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

Drinfeld centers

Markus Thuresson

Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev April 2018

Department of Mathematics Uppsala University

Drinfeld centers Markus Thuresson

Abstract

In the first part of this paper, we describe the structure of the center Z(R-Mod) of the category of left R-modules. Its natural structure as a ring is shown to be isomorphic to the subring Z(R). In the sections that follow, we present the basics of monoidal categories by regarding them as single-object bicategories. The Drinfeld center Z(C ) of a monoidal category C is defined and its basic properties presented. The second half of the paper is devoted to describing the structure of the Drinfeld center of the monoidal categories VectC and Z2-mod. Contents

1 Basics of cateogries ...... 3

2 Categories of modules ...... 6 2.1 The center of R-mod ...... 6 2.2 Tensor product of modules ...... 11 2.2.1 Tensor product of bimodules ...... 17

3 2-categories ...... 21

4 Bicategories ...... 24 4.1 Basics ...... 24 4.2 Coherence ...... 32

5 Monoidal categories ...... 33 5.1 The Drinfeld center ...... 34

6 The Drinfeld center of VectC ...... 37 6.1 Equivalence ...... 45

7 Categories of group representations ...... 45 7.1 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 category theory. 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 objects X,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 morphism idX : 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, called the 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.

0 Remark 1.2. The identity morphism idX is unique for every object X ∈ Ob (C). If idX were another identity morphism, we would have

0 0 idX = idX ◦ idX = idX .

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

Definition 1.4. The terminal category, denoted by 11, is the category having a single object and a single morphism(the identity).

Definition 1.5. Let C be a category. A morphism ϕ ∈ Hom (X,Y ) is called an isomorphism if if 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 if there 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 )) such that

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 each map 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 if each 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 if each 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 natural transformation η : F ⇒ G is a map Ob (C) → HomD (F (X),G(X)) such that the diagram

F (ϕ) F (X) / F (Y )

ηX ηY   G(X) / G(Y ) G(ϕ) commutes for every morphism ϕ ∈ Hom (X,Y ).

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

Definition 1.13. Let C, D be categories. An equivalence of C and D is a functor F : C → D such that there exists another functor G : D → C and two natural isomorphisms η : F ◦G → idD and µ : G ◦ F → idC. Theorem 1.14. Let C, D be categories and F : C → D a functor. Then F is an equivalence if and 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. n Fixing the standard basis of the vector space C , we define a functor F : C → D by mapping n the object of C to the vector space C and mapping each matrix to the linear map which in the standard 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, are isomorphic. 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 the following:

- 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 1 the terminal category. Then we have C × 1 =∼ C (as objects in the category of small categories).

Proof. Consider the map F : C → C × 1 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 × 1 → 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 the class 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 with identity. 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 for the 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 equivalent definition. Now consider the center Z(M) of M. Since M has only one object and the morphisms are elements of M, a natural transformation from the identity functor to itself consists of an element z ∈ M such that the diagram

M x / M

z z   / M x M commutes for all x ∈ M. Since the composition is just multiplication in M, this is equivalent to 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 subring Z(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 map ϕ : 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 pointwise addition and composition. This implies that the componenwise addition and composition of a family of endomorphisms yields another family. We check that these operations preserve naturality. 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)

z Definition 2.4. For any z ∈ Z(R), define a natural transformation η : idR-Mod → idR-Mod z by ηM (x) = z · x, x ∈ M ∈ R-Mod. z z Remark 2.5. The natural transformation η is indeed in the center of R-Mod. Since ηM is given by left multiplication with z, we get:

z z ηN ◦ ϕ(x) = ηN (ϕ(x)) = z ·N ϕ(x)

= ϕ(z ·M x) z = ϕ(ηM (x)) z = ϕ ◦ ηM (x) which is equivalent to commutativity of the diagram

ϕ M / N

z z ηM ηN  ϕ  M / N

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

z Z Proposition 2.7. The homomorphisms ϕ induced by Z(R) form a subring EndR−(R) of EndR−(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.

Z Proposition 2.8. The rings Z(R) and EndR−(R) are isomorphic. Z z Proof. Consider the map F : Z(R) → EndR−(R) defined by F (z) = ϕ . Let z, c ∈ Z(R). We have:

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).

1 We clearly have F (1) = ϕ = idR so F is a homomorphism. For injectivity, note that F (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 induces an 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.  op Corollary 2.11. The rings EndR−(R) and R , are isomorphic.

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

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

Proof of theorem 2.9. Consider ε : Z(R-Mod) → EndR−(R) defined by η 7→ ηR. For every x ∈ 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 have z ηR = ϕ . z z z By definition of ε, η is mapped to ηR. But by the above, ηR = ϕ for any η ∈ Z(R-Mod). z z z Z So in particular, ε(η ) = ηR = ϕ . Thus, the map ε is surjective onto EndR−(R), since ηz ∈ 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 commutative diagrams: Drinfeld centers 10

ϕ ϕ R / M R / M

ηR=µR ηM ηR=µR µM  ϕ   ϕ  R / M R / 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 homomorphism ψ : 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 some z ∈ Z(R).

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

ηM (x) = ηM (ψ(1))

= ψ(ηR(1)) = ψ(z) = z · ψ(1) = z · x.

z z so we see that the map ηM is actually ηM for every module M, so we have η = η .  Markus Thuresson 11

Z Using lemma 2.13 we now see that the map ε : Z(R-Mod) → EndR−(R) is an isomorphism. Z By proposition 2.8 EndR−(R) is isomorphic to Z(R), so in conclusion we get ∼ Z ∼ ∼ Z(R-Mod) = EndR−(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 and let G be an abelian group. Then an R-balanced product is a map ϕ : M × N → G satisfying the following: i) ϕ(m, n + n0) = ϕ(m, n) + ϕ(m, n0)

ii) ϕ(m + m0, n) = ϕ(m, n) + ϕ(m0, n) iii) ϕ(m ·M r, n) = ϕ(m, r ·N n) for all m, m0 ∈ M, n, n0 ∈ 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 a balanced 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 unique group homomorphismϕ ˜ : M ⊗R N → G such thatϕ ˜ ◦ ⊗ = ϕ. In a commutative diagram:

⊗ M × N / M ⊗R N

∃!ϕ ˜ 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. Let  π 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 / M ⊗R N

ϕ 0 &   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 left R-module with module structure given by r · (x ⊗ y) = rx ⊗ y.

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

X X r · xi ⊗ yi = r · 1 ⊗ xiyi X = r · (1 ⊗ xiyi) X = r ⊗ xiyi X = r ⊗ xiyi X = rxi ⊗ yi so our scalar multiplication extends nicely to sums of elements of the form x⊗y. Next we check the module axioms:

i)

 n m  n m X X X X r ·  xi ⊗ yi + xj ⊗ yj = rxi ⊗ yi + rxj ⊗ yj i=1 j=1 i=1 j=1 n m X X = r · xi ⊗ yi + r · xj ⊗ yj. i=1 j=1

ii)

n n X X (r + s) · xi ⊗ yi = (r + s) · 1 ⊗ xiyi i=1 i=1 n ! X = (r + s) · 1 ⊗ xiyi i=1 n X = (r + s) ⊗ xiyi i=1 n n X X = r ⊗ xiyi + s ⊗ xiyi i=1 i=1 n n X X = rxi ⊗ yi + sxi ⊗ yi i=1 i=1 n n X X = r · xi ⊗ yi + s · xi ⊗ yi. i=1 i=1 Markus Thuresson 13 iii)

n n X X (rs) · xi ⊗ yi = (rs)xi ⊗ yi i=1 i=1 n X = r(sxi) ⊗ yi i=1 n X = r · sxi ⊗ yi i=1 n X = r · (s · xi ⊗ yi). i=1

iv)

n n n X X X 1 · xi ⊗ yi = 1xi ⊗ yi = xi ⊗ yi. i=1 i=1 i=1



Proposition 2.18. Let M be a left R-module. Then the left R-modules R ⊗R M and M are isomorphic. n n P P Proof. Consider the map ϕ : R ⊗R M → M defined by xi ⊗ yi 7→ xiyi. i=1 i=1

 n m   n m  X X X X ϕ  xi ⊗ yi + xj ⊗ yj = ϕ  1 ⊗ xiyi + 1 ⊗ xjyj i=1 j=1 i=1 j=1

 n m  X X = ϕ 1 ⊗ xiyi + 1 ⊗ xjyj i=1 j=1

  n m  X X = ϕ 1 ⊗  xiyi + xjyj i=1 j=1

 n m  X X = 1 ·  xiyi + xjyj i=1 j=1 n m X X = xiyi + xjyj i=1 j=1 n !  m  X X = ϕ xi ⊗ yi + ϕ  xj ⊗ yj . i=1 j=1 Drinfeld centers 14

n ! n ! X X ϕ r · xi ⊗ yi = ϕ rxi ⊗ yi i=1 i=1 n X = rxiyi i=1 n X = r · xiyi i=1 n ! X = r · ϕ xi ⊗ yi i=1 so ϕ is a homomorphism. Suppose

n ! n m  m  X X X 0 0 X 0 0 ϕ xi ⊗ yi = xiyi = xjyj = ϕ  xj ⊗ yj . i=1 i=1 j=1 j=1

Then we get:

n n X X xi ⊗ yi = 1 ⊗ xiyi i=1 i=1 n X = 1 ⊗ xiyi i=1 m X 0 0 = 1 ⊗ xjyj j=1 m X 0 0 = 1 ⊗ xjyj j=1 m X 0 0 = xj ⊗ yj j=1 so ϕ is injective. For every m ∈ M, we have ϕ(1 ⊗ m) = m, so ϕ is surjective, hence an isomorphism.

Proposition 2.19. Let M,N be left R-modules and let ϕ : M → N be a homomorphism. Then the map ϕ⊗ : R ⊗R M → R ⊗R N defined by x ⊗ y 7→ x ⊗ ϕ(y) is a homomorphism. Markus Thuresson 15

Proof. n ! n ! X X ϕ⊗ xi ⊗ yi = ϕ⊗ 1 ⊗ xiyi i=1 i=1 n ! X = ϕ⊗ 1 ⊗ xiyi i=1 n ! X = 1 ⊗ ϕ xiyi i=1 n X = 1 ⊗ ϕ(xiyi) i=1 n X = 1 ⊗ xiϕ(yi) i=1 n X = 1 ⊗ xiϕ(yi) i=1 n X = xi ⊗ ϕ(yi) i=1

 n m    n m  X X X X ϕ⊗  xi ⊗ yi + xj ⊗ yj = ϕ⊗ 1 ⊗  xiyi + xjyj i=1 j=1 i=1 j=1

 n m  X X = 1 ⊗ ϕ  xiyi + xjyj i=1 j=1

 n m  X X = 1 ⊗  xiϕ(yi) + xjϕ(yj) i=1 j=1 n m X X = 1 ⊗ xiϕ(yi) + 1 ⊗ xjϕ(yj) i=1 j=1 n m X X = xi ⊗ ϕ(yi) + xj ⊗ ϕ(yj) i=1 j=1 n !  m  X X = ϕ⊗ xi ⊗ yi + ϕ⊗  xj ⊗ yj i=1 j=1 n ! n ! X X ϕ⊗ r xi ⊗ yi = ϕ⊗ rxi ⊗ yi i=1 i=1 n X = rxi ⊗ ϕ(yi) i=1 n X = r · xi ⊗ ϕ(yi) i=1 n ! X = r · ϕ⊗ xi ⊗ yi . i=1 Drinfeld centers 16

Proposition 2.20. Define F on R-Mod by

i) M 7→ R ⊗R M for modules.

ii) ϕ : M → N 7→ ϕ⊗ : R ⊗R M → 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:

n ! n ! X X F (ψ ◦ ϕ) xi ⊗ yi = (ψ ◦ ϕ)⊗ xi ⊗ yi i=1 i=1 n X = xi ⊗ (ψ ◦ ϕ)(yi) i=1 n X = xi ⊗ ψ(ϕ(yi)) i=1 n ! X = ψ⊗ xi ⊗ ϕ(yi) i=1 n !! X = ψ⊗ ϕ⊗ xi ⊗ yi i=1 n ! X = F (ψ) ◦ F (ϕ) xi ⊗ yi . i=1

∼ 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 ⊗R M → M given by

n ! n X X ηM xi ⊗ yi = xiyi. i=1 i=1 This is an isomorphism of modules by proposition 2.18. The diagram

ϕ⊗ R ⊗R M / R ⊗R N

ηM ηN   / M ϕ N Markus Thuresson 17 commutes since

n !! n ! X X ϕ ηM xi ⊗ yi = ϕ xiyi i=1 i=1 n X = xiϕ(yi) i=1 n ! X = ηN xi ⊗ ϕ(yi) i=1 n !! X = ηN ϕ⊗ xi ⊗ yi i=1

so η is a natural isomorphism.

Theorem 2.22. Bimodule endomorphisms of R are exactly the components at R of the natural transformations 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 know that 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 be an S-T -bimodule. Then M ⊗S N is an R-T -bimodule, with scalar multiplication given by

 n n P P  r · mi ⊗ ni = rmi ⊗ ni  i=1 i=1 m m ∀r ∈ R, t ∈ T P P  mj ⊗ nj · t = mj ⊗ nit  j=1 j=1 Drinfeld centers 18

Proof.

 n m  n m X X X X r ·  mi ⊗ ni + mj ⊗ nj = rmi ⊗ ni + rmj ⊗ nj i=1 j=1 i=1 j=1 n m X X = r · mi ⊗ ni + r · mj ⊗ nj i=1 j=1

n n 0 X X (r + r ) mi ⊗ ni = (r + s)mi ⊗ ni i=1 i=1 n X = (rmi + smi) ⊗ ni i=1 n X 0 = rmi ⊗ ni + r mi ⊗ ni i=1 n n X 0 X = r · mi ⊗ ni + r · mi ⊗ ni i=1 i=1

n n 0 X X 0 (rr ) · mi ⊗ ni = (rr )mi ⊗ ni i=1 i=1 n X 0 = r(r mi) ⊗ ni i=1 n X 0 = r · r mi ⊗ ni i=1

n n X X 1R · mi ⊗ ni = 1Rmi ⊗ ni i=1 i=1 n X = mi ⊗ ni i=1

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,

n ! n ! X X r · mi ⊗ ni · t = rmi ⊗ ni · t i=1 i=1 n X = rmi ⊗ nit i=1 n ! X = r · mi ⊗ nit . i=1

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 of the 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 makes the following diagram commute:

⊗T (M ⊗S N) × L / (M ⊗S N) ⊗T L

ϕ 0 )  . (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 n X X (mi ⊗S ni) ⊗T li 7→ mi ⊗S (ni ⊗T li). i=1 i=1 Drinfeld centers 20

 n m  n m X X X X f  (mi ⊗S ni) ⊗T li + (mj ⊗S nj) ⊗T lj = mi ⊗S (ni ⊗T li) + mj ⊗S (nj ⊗T lj) i=1 j=1 i=1 j=1

n !  m  X X = f (mi ⊗S ni) ⊗T li + f  (mj ⊗S nj) ⊗T lj i=1 j=1

n ! n ! X X f r · (mi ⊗S ni) ⊗T li · u = f r(mi ⊗S ni) ⊗T liu i=1 i=1 n ! X = f (rmi ⊗S ni) ⊗T liu i=1 n X = rmi ⊗S (ni ⊗T liu) i=1 n X = r · mi ⊗S (ni ⊗T li) · u i=1 n ! X = r · f (mi ⊗S ni) ⊗T li · u i=1 for any r ∈ R, u ∈ U so f is a homomorphism of bimodules. Moreover, f is clearly invertible with inverse given by

n n X X mi ⊗S (ni ⊗T li) 7→ (mi ⊗S ni) ⊗ li i=1 i=1 so f is an isomorphism.

0 0 Proposition 2.26. Let f :R MS →R MS and g : SNT → SNT be bimodule homomorphisms. 0 0 Then the map f ⊗ g :R (M ⊗S N)T →R (M ⊗S N )T defined by

n n X X mi ⊗ ni 7→ f(mi) ⊗ g(ni) i=1 i=1 is a homomorphism of R-T -bimodules. Markus Thuresson 21

Proof.

 n m  n m X X X X f ⊗ g  mi ⊗ ni + mj ⊗ nj = f(mi) ⊗ g(ni) + f(mj) ⊗ g(nj) i=1 j=1 i=1 j=1

n !  m  X X = f ⊗ g mi ⊗ ni + f ⊗ g  mj ⊗ nj i=1 j=1

n ! n ! X X f ⊗ g r · mi ⊗ ni · t = f ⊗ g rmi ⊗ nit i=1 i=1 n X = f(rmi) ⊗ g(nit) i=1 n X = rf(mi) ⊗ g(ni)t i=1 n X = r · f(mi) ⊗ g(ni) · t i=1 n ! X = r · f ⊗ g mi ⊗ ni · t. i=1

3 2-categories

If we consider a category where the morphism classes are themselves equipped with the structure of a category, we arrive at the notion of a 2-category. Much of this section follows from [1], with more 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. The objects 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 denoted by • and is also called vertical composition.

- for every object X, a functor IX from the terminal category 1 to C(X,X). This functor maps the object of 1 to the identity 1-morphism idX : X → X and the morphism of 1 to the identity 2-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 called horizontal composition. Example 3.2 (Vertical composition). For objects X,Y , 1-morphisms f, g, h and 2-morphisms α, β as in

f

α   / X g YG β  h we get the 2-morphism β • α : f ⇒ h as in f

 β•α X YF

 h Example 3.3 (Horizontal composition). For f, f 0, g, g0, α, β as in f g

  X α YF β ZF

  f 0 g0 we get the 2-morphism β ◦ α : g ◦ f ⇒ g0 ◦ f 0 as in g ◦ f

 X β◦α ZD

 g0 ◦ f 0 Remark 3.4 (Interchange law). Since ◦ is a functor, it commutes with the (vertical) composition of the hom-categories, so we have, for composable 2-morphisms as in f f 0

α α0     / 0 / X g YG g ZG β β0   h h0 Markus Thuresson 23 we have

(β0 ◦ β) • (α0 ◦ α) = (β0 • α0) ◦ (β • α).

Or, in terms of diagrams:

f 0 ◦ f f f 0

α0◦α     0 / = β•α β0•α0 X g ◦ g ZD X YG ZF β0◦β    h0 ◦ h h h0 The left diagram corresponds to the left hand side of the above equation and the right diagram corresponds 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 objects are 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. For 1-morphisms f, g, h and a 2-morphism α we get the following diagram for hα:

f h h ◦ f

   X α YG idh ZG = X idh •α ZC

  g  h h ◦ g

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 triple of maps; a map of objects, a map of 1-morphisms and a map of 2-morphisms, satisfying the following:

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, α0, we have F (α0 ◦ α) = F (α0) ◦ F (α).

iv) For vertically composable 2-morphisms β, β0 we have, F (β0 • β) = F (β0) • F (β). Drinfeld centers 24

Remark 3.8. Note that a 2-functor F : C → D, when applied to the objects and 1-morphisms is an 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 additional structure of a 2-category.

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

F (f) G(f)

  ηY ηX F (X) F (α) F (Y ) / G(Y ) = F (X) / G(X) G(α) G(Y ) A A

  F (g) G(g)

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

F (f) η η G(f) F (X) / F (Y ) Y / G(Y ) = F (X) X / G(X) / G(Y ) which is just the usual naturality square

F (f) F (X) / F (Y )

ηX ηY  G(f)  G(X) / G(Y ) for a natural transformations between the ordinary functors of F and G, so in the same way as with 2-functors, 2-natural transformations can be seen as extensions of ordinary natural transformations to the 2-categorical framework.

4 Bicategories

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

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 1 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 ?XYZ . 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 αXYZW as given in

idB(Z,W ) ×?XYZ B(Z,W ) × B(Y,Z) × B(X,Y ) / B(Z,W ) × B(X,Z) 5

αXYZW ?YZW ×idB(X,Y ) ?XZW

  B(Y,W ) × B(X,Y ) / B(X,W ) ?XYW called the associator. For 1- and 2-morphisms as given in

f g h

   ϕ X YF ψ ZF ξ WF

   f 0 g0 h0 the naturality of α yields the commutative diagram

(ξ?ψ)?ϕ (hg)f / (h0g0)f 0

αhgf αh0g0f0   h(gf) / h0(g0f 0) ξ?(ψ?ϕ) 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 1 × B(X,Y )

∼ IY ×idB(X,Y ) 8 λXY  & B(Y,Y ) × B(X,Y ) / B(X,Y ) ?XYY and B(X,Y ) × 1

∼ idB(X,Y ) ×IX 8 ρXY  & B(X,Y ) × B(X,X) / B(X,Y ) ?XYY called left and right unitors, respectively. So for a 1-morphism f ∈ B(X,Y ), we have invertible 2-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?id / (k(hg))f

α α ~ (kh)(gf) k((hg)f)

α id ?α ' w k(h(gf))

(gI)f α / g(If)

ρ?id id ?λ " | gf Remark 4.2. It is clear that if the natural isomorphisms α, λ, ρ are all identities, in which case the composition is strictly associative, then the definition of bicategory coincides with that of a 2-category. Markus Thuresson 27

Definition 4.3. Let B be a bicategory. An internal equivalence in B consists if a pair of 1-morphisms as given in f ) X i Y g ∼ 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 the following: - 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 ϕXYZ as given in ? B(Y,Z) × B(X,Y ) B / B(X,Z) 5

ϕXYZ FYZ ×FXY FXZ

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

IX 5 FXX ϕX  1 / C(F (X),F (X)) IF (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, denoting the associators in the categories B, C by αB and αC respectively:

ϕ?id ϕ (F (h)F (g))F (f) / F (hg)F (f) / F ((hg)f)

αC F (αB)   F (h)(F (g)F (f)) / F (h)F (gf) / F (h(gf)) id ?ϕ ϕ Drinfeld centers 28

id ?ϕ ϕ F (f) idF (X) / F (f)F (idX ) / F (f idX )

ρF (f) F (ρ ) ' w f F (f)

ϕ?id ϕ idF (Y ) F (f) / F (idY )F (f) / F (idY f)

λ F (λ ) F (f) ' w f F (f)

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

Definition 4.6. If (F, ϕ) is a lax functor such that all the natural transformations ϕXYZ and ϕX 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 ϕXYZ and ϕX 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:

F B(X,Y ) XY / C(F (X),F (Y )) 5 ηXY GXY ηY ◦   C(G(X),G(Y )) / C(F (X),G(Y )) ◦ηX

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

−1 αC id ?ηf αC (G(g)G(f)) ηX / G(g)(G(f)ηX ) / G(g)(ηY F (f)) / (G(g)ηY ) F (f)

ηg?id  (ηZ F (g)) F (f)

ψ?id αC  ηZ (F (g)F (f))

id ?ϕ   G(gf)ηX / ηZ F (gf) ηgf

−1 λC ρC idG(X) ηX / ηX / ηX idF (X)

ψ?id id ?ϕ   / G(idX )ηX η ηX F (idX ) idX

Definition 4.9. If η is a lax natural transformation such that the natural transformations ηXY are 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 that the following diagram commutes:

id ?ΓX G(f)ηX / G(f)µX

ηf µf   ηY F (f) / µY F (f) ΓY ?id 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 would look like this:

0 RMS SMT ϕ ϕ0     / 0 / R RNS SE SNT TE ψ ψ0   0 RLS SLT 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 tensor product behaves nicely with respect to the bimodule structure, so the composite would look like: Drinfeld centers 30

0 R(M ⊗S M )T

ϕ⊗ϕ0  ! 0 / R R(N ⊗S N )T = T ψ⊗ψ0  0 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 the 1-morphisms. So the diagram

f

 α X YF

g  in B becomes the diagram

f

Ö α XYX

g  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-morphisms are modifications. It has a sub-bicategory [B, C] consisting of pseudofunctors, pseudonatural transformations, 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 α  G   B β DCJ  H γ  L

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

α β γ F (X) X / G(X) X / H(X) X / 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 = id α for any transformation α. Similarily, if we have lax functors, natural transformations and modifications as given in

B

α  Γ β

( G Σ F 6> >F  γ  Ω δ

" | C

we get 2-morphisms

ΓX ΣX ΩX αX +3 βX +3 γX +3 δX which now are 2-morphisms of C and again this yields associativity. It is clear that we by the same 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 of psuedofunctors

F ( B i C G together with an internal equivalence idB → GF in [B, B] and an internal equivalence FG → idC in [C, C]. It can be shown that a pseudofunctor F : B → C admits a biequivalence if and only if F 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 is an 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 without proof. Theorem 4.17 (Yoneda lemma for bicategories). Let B be a bicategory and let F : Bop → Cat be 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 this we mean that C is the sub-2-category of [Bop, Cat] whose objects are the objects in the image of 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 a biequivalence. Markus Thuresson 33

5 Monoidal categories

A monoidal category is usually defined as a category equipped with a tensor product. So for a category 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 the hom-category of a bicategory with one single object. Then, taking the tensor product as horizontal composition and the identity 1-morphism as the tensor unit, the associator and unitor isomorphisms together with their coherence axioms yield 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 previous section. 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 details spelled out.

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

βX,Y : X ⊗ Y → Y ⊗ X.

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

β X ⊗ (Y ⊗ Z) / (Y ⊗ Z) ⊗ X 5 α α ) (X ⊗ Y ) ⊗ Z Y ⊗ (Z ⊗ X) 5 β⊗id ) id ⊗β / (Y ⊗ X) ⊗ Z α Y ⊗ (X ⊗ Z)

β (X ⊗ Y ) ⊗ Z / Z ⊗ (X ⊗ Y ) 5 α−1 α−1 ) X ⊗ (Y ⊗ Z) (Z ⊗ X) ⊗ Y 5 id ⊗β ) β⊗id X ⊗ (Z ⊗ Y ) / (X ⊗ Z) ⊗ Y α−1 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 ) as the 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

id ⊗g X ⊗ Y X / X ⊗ Z

ηX,Y ηX,Z   Y ⊗ X / Z ⊗ X g⊗idX 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

f⊗id X ⊗ Z Z / Y ⊗ Z

ηX,Z ηY,Z   Z ⊗ X / Z ⊗ Y idZ ⊗f

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 necessary to make sense of the equations. With the associators spelled out, the conditions amount to commutativity of the following diagrams:

(Y ⊗ X) ⊗ Z α / Y ⊗ (X ⊗ Z) 6 ηX,Y ⊗idZ idY ⊗ηX,Z ( (X ⊗ Y ) ⊗ Z Y ⊗ (Z ⊗ X) O α−1 α−1  X ⊗ (Y ⊗ Z) / (Y ⊗ Z) ⊗ X ηX,Y ⊗Z

−1 X ⊗ (Z ⊗ Y ) α / (X ⊗ Z) ⊗ Y 6 idX ⊗ηY,Z ηX,Z ⊗idY ( X ⊗ (Y ⊗ Z) (Z ⊗ X) ⊗ Y O α α  (X ⊗ Y ) ⊗ Z / Z ⊗ (X ⊗ Y ) ηX⊗Y,Z which in turn yield the accurate equations:

−1 −1 ηX,Y ⊗Z = αYZX (idY ⊗ηX,Z )αYXZ (ηX,Y ⊗ idZ )αXYZ −1 ηX⊗Y,Z = αZXY (ηX,Z ⊗ idY )αXZY (idX ⊗ηY,Z )αXYZ .

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 a monoid. We can impose a tensor product on M by putting • ⊗ • = • for the object of M and 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 the form (•, η•, ). Since • is the only object in M , η•, has only the component η•,•. Denoting this component 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, we require 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 invertible and 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 a strict 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 morphism in 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

ηX⊗Y,Z

id ⊗η η ⊗id ) X ⊗ Y ⊗ Z X Y,Z / X ⊗ Z ⊗ Y X,Z Y / Z ⊗ X ⊗ Y ηX,Y ⊗Z ηX,Z⊗Y ηX,Y ⊗idZ idZ ⊗ηX,Y  ) )  Y ⊗ X ⊗ Z / Y ⊗ Z ⊗ X / Z ⊗ Y ⊗ X idY ⊗ηX,Z ηY,Z ⊗idX 5

ηY ⊗X,Z Markus Thuresson 37 commutes. It does, since the triangles commute by our conditions and the center parallelogram is just a naturality square of ηX, . Left to check are the hexagon identities. In a strict monoidal category, these are equivalent to the diagrams

η η X ⊗ Y ⊗ Z X,Y ⊗Z/ Y ⊗ Z ⊗ X X ⊗ Y ⊗ Z X⊗Y,Z/ Z ⊗ X ⊗ Y 6 6

ηX,Y ⊗idZ idX ⊗ηY,Z  idY ⊗ηX,Z  ηX,Z ⊗idY Y ⊗ X ⊗ Z X ⊗ Z ⊗ Y 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 monoidal category C , its Drinfeld center Z(C ) is a braided monoidal category. The proof, however, is not given here.

6 The Drinfeld center of VectC

Now we consider the category VectC, consisting of finite-dimensional vector spaces over C and linear maps between them. We impose the structure of a monoidal category on VectC using the 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 in Z(VectC). Proof. We need the square id ⊗F V ⊗ W V / V ⊗ X

ΦV,W ΦV,X   W ⊗ V / X ⊗ V F ⊗idV 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 −1 −1 ΦV,W ⊗X = αWXV (idW ⊗ΦV,X )αWVX (ΦV,W ⊗ idX )αVWX .

−1 −1 −1 α (idW ⊗ΦV,X )α(ΦV,W ⊗ idX )α (v ⊗ (w ⊗ x)) = α (idW ⊗ΦV,X )α(ΦV,W ⊗ idX )((v ⊗ w) ⊗ x) −1 = α (idW ⊗ΦV,X )α((w ⊗ v) ⊗ x) −1 = α (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 diagram below. (W ⊗ V ) ⊗ X α / W ⊗ (V ⊗ X) 6 ΦV,W ⊗idX idW ⊗ΦV,X ( (V ⊗ W ) ⊗ X W ⊗ (X ⊗ V ) O α−1 α−1  V ⊗ (W ⊗ X) / (W ⊗ X) ⊗ V ΦV,W ⊗X

To avoid notational clutter, from this point forward, we drop the indices of morphisms when domain and codomain are clear from context. Proposition 6.2.

Hom ((V, Φ), (W, Φ)) = HomVect (V,W ) . Z(VectC) C Proof. By definition, Hom ((V, Φ), (W, Φ)) consists of linear maps F : V → W such that Z(VectC) the diagram F ⊗id V ⊗ X / W ⊗ X

Φ Φ   X ⊗ V / X ⊗ W id ⊗F 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 Hom ((V, Φ), (W, Φ)) by proposition Z(VectC) 6.2 and, similarly, its inverse is in Hom ((W, Φ), (V, Φ)), so G is an isomorphism in Z(VectC) Z(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 the basis {vi} of V . Then {1 ⊗ vi} is a basis of C ⊗ V and {vi ⊗ 1} is a basis of V ⊗ C. Put n = dim V . Let F : V → V be some linear map. Then, the square

id ⊗F C ⊗ V / C ⊗ V

Ψ Ψ   V ⊗ / V ⊗ C F ⊗id 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 must hold 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 given by

1 ⊗ v 7→ λV v ⊗ 1.

We note that the diagram id ⊗F C ⊗ V / C ⊗ W Ψ Ψ   V ⊗ / W ⊗ C F ⊗id 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))

= λW F (v) ⊗ 1 Drinfeld centers 40

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

(V ⊗ ) ⊗ W α / V ⊗ ( ⊗ W ) 6 C C Ψ⊗id id ⊗Ψ ( ( ⊗ V ) ⊗ W V ⊗ (W ⊗ ) C O C α−1 α−1  ⊗ (V ⊗ W ) / (V ⊗ W ) ⊗ C Ψ 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 dim X = 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 seek to generalize the previous arguments higher dimensions. Our first objects of study are elements 2
k of the form C , Ψ . Throughout this section, we fix the standard basis of C . If V is some vector space with a basis {v1, . . . , vn}, we fix the basis

{e1 ⊗ v1, . . . , e1 ⊗ vn, . . . , ek ⊗ v1, . . . , ek ⊗ vn}

k of C ⊗ V and the basis

{v1 ⊗ e1, . . . , vn ⊗ e1, . . . , v1 ⊗ ek, . . . , vn ⊗ ek}

k 2
of V ⊗ C . To help us classify objects of the form C , Ψ in Z(VectC), we have the following lemma. Markus Thuresson 41

2
Lemma 6.6. If C , Ψ ∈ Z(VectC), then the component of Ψ

2 2 ΨX : C ⊗ X → X ⊗ C is given by the (invertible) matrix   aX Idim X bX Idim X [ΨX ] = cX Idim X dX Idim X with respect to the bases chosen above, for any vector space X. In particular, the component of a b  Ψ at the vector space C is given by an invertible 2 × 2 matrix C C . cC dC Proof. Consider the component of Ψ at X. Let n = dim X and let F : X → X be a linear map 2 2 given by the n × n matrix [F ]. Note that C ⊗ X and X ⊗ C have dimension 2n, so the linear isomorphism Ψ is given by a 2n × 2n matrix [Ψ]. We write the matrix of Ψ as a 2 × 2 block AB matrix with blocks of size n × n, so we have [Ψ] = . By naturality of Ψ, the square CD

2 id ⊗F 2 C ⊗ V / C ⊗ V Ψ Ψ   V ⊗ 2 / V ⊗ 2 C F ⊗id C commutes. With respect to the chosen bases, we have

[F ] 0  [id ⊗F ] = [F ⊗ id] = . 0 [F ]

Naturality then amounts to the matrix equation

AB [F ] 0  [F ] 0  AB = CD 0 [F ] 0 [F ] CD 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 = aX In,B = bX In,C = cX In,D = dX In.

Now it is clear that AB a I b I  [Ψ] = = X n X n . CD cX In dX In Drinfeld centers 42

More explicitly, Ψ has the matrix   aX 0 ... bX 0 ...  0 aX 0 bX     ......   . . . .  [Ψ] =   .  cX 0 ... dX 0 ...     0 cX 0 dX   . . . .  ......

 2
Proposition 6.7. If C , Ψ is in Z(VectC) then Ψ = Φ. Proof. By naturality, the square

2 id ⊗F 2 C ⊗ V / C ⊗ W Ψ Ψ   V ⊗ 2 / W ⊗ 2 C F ⊗id C commutes for all vector spaces V,W and every linear map F : V → W . Let n = dim V and m = dim W . Fixing bases {v1, . . . , vn} and {w1, . . . , wm} of V and W respectively, we obtain bases for all involved vector spaces. Denote the component at V by ΨV and the component at W by ΨW . By lemma 6.6, we have     aV In bV In aW Im bW Im [ΨV ] = and [ΨW ] = . cV In dV In cW 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  [F ] 0  [id ⊗F ] = [F ⊗ id] = m×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 ] 0  a I b I  a I b I   [F ] 0  ⇐⇒ m×n V n V n = W m W m m×n 0m×n [F ] cV In dV In cW Im dW Im 0m×n [F ] a [F ] b [F ] a [F ] b [F ] ⇐⇒ V V = W W . cV [F ] dV [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 across all vector spaces, so we may drop the indices. This is an improvement upon the result of lemma 6.6, in which the numbers may depend on the vector space. 2
Since C , Ψ ∈ Z(VectC), the diagram

(V ⊗ 2) ⊗ W α / V ⊗ ( 2 ⊗ W ) 6 C C Ψ⊗id id ⊗Ψ ( ( 2 ⊗ V ) ⊗ W V ⊗ (W ⊗ 2) C O C α−1 α−1  2 ⊗ (V ⊗ W ) / (V ⊗ W ) ⊗ 2 C Ψ C commutes. Putting V = W = C, this diagram becomes

( ⊗ 2) ⊗ α / ⊗ ( 2 ⊗ ) C6 C C C C C Ψ⊗id id ⊗Ψ ( ( 2 ⊗ ) ⊗ ⊗ ( ⊗ 2) C CO C C C C α−1 α−1  2 ⊗ ( ⊗ ) / ( ⊗ ) ⊗ 2 C C C Ψ C C C and now we observe that, with respect to the chosen bases, we have

a b [α] = [α−1] = I and [Ψ ⊗ id] = [id ⊗Ψ] = [Ψ] = . 2 c 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.

2 2 This means that for any vector space, the matrix of Ψ : C ⊗ → ⊗ C with respect to the chosen 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 establish the following proposition. Drinfeld centers 44

k
Proposition 6.8. If C , Ψ is in Z(VectC), then Ψ = Φ. Proof. First, we consider the component of Ψ at some vector space X. Let n = dim X. By naturality, the square k id ⊗F k C ⊗ X / C ⊗ X Ψ Ψ   X ⊗ k / X ⊗ k C F ⊗id C commutes. Writing [Ψ] as a k × k block matrix with block size n × n, we have   A11 ...A1k  . .. .  [Ψ] =  . . .  Ak1 ...Akk and with respect to our bases we have [F ] ... 0   . .. .  [id ⊗F ] = [F ⊗ id] =  . . .  . 0 ... [F ]

X Naturality now yields Aij[F ] = [F ]Aij for all i, j ∈ {1, . . . , k}. This implies Aij = aij In as in the case k = 2. The square k id ⊗F k C ⊗ V / C ⊗ W Ψ Ψ   V ⊗ k / W ⊗ k C F ⊗id C commutes for all vector spaces V,W and every linear map F : V → W . The exact same V W technique as in the case k = 2 can be used to show that aij = aij for all V,W . Putting V = W = C and using the commutativity of the diagram ( ⊗ k) ⊗ α / ⊗ ( k ⊗ ) C6 C C C C C Ψ⊗id id ⊗Ψ ( ( k ⊗ ) ⊗ ⊗ ( ⊗ k) C CO C C C C α−1 α−1  k ⊗ ( ⊗ ) / ( ⊗ ) ⊗ k C C C Ψ C C C we get that the component at a one-dimensional vector space is given by [Ψ] = Ik which implies  0 if i 6= j a = ij 1 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 Drinfeld center 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,

- HomVect (V,W ) 3 f 7→ f ∈ Hom ((V, Φ) , (W, Φ)) for morphisms. C Z(VectC) It is clear from the results of the previous sections that F is a bijection on objects and on morphisms, hence an isomorphism of categories.

Denoting the tensor product of Z(VectC) by ⊗Z to distinguish it from the usual tensor product in 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 follow from [4].

Definition 7.1. Let K be a field, V a vector space over K and let G be a group. Then a representation 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 a G-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 linear map so defining the map ϕ : G × V → V by ϕ(g, v) = ρ(g)(v) defines a G-module structure on V . 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 group G 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 map F : V → W such that

F (g(v)) = g(F (v)) for every g ∈ G and v ∈ V . In other words, the diagram

V F / W

g· g·   V / W F commutes. Definition 7.6. Let V be a G-module and let W ⊂ V be a linear subspace. Then W is a submodule 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. A submodule 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. Then ker(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 that they 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 that any 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-module with 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 yields the condition

g(v ⊗ w + v0 ⊗ w0) = g(v ⊗ w) + g(v0 ⊗ w0).

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 that defining 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 that char(K) - |G|. Then any finite-dimensional G-module can be written as a direct sum of simple modules.

Proof. If V is simple, we are done. If not, we use induction on the dimension of V . Since one-dimensional modules are automatically simple, it suffices to show that any submodule of V has a submodule complement. Let Y be a submodule of V . Let X be a complement of Y , that is, a subspace such that V = X ⊕ Y . Note that X is not necessarily a submodule.Let P0 : V → V be a linear projection onto Y . 1 P −1 Now, define the linear endomorphism P = |G| g P0g. g∈G Markus Thuresson 49

Then, for any y ∈ Y , we have

1 X P (y) = g−1P g(y) |G| 0 g∈G 1 X = g−1P (g(y)) |G| 0 g∈G 1 X = g−1g(y) |G| g∈G 1 X = y |G| g∈G = 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 , where now Z = ker(P ) and Y = im(P ). Next, we note that P is a G-homomorphism. Indeed,

1 X P (h(v)) = g−1P g(h(v)) |G| 0 g∈G 1 X = hh−1g−1P gh(v) |G| g∈G 1 X = h (gh)−1P (gh)(v) |G| g∈G = hP (v).

Now Z is a submodule since it is the kernel of a G-homomorphism, so we have a submodule complement 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 following hold:

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 either ker(F ) = 0 or ker(F ) = V since V is simple. Since F is non-zero, we have ker(F ) = 0 which implies that F is injective. Similarily, since im(F ) is a submodule, we have either im(F ) = 0 or im(F ) = W since W is simple. Since F -is non-zero we have im(F ) = W , so F is surjective. So F is bijective, so it 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, 0 0 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 = 0 which implies F = λ idV .

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

 1 if V =∼ W dim Hom (V,W ) = G-mod 0 if V =6∼ W Proof. If V =6∼ W , then the only G-homomorphism between them is the zero map by Schur’s lemma. Suppose V =∼ W and let F,G : V → W be G-isomorphisms. Then G−1F : V → V is −1 an endomorphism of V , so we have G F = λ idV . Then

−1 −1 G F = λ idV =⇒ G(G F ) = G(λ idV ) −1
=⇒ GG 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 the structure of a monoidal category on G-mod, by taking the tensor product as the usual tensor product of K-vector spaces, with the addition of defining action of G on this tensor product as in proposition 7.9. The tensor unit will be the trivial G-module K and the associator and unitor isomorphisms well be the canonical isomorphisms we are familiar with from the monoidal category 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 canonical isomorphism 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 the diagram id ⊗F V ⊗ W / V ⊗ X

Φz Φz   W ⊗ V / X ⊗ V F ⊗id 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) 6 Φz⊗id id ⊗Φz ( (V ⊗ W ) ⊗ X W ⊗ (X ⊗ V ) O α−1 α−1  V ⊗ (W ⊗ X) / (W ⊗ X) ⊗ V Φz 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

F ⊗id V ⊗ X / W ⊗ X

Φz Φz   X ⊗ V / X ⊗ W id ⊗F 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 have isomorphisms

Ψ: V ⊗ X → X ⊗ V and W ⊗ X → X ⊗ W.

Define Ψ
Θ by

−1 Ψ
Θ := δ (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:

id ⊗F (V ⊕ W ) ⊗ X / (V ⊕ W ) ⊗ Y

ε ε

 (id ⊗F )⊕(id ⊗F )  (V ⊗ X) ⊕ (W ⊗ X) / (V ⊗ Y ) ⊕ (W ⊗ Y )

Ψ⊕Θ Ψ⊕Θ

 (F ⊗id)⊕(F ⊗id)  (X ⊗ V ) ⊕ (X ⊗ W ) / (Y ⊗ V ) ⊕ (Y ⊗ W )

δ−1 δ−1

  X ⊗ (V ⊕ W ) / Y ⊗ (V ⊕ W ) F ⊗id

It is clear that the perimeter is a naturality square of Ψ
Θ. Moreover, the middle square is just the sum of naturality squares of Ψ and Θ, which commute by assumption. Hence, the middle 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 a natural family of isomorphisms. Next, we consider the diagram Drinfeld centers 54

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

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

(Ψ⊕Θ)⊗id ε  s %  (V ⊗ (X ⊗ Y )) ⊕ (W ⊗ (X ⊗ Y )) ((X ⊗ V ) ⊕ (X ⊗ W )) ⊗ Y ((V ⊗ X) ⊗ Y ) ⊕ ((W ⊗ X) ⊗ Y )

ε δ−1⊗id (Ψ⊗id)⊕(Θ⊗id)  +  (X ⊗ (V ⊕ W )) ⊗ Y ((X ⊗ V ) ⊗ Y ) ⊕ ((X ⊗ W ) ⊗ Y )

Ψ⊕Θ α α⊕α   X ⊗ ((V ⊕ W ) ⊗ Y ) (X ⊗ (V ⊗ Y )) ⊕ (X ⊗ (W ⊗ Y ))

δ−1 id ⊗ε (id ⊗Ψ)⊕(id ⊗Θ)   s  ((X ⊗ Y ) ⊗ V ) ⊕ ((X ⊗ Y ) ⊗ W ) X ⊗ ((V ⊗ Y ) ⊕ (W ⊗ Y )) (X ⊗ (Y ⊗ V )) ⊕ (X ⊗ (Y ⊗ W )) e id ⊗(Ψ⊕Θ) δ−1 +  δ−1 X ⊗ ((Y ⊗ V ) ⊕ (Y ⊗ W )) α−1⊕α−1 id ⊗δ−1   (X ⊗ Y ) ⊗ (V ⊕ W ) o X ⊗ (Y ⊗ (V ⊕ W )) α−1 Now the diagram given by the curved arrows together with the first and third columns is just the sum 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 top curved 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. This diagram 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 third column. This implies, that if we follow the perimeter but instead follow the branch into the inner perimeter, 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 2 We write G = Z2 = {e, s} with the relation s = e. It can be shown that Z2 has two simple modules. 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 to ∼ ⊕m ⊕n a direct sum of simples, that is V = Ctriv ⊕ Csign. Fixing the standard basis, we have a basis

e1, . . . , em, em+1, . . . , em+n ⊕m ⊕n of Ctriv ⊕ Csign.

⊕m1 ⊕n1 ⊕m2 ⊕n2 Proposition 8.1. Any G-homomorphism F : Ctriv ⊕ Ctriv → Ctriv ⊕ Ctriv is given by a block matrix 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

⊕m1 ⊕n1 ⊕m2 ⊕n2 F : Ctriv ⊕ Ctriv → Ctriv ⊕ Ctriv

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

[F ] [F ]  [F ] = 11 m2×m1 12 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. In other words, a linear map F as above corresponds to four linear maps

⊕m1 ⊕m2 F11 : Ctriv → Ctriv ⊕n1 ⊕m2 F12 : Csign → Ctriv ⊕m1 ⊕n2 F21 : Ctriv → Csign ⊕n1 ⊕n2 F22 : Csign → Csign. It is clear that F is a G-homomorphism if and only if all of its components are. By additivity of the Hom-functor and Schur’s lemma, we have

 ⊕m ⊕n   ⊕n  dim HomG-mod Ctriv, Csign = m dim HomG-mod Ctriv, Csign

= mn dim HomG-mod (Ctriv, Csign) = 0 for any m, n and

 ⊕m0 ⊕n0  0  ⊕n0  dim Hom Csign, Ctriv = m dim Hom Csign, Ctriv 0 0 = m n dim Hom (Csign, Ctriv) = 0

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

⊕m1 ⊕n1 ⊕m2 ⊕n2 F : Ctriv ⊕ Ctriv → Ctriv ⊕ Ctriv is given by a matrix [F ] 0  [F ] = 11 . 0 [F22]

The element s ∈ Z2 acts as the identity on Ctriv, and as multiplication with −1 on Csign. This ⊕m ⊕n means that on Ctriv ⊕ Csign, the action of s is the linear extension of the map defined on the basis by  ei if 1 ≤ i ≤ m s(ei) = −ei if m + 1 ≤ i ≤ m + n Markus Thuresson 57 so we have

I 0  [s] = m . 0 −In

Denoting the matrix of the action of s on the different modules by [s]1 and [s]2, the condition of F being a G-homomorphism now amounts to the equation [s]2[F ] = [F ][s]1. Using our previous observation we see that         Im2 0 [F11] 0 [F11] 0 Im1 0 [s]2[F ] = [F ][s]1 ⇐⇒ = 0 −In2 0 [F22] 0 [F22] 0 −In1 I [F ] 0  [F ]I 0  ⇐⇒ m2 11 = 11 m1 . 0 −In2 [F22] 0 −[F22]In1

This holds trivially for any matrices [Fii].

Definition 8.2. Let V and W be finite-dimensional G-modules. Define the map Φe : 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 map Φs : 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. Then the component

ΨX : Ctriv ⊗ X → X ⊗ Ctriv has the form ΨX = λX ΦeX for some nonzero λX ∈ C. Moreover,  ∼ λCtriv if X = Ctriv λX = ∼ . λCsign if 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 the respective tensor products. Since ΨX : Ctriv ⊗ X → X ⊗ Ctriv is a linear isomorphism by assumption, we have

ΨX (1 ⊗ x) = λX (x ⊗ 1) = λX ΦeX (1 ⊗ x) Drinfeld centers 58

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

id ⊗ϕ Ctriv ⊗ Ctriv / Ctriv ⊗ X Ψ Ctriv ΨX   triv ⊗ triv / X ⊗ triv C C ϕ⊗id C commutes for any G-homomorphism. In particular, it commutes for any isomorphism ϕ. We then have

λCtriv ϕ(1) ⊗ 1 = (ϕ ⊗ id) (λCtriv (1 ⊗ 1))

= (ϕ ⊗ id)(λCtriv ΦeCtriv (1 ⊗ 1))

= (ϕ ⊗ id)ΨCtriv (1 ⊗ 1) = ΨX (id ⊗ϕ)(1 ⊗ 1)

= ΨX (1 ⊗ ϕ(1))

= λX ΦeX (1 ⊗ ϕ(1))

= λX ϕ(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.

⊕m ⊕n Proof. The component of Ψ at some module V = Ctriv ⊕ Csign is an isomorphism

 ⊕m ⊕n   ⊕m ⊕n  Ψ: Ctriv ⊗ Ctriv ⊕ Csign → Ctriv ⊕ Csign ⊗ Ctriv.

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

A B  [Ψ] = m×m m×n . Cn×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

I 0  [s] = m . 0 −In Markus Thuresson 59

Now, Ψ being a G-homomorphism amounts to the matrix equation [s][Ψ] = [Ψ][s]. This equation yields

I 0  AB AB I 0  [s][Ψ] = [Ψ][s] ⇐⇒ m = m 0 −In CD CD 0 −In  AB  A −B ⇐⇒ = −C −D C −D so we must have B = −B and C = −C which implies B = C = 0. This shows that A 0  [Ψ] = . By proposition 8.1, any endomorphism of the module ⊕m ⊕ ⊕n is given by 0 D Ctriv Csign   F11 0 a matrix [F ] = where F11 is an arbitrary m × m matrix and F22 is an arbitrary 0 F22 n × n matrix. By naturality of Ψ the square     ⊕m ⊕n id ⊗F / ⊕m ⊕n Ctriv ⊗ Ctriv ⊕ Csign Ctriv ⊗ Ctriv ⊕ Csign

Ψ Ψ    ⊕m ⊕n   ⊕m ⊕n  ⊕ ⊗ triv / ⊕ ⊗ triv Ctriv Csign C F ⊗id Ctriv Csign C 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

A 0  F 0  F 0  A 0  [Ψ][F ] = [F ][Ψ] ⇐⇒ 11 = 11 0 D 0 F22 0 F22 0 D AF 0  F A 0  ⇐⇒ 11 = 11 0 DF22 0 F22D

⇐⇒ AF11 = F11A and DF22 = F22D and since Fii is arbitrary we have A = aV Im and D = dV In. ⊕m1 ⊕n1 ⊕m2 ⊕n2 Now let V1 = Ctriv ⊕ Csign and V2 = Ctriv ⊕ Csign. By proposition 8.1 a G-homomorphism F : V1 → V2 is given by a matrix of the form   F11m2×m1 0m2×n1 [F ] =   . 0n2×m1 F22n2×n1

The commutativity of the naturality square

id ⊗F Ctriv ⊗ V1 / Ctriv ⊗ V2

ΨV1 ΨV2   V1 ⊗ triv / V2 ⊗ triv C F ⊗id C Drinfeld centers 60

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

     11    aV2 Im2 0 F11m2×m1 0 Fm2×m1 0 aV1 Im1 0 [ΨV2 ][F ] = [F ][ΨV1 ] ⇐⇒ = 22 0 dV2 In2 0 F22n2×n1 0 Fn2×n1 0 dV1 In1 a F 0  a F 0  ⇐⇒ V2 11 = V1 11 0 dV2 F22 0 dV1 F22 which imples aV1 = aV2 and dV1 = dV2 so we may drop the indices. Consider the commutative diagram α−1 Csign ⊗ (Ctriv ⊗ Ctriv) / (Csign ⊗ Ctriv) ⊗ Ctriv

Ψ⊗id  (Ctriv ⊗ Csign) ⊗ Ctriv

Ψ α  Ctriv ⊗ (Csign ⊗ Ctriv)

id ⊗Ψ   ( triv ⊗ triv) ⊗ triv o triv ⊗ ( triv ⊗ sign) C C C α−1 C C C ∼ 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. This implies a = 1 since a = 0 would contradict Ψ being an isomorphism. Now we consider the diagram

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

Ψ⊗id  (Csign ⊗ Ctriv) ⊗ Csign

Ψ α  Csign ⊗ (Ctriv ⊗ Csign)

id ⊗Ψ   ( sign ⊗ sign) ⊗ triv o sign ⊗ ( sign ⊗ triv) C C C α−1 C C C ∼ 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 the isomorphism Φe, since then we have   Im 0 [Ψ] = = Im+n. 0 In Markus Thuresson 61

If d = −1, then we have

I 0  [Ψ] = m = [s] 0 −In which implies  ei ⊗ 1 if 1 ≤ i ≤ m Ψ(1 ⊗ ei) = −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 Ψ at some direct sum of simples:

 ⊕m ⊕n   ⊕m ⊕n  Ψ: Csign ⊗ Ctriv ⊕ Csign → Ctriv ⊕ Csign ⊗ Csign.

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

AB [Ψ] = . CD

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

−I 0  [s] = m . 0 In

A 0  The isomorphism Ψ commuting with the group action again yields [Ψ] = . Using 0 D proposition 8.1 now gives A = aV Im and D = dV In just like before and naturality then implies aV1 = aV2 and dV1 = dV2 for any modules V1 and V2. Now we consider the commutative diagram

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

Ψ⊗id  (Ctriv ⊗ Csign) ⊗ Ctriv

Ψ α  Ctriv ⊗ (Csign ⊗ Ctriv)

id ⊗Ψ   ( triv ⊗ triv) ⊗ triv o triv ⊗ ( triv ⊗ sign) C C C α−1 C C C Drinfeld centers 62 which shows a = 1 and the diagram

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

Ψ⊗id  (Csign ⊗ Csign) ⊗ Csign

Ψ α  Csign ⊗ (Csign ⊗ Csign)

id ⊗Ψ   ( sign ⊗ sign) ⊗ sign o sign ⊗ ( sign ⊗ sign) C C C α−1 C C C 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 simple modules, so that Vi ∈ {Ctriv, Csign} and Ψi ∈ {Φe, Φs} for all i. Let Vi be the linear span of the vector vi. Then k M (Vi, Ψi) = (V, Ψ) i=1 Lk where V = i=1 Vi and Ψ: V ⊗ X → X ⊗ V is defined by

(v1, . . . , vk) ⊗ x 7→ x ⊗ (c1v1, . . . , ckvk) where  1 if Ψi = Φe ci = −1 if Ψi = Φs

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

−1 (V1, Ψ1) ⊕ (V2, Ψ2) = (V1 ⊕ V2, δ (Ψ1 ⊕ Ψ2)ε) by definition. We have

−1 −1 δ (Ψ1 ⊕ Ψ2)ε((v1, v2) ⊗ x) = δ (Ψ1 ⊕ Ψ2)((v1 ⊗ x, v2 ⊗ x)) −1 = δ ((Ψ1(v1 ⊗ x), Ψ2(v2 ⊗ x))) −1 = δ ((x ⊗ c1v1, x ⊗ c2v2))

= x ⊗ ((c1v1, c2v2))

= Ψ((v1, v2) ⊗ x). Markus Thuresson 63

−1 0 Next, we have (V, Ψ) ⊕ (Vk+1, Ψk+1) = (V ⊕ Vk+1, δ (Ψ ⊕ Ψk+1)ε). Let Ψ denote the isomorphism in the formulation.

−1 −1 δ (Ψ ⊕ Ψk+1)ε(((v1, . . . , vk), vk+1) ⊗ x) = δ (Ψ ⊕ Ψk+1)(((v1, . . . , vk) ⊗ x, vk+1 ⊗ x)) −1 = δ ((x ⊗ (c1v1, . . . , ckvk), x ⊗ ck+1vk+1))

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

= x ⊗ (c1v1, . . . , ckvk, ck+1vk+1) 0 = Ψ ((v1, . . . , vk+1) ⊗ x).

∼ ⊕m ⊕n Lemma 8.8. Let (V, Ψ) be in Z(G-mod) and let W = Ctriv ⊕ Csign 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

[Ψ ⊕m ⊕n ] = [ΨW ]. Ctriv ⊕Csign

k ⊕m ⊕n Proof. Let {vi}i=1 be a basis of V and let ϕ : Ctriv ⊕ Csign → W be a G-isomorphism and . In particular, it is a linear isomorphism so the set {ϕ(ei)} forms a basis of W . Fixing this basis of W , we have [ϕ] = Im+n. By naturality, the square

  id ⊗ϕ ⊕m ⊕n / V ⊗ Ctriv ⊕ Csign V ⊗ W

Ψ ⊕m ⊕ ⊕n Ψ Ctriv Csign W     ⊕m ⊕ ⊕n ⊗ V / W ⊗ V Ctriv Csign ϕ⊗id commutes and since in our bases, we have

[ϕ] ... 0   . .. .  [id ⊗ϕ] = [ϕ ⊗ id] =  . . .  = Ik(m+n) 0 ... [ϕ] the result follows.  ⊕k ⊕k Proposition 8.9. If (Ctriv, Ψ) is in Z(G-mod), then (Ctriv, Ψ) is isomorphic to

⊕k1 ⊕k2 (Ctriv, Φe) ⊕ (Ctriv, Φs) for some k1, k2 such that k1 + k2 = k. ⊕k ⊕m ⊕n Proof. Fix the standard basis {ei} of Ctriv and the standard basis {vi} of Ctriv ⊕ Csign. Then fix bases

{e1 ⊗ v1, . . . , e1 ⊗ vm+n, . . . , ek ⊗ v1, . . . , ek ⊗ vm+n}

{v1 ⊗ 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   A11 ...A1k  . .. .  [Ψ] =  . . .  . Ak1 ...Akk In the module V , we have  vi if 1 ≤ i ≤ m s(vi) = −vi if m + 1 ≤ i ≤ m + n so that we have   Im 0m×n [s]V = . 0n×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 the tensor products is the k×k block matrix with blocks [s]V on the diagonal and zeroes everywhere else. Writing this out we have   [s]V ... 0 . . . [s] ⊕k = [s] ⊕k =  . .. .  . Ctriv⊗V V ⊗Ctriv   0 ... [s]V Since Ψ is a G-isomorphism by assumption, it commutes with the action of the group, so we have Ψ(s(x ⊗ y)) = sΨ(x ⊗ y). In terms of matrices, this is equivalent to the equation [Ψ][s] = [s][Ψ]. Then         A11 ...A1k [s]V ... 0 [s]V ... 0 A11 ...A1k  . .. .   . .. .   . .. .   . .. .   . . .   . . .  =  . . .   . . .  Ak1 ...Akk 0 ... [s]V 0 ... [s]V Ak1 ...Akk which holds if and only if     A11[s]V ...A1k[s]V [s]V A11 ... [s]V A1k  . .. .   . .. .   . . .  =  . . .  . Ak1[s]V ...Akk[s]V [s]V Ak1 ... [s]V Akk

This shows that every matrix Aij must commute with [s]V . We write Aij as a block matrix   Bm×m Cm×n Aij = . Dn×m En×n Markus Thuresson 65

Writing out the condition Aij[s]V = [s]V Aij we get

B C   I 0   I 0  B C  m×m m×n m m×n = m m×n m×m m×n Dn×m En×n 0n×m −In 0n×m −In Dn×m En×n which holds if and only if

B −C  BC = . D −E −D −E

This implies C = −C and D = −D so we must have C = D = 0. Consider the component of ⊕m ⊕n Ψ at the module V = Ctriv ⊕ Csign. By assumption, the diagram     ⊕k ⊕m ⊕n id ⊗F / ⊕k ⊕m ⊕n Ctriv ⊗ Ctriv ⊕ Csign Ctriv ⊗ Ctriv ⊕ Csign

Ψ Ψ       ⊕m ⊕ ⊕n ⊗ ⊕k / ⊕m ⊕ ⊕n ⊗ ⊕k Ctriv Csign Ctriv F ⊗id Ctriv Csign Ctriv commutes for any G-homomorphism F : V → V . With respect to our bases, we know that the matrices of the homomorphisms id ⊗F and F ⊗ id are given by

[F ] ... 0   . .. .  [id ⊗F ] = [F ⊗ id] =  . . .  . 0 ... [F ]

It is now clear that the diagram commutes if and only if this matrix commutes with the matrix of Ψ. This condition is equivalent to Aij[F ] = [F ]Aij for all i, j. By our previous observation the matrices Aij have the form   Bij 0 Aij = 0 0 Bij and by proposition 8.1 the matrix [F ] has the form

F 0  [F ] = 11 . 0 F22

This means that our condition is equivalent to         Bij 0 F11 0 F11 0 Bij 0 0 = 0 0 Bij 0 F22 0 F22 0 Bij which holds if and only if

0 0 BijF11 = F11Bij and BijF22 = F22Bij. Drinfeld centers 66

0 Since the matrices Fii are arbitrary, this implies that Bij = λV Im and Bij = µV In, where ⊕x ⊕y λV , µV ∈ C. Let W = Ctriv ⊕ Csign be another module. By assumption, the diagram

    ⊕k ⊕m ⊕n id ⊗F / ⊕k ⊕x ⊕y Ctriv ⊗ Ctriv ⊕ Csign Ctriv ⊗ Ctriv ⊕ Csign

Ψ Ψ       ⊕m ⊕ ⊕n ⊗ ⊕k / ⊕x ⊕ ⊕y ⊗ ⊕k Ctriv Csign Ctriv F ⊗id Ctriv Csign Ctriv commutes for any G-homomorphism F : V → W . A similar argument now shows that we must W V have Aij [F ] = [F ]Aij for all i, j. This is equivalent to

 W       V  λij Ix 0 F11 0 F11 0 λijIm 0 W = V 0 µij Iy 0 F22 0 F22 0 µijIn

V W which holds if and only if λW F11 = λV F11 and µW F22 = µV F22. This implies λij = λij and V W µij = µij . Now, we consider the commutative diagram

−1   ⊕k α / ⊕k Ctriv ⊗ (Ctriv ⊗ Ctriv) Ctriv ⊗ Ctriv ⊗ Ctriv

Ψ⊗id   ⊕k  Ctriv ⊗ Ctriv ⊗ Ctriv

Ψ α   ⊕k  Ctriv ⊗ Ctriv ⊗ Ctriv

id ⊗Ψ   ⊕k  ⊕k  ( triv ⊗ triv) ⊗ o triv ⊗ triv ⊗ C C Ctriv α−1 C C Ctriv

∼ 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 equivalent 2 to [ΨCtriv ] = [ΨCtriv ] and since [ΨCtriv ] is invertible it follows that [ΨCtriv ] = Ik. This is equivalent to

 1 if i = j λ = . ij 0 if i 6= j Markus Thuresson 67

If we instead consider the commutative diagram

−1   ⊕k α / ⊕k Ctriv ⊗ (Csign ⊗ Csign) Ctriv ⊗ Csign ⊗ Csign

Ψ⊗id   ⊕k  Csign ⊗ Ctriv ⊗ Csign

Ψ α   ⊕k  Csign ⊗ Ctriv ⊗ Csign

id ⊗Ψ   ⊕k  ⊕k  ( sign ⊗ sign) ⊗ o sign ⊗ sign ⊗ C C Ctriv α−1 C C Ctriv ∼ Since Csign ⊗ Csign = Ctriv we may assume that the left arrow is given by Ik, by lemma 8.8. 2 Commutativity of the diagram is then equivalent to [ΨCsign ] = Ik. Such a matrix must have eigenvalues ±1 and be diagonalizable. This means that there is a ⊕k basis of Csign such that   Ip 0 [ΨCsign ] = 0 −Iq with respect to this basis, where p + q = k. At first sight, it seems like this change of basis could 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.

⊕k - The form of the module Ctriv is not affected by a change of basis. So, we can safely make this change of basis, which shows that we may assume

 ±1 if i = j µ = ij 0 otherwise and moreover,

 1 if 1 ≤ i ≤ p µ = . ii −1 if p + 1 ≤ i ≤ k

⊕k ⊕m ⊕n ⊕m ⊕n ⊕k This shows that Ψ : Ctriv ⊗ (Ctriv ⊕ Csign) → (Ctriv ⊕ Csign)Ctriv is defined by

ei ⊗ vj 7→ vj ⊗ ei if 1 ≤ i ≤ p ei ⊗ 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

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

⊕k1 ⊕k2 (Ctriv, Φe) ⊕ (Ctriv, Φs) 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 the ⊕m ⊕n component of Ψ at the module V = Ctriv ⊕ Csign. The first point where we differ is the action of 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   Im 0m×n [s]V = 0n×m −In but now we get a different matrix representing the action of s in the tensor products, namely:   −[s]V ... 0 . . . [s] ⊕k = [s] ⊕k =  . .. .  . Csign⊗V V ⊗Csign   0 ... −[s]V

If we write   A11 ...A1k  . .. .  [Ψ] =  . . .  . Ak1 ...Akk

The assumption that Ψ be a G-homomorphism again yields [s]V Aij = Aij[s]V for all i, j. Just like in the previous proposition, this implies that the matrices Aij have the form   Bij 0 Aij = 0 . 0 Bij Moreover, it follows in the same exact way that

V 0 V Bij = λijIm and Bij = µijIn, λV , µV ∈ C.

Next, naturality shows that if W is some other module, then λV = λW and µV = µW . This means that so far, we know that the matrix representing the component of Ψ at V is given by   A11 ...A1k  . .. .  [Ψ] =  . . .  Ak1 ...Akk Markus Thuresson 69

where the matrices Aij have the form   λijIm 0 Aij = . 0 µijIn

Now, we consider the commutative diagram

−1   ⊕k α / ⊕k Csign ⊗ (Ctriv ⊗ Ctriv) Csign ⊗ Ctriv ⊗ Ctriv

Ψ⊗id   ⊕k  Ctriv ⊗ Csign ⊗ Ctriv

Ψ α   ⊕k  Ctriv ⊗ Csign ⊗ Ctriv

id ⊗Ψ   ⊕k  ⊕k  ( triv ⊗ triv) ⊗ o triv ⊗ triv ⊗ C C Csign α−1 C C Csign

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

[ΨCtriv ] which implies [ΨCtriv ] = Ik. This is equivalent to  1 if i = j λ = . ij 0 if i 6= j

Now, we consider the commutative diagram

−1   ⊕k α / ⊕k Csign ⊗ (Csign ⊗ Csign) Csign ⊗ Csign ⊗ Csign

Ψ⊗id   ⊕k  Csign ⊗ Csign ⊗ Csign

Ψ α   ⊕k  Csign ⊗ Csign ⊗ Csign

id ⊗Ψ   ⊕k  ⊕k  ( sign ⊗ sign) ⊗ o sign ⊗ sign ⊗ C C Csign α−1 C C Csign ∼ Since Csign ⊗ Csign = Ctriv we may assume that the left arrow is given by Ik, by lemma 8.8. 2 Commutativity of the diagram is then equivalent to [ΨCsign ] = Ik. Now we use the same argument as in the proof of proposition 8.9 to arrive at the conclusion. Drinfeld centers 70

⊕k ⊕l ⊕k ⊕l Proposition 8.11. If (Ctriv ⊕ Csign, Ψ) is in Z(G-mod), then (Ctriv ⊕ Csign, Ψ) is isomorphic ⊕k ⊕l to (Ctriv, Ψ1) ⊕ (Csign, Ψ2). ⊕m ⊕n Proof. Considering the component of Ψ at V = Ctriv ⊕ Csign we get the commutative diagram         ⊕k ⊕l ⊕m ⊕n id ⊗F / ⊕k ⊕l ⊕m ⊕n Ctriv ⊕ Csign ⊗ Ctriv ⊕ Csign Ctriv ⊕ Csign ⊗ Ctriv ⊕ Csign

Ψ Ψ           ⊕m ⊕ ⊕n ⊗ ⊕k ⊕ ⊕l / ⊕m ⊕ ⊕n ⊗ ⊕k ⊕ ⊕l Ctriv Csign Ctriv Csign F ⊗id Ctriv Csign Ctriv Csign

We write [Ψ] as a (k + l) × (k + l) block matrix with block size (m + n) × (m + n). So we have   A11 ...A1(k+l)  . .. .  [Ψ] =  . . .  A(k+l)1 ...A(k+l)(k+l)

⊕k ⊕l ⊕m ⊕n Let {ui} denote the standard basis of Ctriv ⊕Csign and {vi} the standard basis of Ctriv ⊕Csign. Fix the basis

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

 ⊕k ⊕l   ⊕m ⊕n  of the module Ctriv ⊕ Csign ⊗ Ctriv ⊕ Csign and the basis

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

 ⊕m ⊕n   ⊕k ⊕l  of the module Ctriv ⊕ Csign ⊗ Ctriv ⊕ Csign . Now we note that   ui if 1 ≤ i ≤ k vi if 1 ≤ i ≤ m s(ui) = , s(vi) = . −ui if k + 1 ≤ i ≤ k + l −vi if m + 1 ≤ i ≤ m + n

I 0  Letting E = m we get the matrices for the action of s in U ⊗ V and in V ⊗ U as 0 −In (k + l) × (k + l) block matrices with blocks of the form E or −E on the diagonal and zeroes everywhere else. Clearly, there are k + l such blocks. Then, the first k blocks are E and the last l blocks are −E. E... 0   . .. .  [s]U⊗V =  . . .  = [s]V ⊗U . 0 ... −E

Now Ψ being a G-homomorphism is equivalent to [Ψ][s]U⊗V = [s]V ⊗U [Ψ]. Writing the condition out we have Markus Thuresson 71

        A11 ...A1(k+l) E... 0 E... 0 A11 ...A1(k+l)  . .. .   . .. .   . .. .   . .. .   . . .   . . .  =  . . .   . . .  A(k+l)1 ...A(k+l)(k+l) 0 ... −E 0 ... −E A(k+l)1 ...A(k+l)(k+l) which holds if and only if   EA11 ...EA1(k+l)  . .     . .  A11E...A1kE −A1(k+1)E... −A1(k+l)E   . . . .  EAk1 ...EAk(k+l)   . . . .  =   .   −EA(k+1)1 ... −EA(k+1)(k+l) A 1E...A kE −A E... −A E   k+l k+l (k+l)(k+1) (k+l)(k+l)  . .   . .  −EA(k+l)1 ... −EA(k+l)(k+l)

This means that we have four possibly different conditions on the matrices Aij.   B11 B12 1) If i ≤ k, j ≤ k, we require EAij = AijE. Writing Aij = we get B21 B22

I 0  B B  B B  I 0   B B  B −B  m 11 12 = 11 12 m ⇐⇒ 11 12 = 11 12 0 −In B21 B22 B21 B22 0 −In −B21 −B22 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  B B  −B B  11 12 = 11 12 −B21 −B22 −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   A11 ...A1(k+l)  . .. .  [Ψ] =  . . .  A(k+l)1 ...A(k+l)(k+l)     B11 0 0 B12 where the blocks Aij have either the form or , according to the cases 0 B22 B21 0 above. We return to the naturality square         ⊕k ⊕l ⊕m ⊕n id ⊗F / ⊕k ⊕l ⊕m ⊕n Ctriv ⊕ Csign ⊗ Ctriv ⊕ Csign Ctriv ⊕ Csign ⊗ Ctriv ⊕ Csign

Ψ Ψ           ⊕m ⊕ ⊕n ⊗ ⊕k ⊕ ⊕l / ⊕m ⊕ ⊕n ⊗ ⊕k ⊕ ⊕l Ctriv Csign Ctriv Csign F ⊗id Ctriv Csign Ctriv Csign Drinfeld centers 72 and just like before

[F ] ... 0   . .. .  [id ⊗F ] = [F ⊗ id] =  . . .  . 0 ... [F ]

By the commutativity of the diagram, we get Aij[F ] = [F ]Aij for all i, j. Note that     F11 0 B11 0 [F ] = by proposition 8.1. If Aij has the form Aij = then we have 0 F22 0 B22         B11 0 F11 0 F11 0 B11 0 Aij[F ] = [F ]Aij ⇐⇒ = 0 B22 0 F22 0 F22 0 B22 B F 0  F B 0  ⇐⇒ 11 11 = 11 11 0 B22F22 0 F22B22

V V which implies B11 = aijIm and B22 = dijIn since the matrices Fii are arbitrary. If Aij has the   0 B12 form Aij = then we have B21 0         0 B12 F11 0 F11 0 0 B12 Aij[F ] = [F ]Aij ⇐⇒ = B21 0 0 F22 0 F22 B21 0  0 B F   0 F B  ⇐⇒ 12 22 = 11 12 B21F11 0 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 M 0  Ψ as a 2 × 2 block matrix [Ψ] = where all entries are block matrices with block size 0 N (m + n) × (m + n), where M is a k × k block matrix and N is an l × l block matrix. We see ⊕k ⊕k ⊕l ⊕l that M corresponds to a map Ctriv ⊗ V → V ⊗ Ctriv and N to a map Csign ⊗ V → V ⊗ Csign, 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

 Hom (V,W ) if V = W and Ψ = Θ Hom ((V, Ψ), (W, Θ)) = G-mod Z(G-mod) 0 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 sum of 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, New York, 1995.

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