On the linear independency of monoidal natural transformations

Let $F, G: \mathcal{I} \to \mathcal{C}$ be strong monoidal functors from a skeletally small monoidal category $\mathcal{I}$ to a tensor category $\mathcal{C}$ over an algebraically closed field $k$. The set $Nat(F, G)$ of natural transformations $F \to G$ is naturally a vector space over $k$. We show that the set $Nat_\otimes(F, G)$ of monoidal natural transformations $F \to G$ is linearly independent as a subset of $Nat(F, G)$. As a corollary, we can show that the group of monoidal natural automorphisms on the identity functor on a finite tensor category is finite. We can also show that the set of pivotal structures on a finite tensor category is finite.


Introduction
Monoidal categories [3] arise in many contexts in mathematics. In this note, we prove a basic fact on monoidal natural transformations between strong monoidal functors. To describe our result, we recall some definitions. Throughout, we work over an algebraically closed field k. By a tensor category over k we mean a k-linear abelian rigid monoidal category C satisfying the following conditions: • The unit object 1 ∈ C is simple.
• The tensor product ⊗ : C × C → C is k-linear in both variables.
• Every object of C is of finite length.
• Every hom-set in C is finite-dimensional over k. Let C be a tensor category over k and let I be a skeletally small monoidal category. If F, G : I → C are functors, the set Nat(F, G) of natural transformations F → G is naturally a vector space over k. Now we suppose that both F and G are strong monoidal functors. Then we can consider the set Nat ⊗ (F, G) of monoidal natural transformations. Our main result in this note is the following: A k-linear abelian category A is said to be finite if it is k-linearly equivalent to the category of finitely-generated modules over a finite-dimensional k-algebra. We present some applications of Theorem 1.1 to finite tensor categories [2].
By a tensor functor we mean a k-linear strong monoidal functor between tensor categories. Let F, G : C → D be right exact tensor functors from a finite tensor category C to a tensor category D. Then it turns out that Nat(F, G) is finite-dimensional (see Lemma 2.5). Hence: Let Aut ⊗ (F ) ⊂ Nat ⊗ (F, F ) denote the group of monoidal natural automorphism on a monoidal functor F : C → D. As an immediate consequence of Corollary 1.2, we have the following: Corollary 1.3. Let C be a finite tensor category and let F : C → C be a right exact tensor functor. Then Aut ⊗ (F ) is finite. In particular, Aut ⊗ (id C ) is finite.
We give some remarks on the structure of Aut ⊗ (id C ) in Section 3. We note that the finiteness of Aut ⊗ (id C ) is well-known in the case where C is the category Rep(H) of finite-dimensional representations of a finite-dimensional Hopf algebra H. In fact, then Aut ⊗ (id C ) is isomorphic to the group of central grouplike elements of H.
There is another important corollary of Corollary 1.2. A pivotal structure on a rigid monoidal category C is an element of Piv(C) := Nat ⊗ (id C , (−) * * ), where (−) * : C → C is the left duality functor.
Also this corollary is well-known in the case where C = Rep(H) for some finitedimensional Hopf algebra H. In fact, then Piv(C) is in one-to-one correspondence between grouplike elements g ∈ H such that S 2 (x) = gxg −1 for all x ∈ H, where S is the antipode of H. Remark 1.5. We should remark that every finite tensor category is equivalent as a tensor category to the category of finite-dimensional representations of a weak quasi-Hopf algebra [2]. It will be interesting to interpret our results in terms of "grouplike elements" of weak quasi-Hopf algebras (cf. [4]).
2. Proof of the main theorem and its corollary 2.1. Proof of Theorem 1.1. Let C be a tensor category over k. Without loss of generality, we may suppose C to be strict. Let V ∈ C be an object. The tensor product of C defines two k-linear endofunctors V ⊗ (−) and (−) ⊗ V on C. Both these functors are exact since C is rigid [1]. Proof. Suppose that V = 0. Then the evaluation morphism d V : V * ⊗ V → 1 is an epimorphism. Indeed, otherwise, since 1 is simple, d V = 0. It follows from the rigidity axiom that id V = 0, and hence V = 0, a contradiction. Let X, Y ∈ C be objects. Consider the linear map If id V ⊗f = 0, then, by the above equation, we have f • (d V ⊗ id X ) = 0. As we observed above, d V is an epimorphism. Since the tensor product of C is exact, also d V ⊗ id X is an epimorphism. Therefore, we conclude that f = 0. This means that the functor V ⊗ (−) is faithful.
The faithfulness of (−) ⊗ V can be proved in a similar way.
Let I be a skeletally small category and let F, G : I → C be functors. Then the set Nat(F, G) of natural transformations F → G is a vector space over k. Let V, W ∈ C be objects. The tensor product of C defines a linear map

Lemma 2.2. The above map is injective.
Proof. Let f 1 , · · · , f m ∈ Nat(F, G) be linearly independent elements. It suffices to show that if c 1 , · · · , c m : V → W are morphisms in C such that for all X ∈ I, then c i = 0 for i = 1, · · · , m. We show the above claim by induction on the length ℓ(V ) of V . If ℓ(V ) = 0, our claim is obvious. Suppose that ℓ(V ) ≥ 1. Then there exists a simple subobject of V , say L. Let K be the image of the morphism By the linear independence of f i 's, we have λ ij = 0 for all i and j. This means that c i | L = 0 for all i.
Let p : V → V /L be the projection. By the above observation, for each i, there exists a morphism c i : In the case where F and G are constant functors sending all objects of I respectively to X ∈ C and Y ∈ C, Lemma 2.2 states that the map induced from the tensor product is injective. Now we generalize Lemma 2.2. Let I i (i = 1, 2) be skeletally small categories and let F i : I i → C (i = 1, 2) be functors. We denote by F 1 ⊗ F 2 the functor induced from the tensor product is injective.
Proof. Let f 1 , · · · , f m ∈ Nat(F 2 , G 2 ) be linearly independent elements. Suppose that c 1 , · · · , c m ∈ Nat(F 1 , G 1 ) are elements such that If we fix X ∈ I 2 , we can apply Lemma 2.2 and obtain that c i | X = 0 for i = 1, · · · , m. By letting X run through all objects of I 1 , we have that c i = 0 for i = 1, · · · , m. Thus the map under consideration is injective. Now we can prove Theorem 1.1. Our proof is based on a proof of the linear independence of grouplike elements in a coalgebra over a field.
Proof of Theorem 1.1. Recall the assumptions: I is a skeletally small monoidal category, C is a tensor category over k and F and G are strong monoidal functors from I to C.
By the definition of monoidal natural transformations, we have for all X, Y ∈ I. By Lemma 2.4, m i=1 λ j λ i g i = λ j g j for each j. By the linear independence of g i 's, we have that λ i = 0 for i = h. Therefore, g = g h . This is a contradiction.

2.2.
Proof of Corollary 1.2. In view of Theorem 1.1, it is interesting to know when Nat(F, G) is finite-dimensional. We conclude this section by proving the following lemma, which completes the proof of Corollary 1.2. Note that a finite k-linear abelian category satisfies the assumptions on A in this lemma. Therefore Corollary 1.2 follows immediately from Theorem 1.1.
Proof. Let f ∈ Nat(F, G). By the assumption, for every X ∈ A, there exists an exact sequence P ⊕m → P ⊕n → X → 0. By applying F and G to this sequence, we have a commutative diagram in B with exact rows. This means that f | X is determined by f | P , and hence the map f → f | P is injective.
3. Some remarks on Aut ⊗ (id C ) 3.1. Bound of the order. Let C be a finite tensor category over k. In this section, we give some remarks on the structure of the group G(C) = Aut ⊗ (id C ). We first note that by the definition of natural transformations, G(C) is abelian.
Let I be the set of isomorphism classes of simple objects of C. For each i ∈ I, we fix S i ∈ i. Let P i be the projective cover of S i . Then P = i∈I P i is a projective generator. Applying Lemma 2.5, we have an injective linear map Nat(id C , id C ) → End C (P ). By the definition of natural transformations, the image of this map must be contained in the center of End C (P ). Hence we obtain an injective linear map Nat(id C , id C ) → Z(End C (P )), η → η| P . Remark 3.1. As every object of C is of finite-length, C is k-linearly equivalent to the category of finite-dimensional right End C (P )-modules, and hence the above map is actually an isomorphism of k-algebras. We omit the detail here since the surjectivity will not be used. Theorem 1.1 states that G(C) ⊂ Nat(id C , id C ) is linearly independent. Therefore, we have the following bound on the order of G(C).
In the case where C = Rep(H) for some finite-dimensional Hopf algebra H, this proposition is obvious since the right-hand side of the inequality is equal to the dimension of the center of H.

Values on simple objects. Let us consider the map
This map is not injective any more unless C is semisimple. Since k is algebraically closed, we can identify End C (S i ) with k. The above map induces a group homomorphism We show that ϕ is injective if k is of characteristic zero. To describe the kernel of ϕ, we introduce some subgroups of G(C). If p = char(k) > 0, then we set G(C) p = {g ∈ G(C) | g p k = 1 for some k ≥ 0}, The order of g is relatively prime to p}. Otherwise we set G(C) p = {1} and G(C) ′ p = G(C). By the fundamental theorem of finite abelian groups, we have a decomposition G(C) = G(C) p × G(C) ′ p . Lemma 3.3. Let X ∈ C be an indecomposable object.
(a) If g ∈ G(C) p , then g| X is unipotent.
. As X is indecomposable, End C (X) is a local algebra. Since k is algebraically closed, g| X can be written uniquely in the form where m is the maximal ideal of End C (X). If r = 0, then there exists k ≥ 1 such that r ∈ m k−1 but r = m k . By the binomial formula, we have (g| X ) n ≡ λ n id X +nr (mod m k ) for every n ≥ 0. This implies that if (g| X ) n = id X , then λ n = 1 and nr = 0.
(a) Suppose that g ∈ G(C) p . Since the claim is obvious for p = 0, we assume that p > 0. Then the order of g| X is a power of p. Thus, by the above observation, λ = 1. This implies that g| X is unipotent.
(b) Suppose that g ∈ G(C) ′ p . Then the order of g| X is nonzero in k. Thus, by the above observation, r = 0. This implies that g| X = λ id X .
In what follows, we denote λ in equation (3.1) by λ g (X). We remark the following easy but important property of λ g (X).
Lemma 3.4. Let X and Y be indecomposable objects of C. If X and Y belong to the same block, then λ g (X) = λ g (Y ) for every g ∈ G(C) ′ p . Here, a block is an equivalence class of indecomposable objects of C under the weakest equivalence relation such that two indecomposable objects X and Y of C are equivalent whenever Hom C (X, Y ) = 0.
Proof. Let g ∈ G(C) ′ p . By the definition of blocks, it is sufficient to prove in the case when there exists a nonzero morphism f : X → Y . By the naturality of g, Now we have a description of the kernel of ϕ as follows: Proof. Let g ∈ G(C) p . Then, by Lemma 3.3, g| S = id S for every simple object S of C and hence g ∈ Ker(ϕ). This implies that G(C) p ⊂ Ker(ϕ).
Next let g ∈ G(C) ′ p ∩ Ker(ϕ). Let X be an indecomposable object of C and fix a simple subobject S of X. Then, by Lemma 3.4, λ g (X) = λ g (S). On the other hand, λ g (S) = 1 since g ∈ Ker(ϕ). This implies that g| X = id X for all indecomposable object X and hence g = 1. Therefore G(C) ′ p ∩ Ker(ϕ) = {1}. Now we recall that G(C) = G(C) p × G(C) ′ p . Our claim follows immediately from the above observations. It is interesting to characterize the image of ϕ. Let N k ij (i, j, k ∈ I) be the multiplicity of S k as a composition factor of S i ⊗ S j . In the case where C is semisimple, it is known that the image of ϕ is the set of functions λ : I → k × such that (3.2) λ(i)λ(j) = λ(k) whenever N k ij = 0 (i, j, k ∈ I). In general, the image of ϕ is smaller than the set of such functions.
Proposition 3.6. The image of ϕ is the set of all functions satisfying (3.2) and the following condition: λ(i) = λ(j) whenever S i and S j belong to the same block (i, j ∈ I).
Proof. We remark that if an indecomposable object X ∈ C has S i as a composition factor, then S i and X belong to the same block. Indeed, then there exists a nonzero morphism P i → X and hence P i and X belong to the same block. On the other hand, since S i is a quotient of P i , S i and P i belong to the same block. Therefore the claim follows.
Let λ = ϕ(g). By Proposition 3.5, we may assume g ∈ G(C) ′ p . (3.3) follows from Lemma 3.4. Thus we check that λ satisfies (3.2). Let i, j ∈ I. By the definition of monoidal natural transformations, Suppose that N k ij = 0. This means that S i ⊗ S j has S k as a composition factor. Let X be an indecomposable direct summand of S i ⊗ S j having S k as a composition factor. By the above equation, g| X = λ(i)λ(j) id X . On the other hand, since X and S k belong to the same block, g| X = λ(k) id X . Therefore λ(k) = λ(i)λ(j).
Conversely, given a function λ : I → k × satisfying (3.2) and (3.3), we define a natural automorphism g : id C → id C as follows: If X is an indecomposable object of C, then g| X = λ(i) id X where i ∈ I is such that X has S i as a composition factor. As λ satisfies (3.3), this does not depends on the choice of i. We can extend g to all objects of C, since they are direct sums of indecomposable objects. Now we need to show that g ∈ G(C), that is, g| X⊗Y = g| X ⊗ g| Y for all objects X, Y ∈ C. We may assume that X and Y are indecomposable. Suppose that X = i∈I m i S i (m i ∈ Z ≥0 ) and Y = j∈I n j S j (n i ∈ Z ≥0 ) in the Grothendieck ring K(C). Then . This equation means that if X ⊗ Y has S k as a composition factor, then there exist i, j ∈ I such that X has S i as a composition factor, Y has S j as a composition factor and N k ij = 0. By the definition of g and (3.2), we have g| X⊗Y = g| X ⊗ g| Y .
It is obvious that λ = ϕ(g). The proof is completed.
The following theorem is a direct consequence of Proposition 3.5 and 3.6.