U.U.D.M. Project Report 2019:34

Examensarbete i matematik, 15 hpHandledare: Volodymyr MazorchukExaminator: Martin HerschendJuni 2019

Department of MathematicsUppsala University

The Nichols-Zoeller Theorem for Quasi-Hopf Algebras

Isak Matsson

The Nichols-Zoeller Theorem

for Quasi-Hopf Algebras

Isak Matsson

June 19, 2019

Abstract

We prove the quasi-Hopf algebra equivalent of the Nichols-Zoeller theoremas a corollary in the theory of tensor categories. Namely, a finite dimen-sional quasi-Hopf algebra is free over any of its quasi-Hopf subalgebras.

Contents

1 Introduction 3

2 Abelian categories 32.1 Additive categories . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Finite abelian categories . . . . . . . . . . . . . . . . . . . . . . . 7

3 Tensor categories 93.1 Monoidal categories . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Tensor categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Frobenius-Perron dimensions 124.1 Grothendieck rings . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Frobenius-Perron dimensions . . . . . . . . . . . . . . . . . . . . 13

5 Hopf algebras 145.1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Quasi-Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Main results 19

7 References 21

2

1 Introduction

Freeness theorems are highly important in the field of algebra. A prime exam-ple of this is the Nichols-Zoeller theorem for Hopf algebras, which states thata finite dimensional Hopf algebra is a free (left or right) module over any ofits Hopf subalgebras. This can be thought of as a Hopf algebra analogue ofLagrange’s theorem for finite groups, which in fact follows immediately becauseof the strong connection between Hopf algebras and groups.

The original proof was done by showing that every left (H,B)-Hopf moduleis free as a left B-module (Nichols, Zoeller, 1989, [3]). Using a similar argu-ment, the equivalent theorem for finite dimensional quasi-Hopf algebras andtheir quasi-Hopf subalgebras was proven true by Schauenburg (2004, [4]), whichis the theorem that this paper is concerned with.

This does not hold in the infinite dimensional case, neither for quasi-Hopfnor for Hopf algebras, as shown by Oberst and Schneider (1974, [6]) who con-structed a Hopf algebra counterexample.

This paper will prove the quasi-Hopf algebra equivalent of the Nichols-Zoellertheorem by first proving a freeness theorem for finite tensor categories, as doneby Etingof el al. (2015, [1]).

We will start by defining abelian, monoidal and tensor categories, Frobenius-Perron dimensions, and Hopf and quasi-Hopf algebras themselves, which areneeded for the proof. Familiarity with the basics of category theory is assumed.

2 Abelian categories

Abelian categories, like the name suggests, are categories that have propertiessimilar to the category Ab of abelian groups. Before defining them, we will firstintroduce the concept of additive categories, which has a less strict definition.

There are also what’s known as preadditive and preabelian categories, whichwe will not discuss in this paper.

2.1 Additive categories

Definition 2.1. An additive category is a category satisfying the following:

(A1) It is enriched over the category Ab of abelian groups, meaning that everyhom-set has the structure of an abelian group (written additively) suchthat morphism composition is bilinear with respect to this structure.

(A2) It has a zero object (denoted 0).

(A3) Every pair of objects X,Y has a biproduct (denoted X⊕Y ).

Since the dual of every axiom is itself an axiom, if C is additive, then so is Cop.

3

We denote the group operation in (A1) as +, which gives us:

f ◦ (g + h) = (f ◦ g) + (f ◦ h) and (f + g) ◦ h = (f ◦ h) + (g ◦ h)

We also denote f + (−g) as f − g.By (A2) we mean an object that is both terminal and initial. We also denote

the zero morphisms as simply 0, which are the hom-set identities in (A1).Also note that (A3) is sometimes written as requiring coproducts, but for

any category that is enriched over Ab1, products and coproducts coincide (MacLane, 1998, p. 194, [2]).

Example 2.2. The category Ab of abelian groups is additive, with (A1) beingpointwise addition of morphisms, (A2) being the trivial group {e}, and (A3)being the usual direct sum of groups.

1. Given abelian groups X,Y and f(x), g(x) : X −→ Y we define:

(f + g)(x) := f(x) + g(x) = g(x) + f(x)

and we get an abelian group structure with inverses −f(x) = f(−x), theidentity being the morphism 0(x) that maps all of X to 0Y , and we canverify that the usual morphism composition:

◦ : Hom(Y, Z)×Hom(X,Y ) −→ Hom(X,Z)

is bilinear with respect to this structure.

2. Given an abelian group X, we have a unique f(x) : X −→ {e}, given byf(x) = e, so {e} is terminal. We also have a unique g(x) : {e} −→ X,given by g(e) = 0X , so {e} is initial. By definition we have a zero object.

3. Given abelian groups X,Y , their direct sum X⊕Y with the addition

operation defined as (x, y) + (x′, y′) = (x + x′, y + y′) is also an abeliangroup and defines a categorical biproduct.

Remark 2.3. The category Grp of groups contains groups that aren’t com-mutative, so we cannot define an abelian group structure its hom-sets.

The category Set of sets has ∅ as a terminal object and the singleton set asan initial object but no zero objects, so we quickly see that it is not additive.

The category FinGrp of finite groups doesn’t have coproducts, which isharder to show and means that it is not additive.

Definition 2.4. Let C,D be additive categories. A functor F : C −→ D iscalled an additive functor if the maps

HomC(X,Y ) −→ HomD(F (X), F (Y ))

are homomorphisms of abelian groups for any X,Y ∈ C.1More generally, enrichment over CMod of commutative monoids is sufficient.

4

2.2 Abelian categories

Definition 2.5. The kernel (if it exists) of a morphism f : X −→ Y in anadditive category is an object K and a morphism k : K −→ X such that fk = 0and for every other candidate (K ′, k′) with fk′ = 0, there is a unique morphisml : K ′ −→ K such that k′ = kl. We sometimes write (K, k) as K for simplicity.

X

K ′ K Y

fk′

l

0

k

0

Example 2.6. We know that the kernel in Ab of the canonical mapf : Z −→ Z/3Z is (3Z, π). Consider the candidate K = 0. Since there is nol : 3Z −→ 0 such that π = 0l = 0, we see that ker(f) 6= 0 even though f0 = 0.

Proposition 2.7. The kernel is monic.

Proof. Given K ′l1−−→−−→l2

Kk−→ X with kl1 = kl2 = k′, then fk′ = 0 and (K ′, k′)

is a candidate. Thus l1 = l2 by uniqueness and k is monic.

Proposition 2.8. The morphism f is monic if and only if ker(f) = 0.

Proof. If f is monic, then given the kernel (K, k) we get fk = 0 = f0, andbecause f is monic, k = 0. By definition we need a unique morphism from allcandidates (K ′, 0) to K, so K must be 0.

If ker(f) = 0 then for any g such that fg = 0, we have a unique morphism lsuch that g = kl = 0l = 0, thus fg = 0 =⇒ g = 0. Now let fg1 = fg2, sinceour category is additive we can consider f ◦ (g1− g2) = (fg1)− (fg2) = 0. Thenf ◦ (g1 − g2) = 0 =⇒ g1 − g2 = 0 ⇐⇒ g1 = g2 which shows f is monic.

Definition 2.9. The cokernel (if it exists) of a morphism f : X −→ Y in anadditive category is an object C and a morphism c : Y −→ C such that cf = 0and for every other candidate (C ′, c′) with c′f = 0, there is a unique morphisml : C −→ C ′ such that c′ = lc. We sometimes write (C, c) as C for simplicity.

Y

C ′ C X

c′c

l

f

0

0

As can be seen, cokernels are duals of kernels, which means that a cokernel ina category C is a kernel in the opposite category Cop. Since epimorphisms areduals of monomorphisms, we automatically get the following two propositions:

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Proposition 2.10. The cokernel is epic.

Proposition 2.11. The morphism f is epic if and only if coker(f) = 0.

In most of the common categories that we will consider, the cokernel of a mor-phism f : X −→ Y will correspond to the quotient of Y by the image of f .

Definition 2.12. An abelian category is a category satisfying the following:

(A1-3) It is additive.

(A4) Every morphism has a kernel and a cokernel.

(A5) Every mono-/epimorphism is normal, meaning that it is a kernel/cokernel,respectively, of some morphism in the category.

Like additive categories, abelian categories are self-dual since the dual of everyaxiom is an axiom, so if C is additive then so is Cop.

Example 2.13. The category Vectk of vector spaces over a field k is abelian,with (A1) pointwise addition of linear maps, (A2) the zero vector space {0}, and(A3) the usual direct sum of vector spaces. To verify (A1-3) see Example 2.2.Shown similarly, the category R-Mod of left-modules over a ring is abelian.

4. For f : V −→W , ker(f) is the preimage of 0 and coker(f) = W/ im(f).

5. Given any f : V −→ W , we have the following canonical decomposition:

f : Vi−→ im(f)

j−→ W where i is epic and j is monic. Looking at (A4),

we also have Kk−→ V

i−→ im(f)j−→W

c−→ C where K is the kernel andC is the cokernel. Since C = W/ im(f), we have ker(c) = im(f) = (I, j).If f is monic, then f = j = ker(c), and f is a kernel.

By the first isomorphism theorem, im(f) ∼= X/ ker(f) = X/ im(k) =coker(k) = (I, i). If f is epic, then f = i = coker(k) so f is a cokernel.

Remark 2.14. The category of even-dimensional vector spaces has morphismswithout kernels, since the kernel of a linear map with odd rank is also odd.

The category of filtered modules over a ring has mono-/epimorphisms thataren’t normal, so it is not abelian.

Definition 2.15. An additive category is called k-linear if it is enriched overVectk, meaning that every hom-set has the structure of a vector space over ksuch that usual morphism composition is bilinear with respect to this structure.

Since vector spaces are also abelian groups, k-linear implies Ab-enriched.

Definition 2.16. Let C,D be k-linear categories. A functor F : C −→ D iscalled k-linear if the maps

HomC(X,Y ) −→ HomD(F (X), F (Y ))

are k-linear homomorphisms of abelian groups for any X,Y ∈ C.

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2.3 Finite abelian categories

Note that by finite abelian categories, we don’t mean finite in the sense of havingfinitely many morphisms or (locally) that every hom-set is finite, but rather inthe sense of having finitely many simple objects.

Definition 2.17. A subobject of an object Y is an object X together with amonomorphism i : X −→ Y . We denote it X ⊆ Y or X ⊂ Y when i isn’t epic.

An object that cannot be expressed as a non-trivial direct sum of its subob-jects is called indecomposable. Note that 0 is not indecomposable.

In Ab we have subobjects such as the canonical map i : G/N −→ N where Nis a normal subgroup of G, but also the identity idG and f : 2Z −→ Z, etc.

Definition 2.18. The dual notion of a subobject is a quotient object, whichis an object Z together with an epimorphism p : Y −→ Z.

Given an object Y and a subobject X ⊆ Y with i : X −→ Y , the cokernel of iis a quotient object of Y and we write coker(i) = Y/X.

Definition 2.19. A simple object in an abelian category is a non-zero objectthat only has 0 and itself as subobjects.

The more common definition of simple objects is with quotient objects insteadof subobjects, but since abelian categories are self-dual we get the same result.

Example 2.20. The simple objects in Ab are the simple groups, and the simpleabelian groups are exactly the cyclic groups of prime order.

The simple objects in Vectk are the one-dimensional vector spaces, whichare isomorphic to k itself.

The simple objects in the category Repk(G) of k-representations of a groupG are its irreducible representations.

Definition 2.21. An object of finite length in an abelian category is an objectX for which there exists a filtration:

0 = X0 ⊂ X1 ⊂ · · · ⊂ Xn−1 ⊂ Xn = X

Such that the cokernel Xi/Xi−1 is simple for all i > 0.

Example 2.22. In Vectk, for any kn we have the filtration:

0 ⊂ k ⊂ k2 ⊂ · · · ⊂ kn−1 ⊂ kn

Where every ki/ki−1 ∼= k which is simple. Similarly, for any any finite dimen-sional k-vector space, one can consider the filtration over its basis elements.Thus every finite dimensional vector space in Vectk has finite length.

This leads us to a famous theorem in group theory which we can apply to abeliancategories. We omit the proof for sake of brevity.

7

Theorem 2.23 (Krull-Schmidt). Let X be an object of finite length in anabelian category. Then X has a unique decomposition into a direct sum ofindecomposable objects, up to isomorphism.

Definition 2.24. An exact sequence is a sequence of morphisms:

· · · −→Xi−1fi−1−→ Xi

fi−→ Xi+1 −→ · · ·

Where im(fi−1) = ker(fi).A short exact sequence, is an exact sequence of the form:

0 −→ Xf−→ Y

g−→ Z −→ 0

Since ker(f) = im(k) = 0, f is monic, and since im(g) = ker(c) = C, g is epic.A functor F : C −→ D is called exact if for any short exact sequence in C

(written like above), the following sequence is exact in D:

0 −→ F (X)F (f)−→ F (Y )

F (g)−→ F (Z) −→ 0

Definition 2.25. A projective object in an abelian category C is an objectP for which the hom-functor HomC(P,−) : C −→ Ab is exact.

We say that an object X has a projective cover P if there exists a projec-tive object together with an epimorphism p : P −→ X such that for every otherprojective object N and epimorphism g : N −→ X there exists an epimorphismh : N −→ P such that ph = g.

Every projective object automatically has itself as a projective cover with theepimorphism being the identity morphism.

Example 2.26. Free modules are projective in R-Mod, in fact the projectiveobjects are exactly the direct summands of the free modules.

Since all vector spaces are free k-modules (every vector space has a basis),every object in Vectk is projective and therefore also has a projective cover.

Definition 2.27. A k-linear abelian category is called finite if:

(F1) Every hom-set (which is a k-vector space) has finite dimension.

(F2) Every object has finite length.

(F3) Every simple object has a projective cover.

(F4) There are only finitely many simple objects (up to isomorphism).

Example 2.28. The category FinVectk of finite dimensional vector spacesover a field k is a finite abelian category. That it is k-linear and abelian can beverified as in Example 2.13 and conditions (F1-4) have been verified previously.

For a finite k-linear abelian category C, given any X,Y ∈ C where X is simple:

dimk HomC(P (X), Y ) = [Y : X]

Where P (X) is a projective cover of X and [Y : X] denotes the number ofoccurrences of X in the filtration of Y .

8

3 Tensor categories

There are a lot of criteria for a category to follow our definition of a tensorcategory. Most importantly though, it is equipped with a bifunctor called thetensor product, making it a monoidal category which we will define first.

3.1 Monoidal categories

Definition 3.1. A monoidal category is a category C satisfying the following:

(M1) It has bifunctor⊗

: C × C −→ C, called the tensor product.

(M2) It has a natural isomorphism a : (−⊗−)

⊗− ∼−→ −

⊗(−

⊗−),

called the associativity constraint, with components of the form

aX,Y,Z : (X⊗Y )

⊗Z∼−→ X

⊗(Y

⊗Z).

(M3) It has an object called the unit object, denoted 1.

(M4) It has a natural isomorphism l : 1⊗− ∼−→ − called the left unit constraint

with components of the form lx : 1⊗X

∼−→ X.

(M5) It has a natural isomorphism r : −⊗

1∼−→ − called the right unit con-

straint with components of the form rx : X⊗

1∼−→ X.

(M6) The following triangle diagram commutes for all objects X,Y :

(X ⊗ 1)⊗ Y X ⊗ (1⊗ Y )

X ⊗ YrX⊗idY

aX,1,Y

idX ⊗lY

(M7) The following pentagon diagram commutes for all objects X,Y, Z:

((W ⊗X)⊗ Y )⊗ Z

(W ⊗ (X ⊗ Y ))⊗ Z (W ⊗X)⊗ (Y ⊗ Z)

W ⊗ ((X ⊗ Y )⊗ Z) W ⊗ (X ⊗ (Y ⊗ Z))

aW,X,Y ⊗idZaW⊗X,Y,Z

aW,X⊗Y,Z aW,X,Y⊗Z

idW ⊗aX,Y,Z

One can quickly check that Set is monoidal with the cartesian product as thetensor product, the singleton set as the unit object, and with obvious choices fora, l, r. It also holds for the category FinSet of finite sets and can be generalizedto categories of sets with some additional structure, such as the category Grp.

More generally, every category with a product and a terminal object ismonoidal, in particular every additive category is monoidal with the direct sum

9

and zero object as tensor product and unit object. However, this choice of ten-sor and unit is not unique and often other choices are more useful since theyreveal more about the structure of the category.

A good example for illustrating this is Vectk.

Definition 3.2. The standard tensor product of vector spaces U and Vover a field k, denoted

⊗k, is the quotient space U ⊗ V := Z/E, where

Z := span{(u, v) | u ∈ U, v ∈ V }

And E is the subspace of Z generated by vectors of the form:

(u, v1 + v2)− (u, v1)− (u, v2),

(u1 + u2, v)− (u1, v)− (u2, v),

(λu, v)− λ(u, v), λ ∈ k(u, λv)− λ(u, v)

The motivation is that bilinear maps U × V −→ W are equivalent to linearmaps U ⊗V −→W , which comes from the fact that we are taking a quotient ofE, so vectors in E will be zero in Z/E. This means that vectors u⊗ v in U ⊗Vhave the following properties:

u⊗ v1 + u⊗ v2 = u⊗ (v1 + v2)

u1 ⊗ v + u2 ⊗ v = (u1 + u2)⊗ v(λu)⊗ v = λ(u⊗ v) = u⊗ (λv)

Given vectors v = (v1, . . . , vn), w = (w1, . . . , wm) in U, V their tensor productv⊗w is equivalent to their outer product vwT . Take for example (2, 6), (4, 5, 3)in R2,R3 and we get (2, 6)⊗ (4, 5, 3) = ( 8 10 6

24 30 18 ).

Remark 3.3. If (e1, . . . , en) is a basis for U and (f1, . . . , fm) a basis for V then{ei ⊗ fj} is a basis for U ⊗ V (where i, j go from 1 to n,m respectively).

Example 3.4. The category Vectk is monoidal with the one dimensional vectorspace k as the unit object, the standard tensor product of vector spaces

⊗k and

obvious choices for a, l, r.This works since

⊗k is associative, which is easily checked, and if V is a

vector space over k, then k⊗ V ∼= V ∼= k⊗ V .To see this, consider that k is a one-dimensional vector space over itself and

therefore has a basis {e}. If (f1, . . . , fm) is a basis for V then {e⊗ fj} is a basisfor k⊗ V and {fj ⊗ e} a basis for V ⊗ k, where 1 ≤ j ≤ m.

We will hereafter consider this particular structure when dealing with Vectkas a monoidal category.

Definition 3.5. The left dual of an object X in a monoidal category is anobject X∗ for which there exists a pair of morphisms evX : X∗

⊗X −→ 1 and

10

coevX : 1 −→ X⊗X∗, called the evaluation and coevaluation morphisms, such

that the following two diagrams commute:

X

1⊗X X ⊗ 1

(X ⊗X∗)⊗X X ⊗ (X∗ ⊗X)

lX

coevX ⊗ idX

rX

aX,X∗,X

idX ⊗ evX

X∗

1⊗X∗ X∗ ⊗ 1

(X∗ ⊗X)⊗X∗ X∗ ⊗ (X ⊗X∗)

lX rX

idX∗ coevXevX ⊗ idX∗

a−1X∗,X,X∗

Here X is the right dual of X∗. We denote a right dual of an object Y as ∗Y .One can make a definition with similar diagrams using the morphismsev′X : X

⊗∗X −→ 1 and coev′X : 1 −→∗ X

⊗X, which we will omit here.

An object that has both left and right duals is called rigid. This is equivalentto saying that a rigid object has a left dual and is itself a left dual to some object.

Example 3.6. In Vectk, the left and right duals coincide and exist for vectorspaces V that are finite-dimensional, which also means that they are the rigidobjects. In that case the dual V ∗ is the standard notion of a dual vector space,which is the set of all linear maps f : V −→ k with pointwise addition and scalarmultiplication with scalars from k, sometimes written simply as Hom(V,k).

3.2 Tensor categories

Definition 3.7. A tensor category is a category satisfying the following:

(T1-5) It is abelian.

(T6) It is k-linear.

(T7) It is locally finite.

(T8-14) It is monoidal.

(T15) The tensor product⊗

is bilinear on morphisms.

(T16) End(1) ∼= k.

(T17) It is rigid, meaning that every object in the category is rigid.

The fact that a tensor category is a rigid, monoidal and abelian implies thatthe tensor product

⊗is biexact (Etingof et al., 2015, p. 66, [1]).

11

Example 3.8. The category FinVectk of finite dimensional vector spaces isalso a tensor category, with (T1-14) and (T17) being shown in Chapter 2, (T15)follows from the definition of

⊗k and (T16) follows from the fact that 1 = k.

We have also previously shown that it is finite, making it a finite tensor category.The category FinRepk(G) of finite dimensional k-representations of a group

G is also a fintie tensor category. Note that FinVectk ∼= FinRepk({e}).

Definition 3.9. Let C,D be tensor categories and let F : C −→ D be an exactand faithful k-linear functor. We call F a quasi-tensor functor if it mapsF (1C) = 1D and it is equipped with a functorial isomorphism

J : F (−)⊗ F (−)∼−→ F (−⊗−).

Remark 3.10. In other definitions of a quasi-tensor functor, the categoriesC,D need not be rigid, but we would then need to make sure that their tensorproducts are still bilinear. Since we are not affected by this, we choose to makethis simpler definition instead.

4 Frobenius-Perron dimensions

Frobenius-Perron dimensions provide us with a way to assign a value to objectsin tensor categories. It is named after the following classical theorem in linearalgebra, which we will state but not prove here.

4.1 Grothendieck rings

Theorem 4.1 (Frobenius-Perron). Let A be a square matrix with non-negative real entries. Then the following holds:

1. A has a non-negative real eigenvalue. In particular, the spectral radiusλ is an eigenvalue, meaning that |µ| ≤ λ for all other eigenvalues µ.Additionally, the eigenvector corresponding to λ has non-negative entries.

2. If A has an eigenvector v with strictly positive entries, then the corre-sponding eigenvalue is λ.

3. If A has strictly positive entries, then λ is simple, meaning that |µ| < λfor all other eigenvalues µ and the corresponding eigenvector v has strictlypositive entries.

Definition 4.2. Let A be a ring that is free as a Z-module. If A has a basisB = {bi}i∈I such that bibj =

∑k∈I c

kijbk, ckij ∈ Z+, then A is called a Z+-ring.

If 1 is a basis element, then we call A a unital Z+-ring.If for any X,Z in B, there exist elements Y1, Y2 such that XY1 and Y2X

contain Z with a non-zero coefficient, then we call A a transitive Z+-ring.

A very important example of a transitive unital Z+-ring is the following:

12

Definition 4.3. Let C be a finite tensor category. The Grothendieck ring,Gr(C), is the free abelian group generated by isomorphism classes of simpleobjects Xi. The class of any object Y in C is given by

[Y ] =∑i

[Y : Xi]Xi

Where [Y : Xi] denotes the number of occurrences of Xi in the filtration of Y .

Now we will show that it is in fact a transitive unital Z+-ring.It is an abelian group with [X] = [A] + [B] for every short exact sequence

0 −→ A −→ X −→ B −→ 0.

It has multiplicative unit [1], since 1 is simple, and multiplication is defined by

XiXj := [Xi ⊗Xj ] =∑k

[Xi ⊗Xj : Xk]Xk

Where Xi,j,k are representatives of the simple objects in C.The exactness of the tensor product gives us the following two relations:

[(Xi ⊗Xj)⊗Xm : Xl] =∑k

[Xi ⊗Xj : Xk][Xk ⊗Xm : Xl],

[Xi ⊗ (Xj ⊗Xm) : Xl] =∑k

[Xj ⊗Xm : Xk][Xi ⊗Xk : Xl]

Which gives us an isomorphism (Xi ⊗Xj)⊗Xm∼= Xi ⊗ (Xj ⊗Xm) and we see

that the multiplication defined above is associative.Lastly, transitivity comes from the fact that 1 is simple and C is rigid, so for

any simple objects X,Z in C, Z is contained in X ⊗X∗⊗Z, since X ⊗X∗ = 1.Thus one can always pick a simple object Y1 contained in X∗ ⊗ Z so thatX ⊗ Y1 = XY1 contains Z. One similarly shows the case for Y2.

From this we can start to talk about Frobenius-Perron dimensions.

4.2 Frobenius-Perron dimensions

Definition 4.4. Let A be a transitive unital Z+-ring of finite rank and let B beits basis. We define the Frobenius-Perron dimension, FPdim, as the grouphomomorphism FPdim : A −→ C, so that for any X in A, FPdim(X) is thelargest eigenvalue of the matrix NX of left multiplication by X, given by

XXi =∑j

(NX)ijXj , for Xi, Xj ∈ B

Since we only have strictly positive entries, by the Frobenius-Perron theoremthe eigenvalue will correspond to the spectral radius of the matrix and one canalso show that FPdim(X) ≥ 1.

FPdim(Y ) for Y ∈ A is extended from the basis elements by additivity.

13

For a finite tensor category C with simple objects Xi, the Frobenius-Perrondimension of any object X is given by its the Frobenius-Perron dimension of itscorresponding object in Gr(C). The left multiplication matrix NX of an objectX would be given by (NX)ij = [X ⊗Xi : Xj ] for simple objects Xi, Xj and inthe Grothendieck ring we get XXi =

∑j(NX)ijXj .

Example 4.5. In the category FinRepk(G) of finite dimensional k-representationsof a group G, we have that FPdim(X) = dimk(X) for all X.

Definition 4.6. Let C be a finite tensor category. We say that C is integral ifthe Frobenius-Perron dimensions of all elements in Gr(C) are integers.

5 Hopf algebras

Hopf algebras are structures with strong connections to groups. One may viewthem as an abstraction of the group algebra of a group over a field. They arecommon in representation theory, since they have the nice property that in therepresentation category of a Hopf algebra, the tensor product of two represen-tations is a representation, as is the dual vector space of a representation. Wewill only consider representation categories of quasi-Hopf algebras, since theygive us a connection to tensor categories.

5.1 Algebras

In field theory, an algebra over a field is a vector space equipped with a bilin-ear product, called multiplication, meaning that it is distributive over additionand compatible with scalar multiplication. One can extend this definition toalgebras over commutative rings, but we will focus on algebras over fields andfurthermore, algebras where the multiplication is associative.

Canonical examples of algebras are the real numbers and the complex num-bers, with regular addition and multiplication. But in fact, every field is analgebra over itself. In categorical terms we will consider k-algebras as objectsin Vectk, or more formally:

Definition 5.1. A unital associative algebra over a field k is a k-vectorspace A together with two linear maps, the multiplication µ : A

⊗A −→ A,

and the unit ι : k −→ A that satisfy both the associative law,

µ ◦ (idA⊗µ) = µ ◦ (µ⊗ idA)

and the unit law:µ ◦ (idA⊗ι) = µ ◦ (ι⊗ idA) = idA

Dually, a counital coassociative coalgebra over a field k is a k-vectorspace C together with two linear maps, the comultiplication ∆ : C −→ C

⊗C,

and the counit ε : C −→ k that satisfy both the coassociative law,

(∆⊗ idC) ◦∆ = (idc⊗∆) ◦∆

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and the counit law:

(ε⊗ idC) ◦∆ = (idC ⊗ε) ◦∆ = idC

We will from now on call them simply algebras and coalgebras. A k-vectorspace B that is both an algebra and a coalgebra is called a bialgebra.

Example 5.2. The group algebra of a group G over a field k, written kG,which is the vector space over k that is freely generated by elements g in theunderlying set of G, is a bialgebra. Denote the scalars λ ∈ k.

1. It is a vector space by definition with elements of the form:

v =∑g∈G

λg · g

.

2. The multiplication is given on basis elements by the group operation:

v · u =∑gv∈G

∑gu∈G

(λgvλgu)(gv · gu)

3. We have the group identity as the unit, thus ι(λ) = eG.

4. Checking the associative and the unit laws are equivalent to checking thatthe following two diagrams commute (we write kG as B to save space):

B ⊗B ⊗B B ⊗B

B ⊗B B

idB ⊗µ

µ⊗idB µ

µ

,

B ⊗ k B ⊗B k⊗B

B B B

idB ⊗ι

rB µ

ι⊗idB

lB

Rewriting the diagrams with elements instead and writing the multiplica-tion as µ(u, v) = uv gives us the following two diagrams:

< u, v, w > < u, vw >

< uv,w > (uv)w = u(vw)

< u, λ > < u, eG > < eG, u > < λ, u >

u u u u

These are fulfilled by (1.) and (2.), thus kG follows the categorical defini-tion of an algebra.

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5. The comultiplication is given by ∆(g) = g⊗ g. One can see that the coas-sociative law holds by confirming that the following diagram commutes:

B B ⊗B

B ⊗B B ⊗B ⊗B

∆

∆ idB ⊗∆

∆⊗idB

6. The counit is given by ε(g) = 1k. The counit law is equivalent to thefollowing commuting diagram:

B ⊗B B B ⊗B

k⊗B B B ⊗ k

ε⊗idB

∆∆

idB idB ⊗ε

∼ ∼

Thus kG is also a coalgebra, and therefore a bialgebra.

Note that our comultiplication is bilinear, so for any vector αg + βh ∈ B withα, β ∈ k and g, h ∈ G, we get ∆(αg+βh) = α∆(g)+β∆(h) = α(g⊗g)+β(h⊗h).

Remark 5.3. The definition of a bialgebra is self-dual, so if B is a bialgebra,then its dual B∗ is also a bialgebra. Given the above example of group algebras,we get that the dual of a group algebra of a group G over a field k, written(kG)∗, is a bialgebra.

Definition 5.4. An antipode S on a bialgebra (B,µ, ι,∆, ε) is a linear mapS : B −→ B such that:

µ ◦ (idB ⊗S) ◦∆ = ι ◦ ε = µ ◦ (S ⊗ idB) ◦∆

Antipodes are self dual, so if B is a finite-dimensional bialgebra with an antipodeSB , then the dual bialgebra B∗ has an antipode SB∗ = S∗B .

Example 5.5. In the example of group algebras, we want to find an antipodesuch that S(g) · g = g · S(g) = eG, so our antipode is the inverse S(g) = g−1.

Proposition 5.6. If a bialgebra B has an antipode, then it is unique.

Proof. Let S, S′ be two antipodes, then using the properties of antipodes, theassociativity of µ, and the coassociativity of ∆ we get:

S = µ ◦ (S ⊗ (µ ◦ (idB ⊗S′) ◦∆)) ◦∆

= µ ◦ (idB ⊗µ) ◦ (S ⊗ idB ⊗S′) ◦ (idB ⊗∆) ◦∆

= µ ◦ (µ⊗ idB) ◦ (S ⊗ idB ⊗S′) ◦ (∆⊗ idB) ◦∆

= µ ◦ ((µ ◦ (S ⊗ idB) ◦∆)⊗ S′) ◦∆ = S′

Note that this does not hold for antipodes of quasi-bialgebras, whose antipodesneed not be unique.

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Definition 5.7. A Hopf algebra over k is a k-bialgebra with an invertibleantipode, meaning that there is a S−1 such that SS−1 = idB = S−1S.

The group algebra is a Hopf algebra, with S−1 = S.Since both bialgebras and antipodes are self dual, so are Hopf algebras,

meaning that the dual of a group algebra is also a Hopf algebra.

Definition 5.8. For any Hopf algebra H, we call the elements g in H grouplikeif g 6= 0 and ∆(g) = g ⊗ g.

The grouplike elements in a Hopf algebra H form a group G(H) under themultiplication inherited from H. This is because 1⊗ 1 is always grouplike, theinherited multiplication is closed and associative and we get inverses from theantipode S(g).

5.2 Quasi-Hopf algebras

If we relax the requirements of coassociativity and instead introduce an invert-ible element Φ that controls the non-coassociativity, we can generalize bialgebrasand Hopf algebras to the following:

Definition 5.9. A quasi-bialgebra B over k is an algebra over k togetherwith a comultiplication ∆ : B −→ B⊗B, a counit ε : B −→ k and an invertibleelement Φ ∈ B ⊗B ⊗B that satisfies the following identities:

(idB ⊗∆) ◦ (∆(b)) = Φ ◦ (∆⊗ idB) ◦ (∆(b)) ◦ Φ−1, b ∈ B(idB ⊗ idB ⊗∆) ◦ Φ ◦ (∆⊗ idB ⊗ idB) ◦ Φ = (1⊗ Φ) ◦ (idB ⊗∆⊗ idB) ◦ Φ ◦ (Φ⊗ 1)

(ε⊗ idB) ◦ (∆(b)) = b = (idB ⊗ε) ◦ (∆(b))

(idB ⊗ε⊗ idB) ◦ Φ = 1⊗ 1

Every bialgebra is also a quasi-bialgebra with Φ = 1⊗ 1⊗ 1.

Definition 5.10. A quasi-Hopf algebra H over k is a quasi-bialgebra thathas a bijective antipode S : H −→ H and elements α, β ∈ H such that:

For Φ =∑i

Φ1i ⊗ Φ2

i ⊗ Φ3i , and Φ−1 =

∑i

Φ1i ⊗ Φ2

i ⊗ Φ3i

We have∑i

Φ1iβS(Φ2

i )αΦ3i = 1, and

∑i

S(Φ1i )αΦ2

iβS(Φ3i ) = 1

And if ∆(a) =∑

bi ⊗ ci, for a ∈ H

then∑i

S(a1i )αa

2i = ε(a)α, and

∑i

a1iβS(a2

i ) = ε(a)β

Every Hopf algebra is also a quasi-Hopf algebra with α = β = 1.

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Example 5.11. Let k be a field containing a primitive fourth root of unity ω,meaning that the smallest integer a such that ωa = 1 is a = 4. Let H denotethe algebra over k generated by the elements x, y, z with the following relations:

x2 = 1, y4 = z4 = 0, xy = −yx, xz = −zx, zy = ωyz

Let p = 12 (1 + x) and q = 1

2 (1 − x). We then define comultiplication and thecounit operation on H as follows:

∆(x) = x⊗ x, ε(x) = 1

∆(y) = y ⊗ (p+ ωq) + 1⊗ py + x⊗ qy, ε(y) = 0

∆(z) = z ⊗ (p− ωq) + 1⊗ pz + x⊗ qz, ε(y) = 0

Our antipode is given by:

S(x) = x, S(y) = −y(p+ ωq), and S(z) = −z(p− ωq)

And lastly α = x, β = 1 and Φ = 1⊗ 1⊗ 1− 2q ⊗ q ⊗ q.This does indeed define a quasi-Hopf algebra, and since the comultiplication

is not coassociative, it is not a Hopf algebra (Bulacu et al., 2019, p. 124, [7]).

Definition 5.12. We call K a quasi-Hopf subalgebra of a quasi-Hopf algebraH if it is a subalgebra K ⊆ H that is closed under the operations ∆, ε and S,and where Φ,Φ−1 ∈ K ⊗K ⊗K.

From the above one can also deduce the meaning of a Hopf subalgebra,namely a sub-bialgebra that is closed under the antipode.

Theorem 5.13. A finite tensor category is integral if and only if it is equivalentto the category of representations of a finite-dimensional quasi-Hopf algebra.

This is the link that connects tensor categories with quasi-Hopf algebras and isa key part in our proof of the Nichols-Zoeller theorem for quasi-Hopf algebras.The proof is omitted for sake of brevity (Etingof el al., 2015, p. 121, [1]).

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6 Main results

Now we will begin to prove the Nichols-Zoeller theorem for quasi-Hopf algebras.

Definition 6.1. Let C be a finite k-linear abelian category and let K(C) bethe free abelian group generated by the indecomposable projective objects ofC, up to isomorphism. Then elements of K(C) ⊗Z k will be called the virtualprojective objects of C.

Definition 6.2. Let C be a finite tensor category. Then the regular objectof C is the virtual projective object

RC =∑i∈I

FPdim(Xi)Pi

where Xi, Pi ∈ C, Xi are simple and Pi are indecomposable projective objects.

Note that some authors define FPdim(C) := FPdim(RC).

Definition 6.3. Let C,D be locally finite abelian categories and let F : C −→ Dbe an additive functor. We say that F is surjective if every simple object inD is a subquotient of some object F (X), X ∈ C.

This definition is not equivalent to the usual definition of an essentially surjectivefunctor, which says that every object of D is isomorphic to some object F (X).

Lemma 6.4. Let C,D be finite tensor categories, and F : C −→ D a surjectivequasi-tensor functor. Then F maps projective objects to projective objects.

This follows from the surjectivity and exactness of F (Etingof el al., 2015, p.122, [1]). We will use it prove the following theorem:

Theorem 6.5. Given finite tensor categories C,D with regular objects RC , RD,

and a surjective quasi-tensor functor F : C −→ D, we have F (RC) = FPdim(RC)FPdim(RD)RD.

Proof. By the above lemma, F (RC) is a virtually projective object. If wewrite F (RC) and RD in the basis of Pi, then they are both eigenvectors ofa multiplication matrix with strictly positive entries. Since F (RC) and RDalso have strictly positive entries, by the Frobenius-Perron theorem they bothhave the same eigenvalue, the spectral radius, which is simple for matrices withstrictly positive entries. Therefore, they must be proportional, and we get theproportionality constant by computing their Frobenius-Perron dimensions.

Lemma 6.6. Let C,D be finite tensor categories, and F : C −→ D a surjectivequasi-tensor functor. If C is integral, then so is D, and F (RC) is free of rankFPdim(RC)FPdim(RD) , where FPdim(RD) divides FPdim(RC).

Proof. Since C is integral, we have that FPdim(RC) is an integer, by assumption.Thus the multiplication matrix of F (RC) has integer eigenvalue FPdim(RC).By the above theorem, F (RC) is proportional to RD so the Frobenius-Perron

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dimensions of the simple objects of D correspond to the coordinates of theeigenvector with strictly positive entries given by the multiplication matrix ofF (RC). Therefore, all coordinates are rational numbers, but since they are alsoeigenvalues of some integer matrix, they are algebraic integers and the rationalalgebraic integers are integers.

This means that D is integral, which we know by Theorem 5.13 to be equiv-alent to the category of representations of a quasi-Hopf algebra B. Here RDcorresponds to the free module of rank 1 over B, and so multiples of RD are

free B-modules of finite rank. In particular, F (RC) is free of rank FPdim(RC)FPdim(RD) ,

where FPdim(RD) divides FPdim(RC).

The above lemma together with Theorem 5.13 imply the following:

Theorem 6.7. Let H be a finite-dimensional quasi-Hopf algebra over a field k,with a quasi-Hopf subalgebra B. Then H is free as a left (and right) B-module,and dimB divides dimH.

This is the Nichols-Zoeller theorem for quasi-Hopf algebras, with the Hopf al-gebra version following immediately.

Worth noting is that in the Hopf algebra case, there exists a nice choice ofthe basis for H as a B-module (Schneider, 1992, [5]).

Corollary 6.8. Let H be a quasi-Hopf algebra. If dim(H) = p, where p is aprime, then H has no proper nontrivial quasi-Hopf subalgebras.

Now consider the case when H is the group algebra kG of a finite group G. Anysubgroup N of G gives rise to a Hopf subalgebra kN of H and so dim(kN) = |N |divides dim(kG) = |G|. Thus, for any finite group G the order of its subgroupsdivide the order of G. This is the classic Lagrange’s theorem.

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7 References

[1] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik. TensorCategories. Mathematical Surveys and Monographs, American Mathemati-cal Society, 2015, Volume 205.

[2] Saunders Mac Lane. Categories for the working mathematician. Graduatetexts in mathematics, Springer, 1998, 2nd ed.

[3] Warren D. Nichols and M. Bettina Zoeller. A Hopf Algebra Freeness Theo-rem. American Journal of Mathematics, 04/1989, Volume 111, Issue 2.

[4] Peter Schauenburg A quasi-Hopf algebra freeness theorem. Proceedings ofthe American Mathematical Society, 04/2004, Volume 132, Issue 4.

[5] Hans-Jurgen Schneider Normal basis and transitivity of crossed products forHopf algebras. Journal of Algebra, 11/1992, Volume 152, Issue 2.

[6] Ulrich Oberst and Hans-Jurgen Schneider Untergruppen formeller gruppenvon endlichem index. Journal of Algebra, 07/1974, Volume 31, Issue 1.

[7] Daniel Bulacu, Stefaan Caenepeel, Florin Panaite, and Freddy Van Oys-taeyen Quasi-hopf algebras: A categorical approach. Cambridge UniversityPress, 2019, Volume 171.

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