Proposition 8.12. Let (V,Ψ) and (W,Θ) where V and W are simple modules be in Z(G-mod).Then

HomZ(G-mod) ((V,Ψ), (W,Θ)) =

HomG-mod(V,W ) if V = W and Ψ = Θ0 otherwise

Proof. This follows immediately from Schur’s lemma and proposition 7.19.

We summarize the results of this section in the following theorem:

Markus Thuresson 73

Theorem 8.13. Each object of the Drinfeld center Z(G-mod) is isomorphic to a direct sumof the following objects, with some multiplicites:

(Ctriv,Φe), (Ctriv,Φs), (Csign,Φe) and (Csign,Φs).

Morphisms are given by proposition 8.12.

References

[1] S. Mac Lane Categories for the Working Mathematician, Second Edition.S. Mac Lane.Categories for the Working Mathematician. Second Edition. Springer Verlag, 1971.

[2] T. Leinster Basic Bicategories. Department of Pure Mathematics, University of Cambridge,1998.

[3] C. Kassel. Quantum Groups. Graduate Texts in Mathematics 155. Springer Verlag, NewYork, 1995.

[4] B.Sagan. The Symmetric Group. Graduate Texts in Mathematics 203. Springer Verlag,New York, 2001.