K-theory of braided monoidal categories

A.A.Davydov

August 22, 2001

Department of Mathematics, Division of Information and Communication Sciences, Macquarie University, Sydney, NSW 2109, Australia

davydov@math.mq.edu.au

Abstract

The commutativity constraint of braided category gives rise to an algebraic structure on its K-theory known as Gerstenhaber algebra. If, in addition, the braiding has a compatible balanced structure the Gerstenhaber bracket on K-theory is generated by a Batalin-Vilkovisky differential.

We use these algebraic structures to prove a generalization of Anderson-Moore-Vafa theorem Anderson-Moore-Vafa which says that the order of the twist in a semi-simple ribbon (tortile) category with finitely many simple objects is finite.

1 Introduction

Categories equipped with tensor products, called monoidal or tensor categories, have been studied and used extensively (see [9] and references therein). Informally speaking, monoidal category is a ”categorification” of the notion of monoid (or algebra in the linear case). Thinking of elements of the set an objects of a category we have to replace the operation on the set by a functor between appropriate categories; the relations for the operation give rise to isomorphisms of functors (constraints). The remarkable thing about this categorification process is that the constraints can be chosen to be coherent, i.e. satisfy to natural self-compatibility conditions. The presence of coherent constraints explains the following phenomenon: going back with inverse ”decategorification” process such as K-theory we will have richer structure than we started with. In this paper we examine the extra structure appearing on the K-theory of commutative (braided) monoidal categories. It was proved in [8] that the K-theory spectrum of a braided monoidal category is an E_{2}-ring spectrum, i.e. an algebra over the little cube operad e_{2}. Here we note that the K-groups of a braided monoidal category (the stable homotopy of the corresponding K-theory spectrum) form an algebra over the homology of the operad e_{2}.

The commutativity constraint (braiding) for a monoidal category was introduced in [9] (which circulated as a preprint since mid eigties). Immediately after, it found applications in the theory of quantum groups [7], low-dimensional topology and mathematical phisics (see [17]). The algebraic structure on K-theory induced by braiding appeared first in the deformation theory of associative algebras [13, 14] and now known as Gerstenhaber algebra. A Gerstenhaber algebra is a graded commutative algebra equipped with the bracket { , } of degree 1 which gives a Lie algebra structure on the suspension

{a, b} = −(−1)^{(|a|+1)(|b|+1)}{b, a},
{a, {b, c}} = {{a, b}, c} + (−1)^{(|a|+1)(|b|+1)}{b, {a, c}}

and obey a version of graded Leibniz identity:

{a, bc} = {a, b}c + (−1)^{(|a|+1)|b|}b{a, c}.  (1)

When the braiding has a compatible balanced structure (which is a sort of coboundary condition) the Gerstenhaber bracket on K-theory extends to a richer structure appeared first in mathematical physics [4, 5] and known as Batalin-Vilkovisky algebra. A Batalin-Vilkovisky or BV-algebra is a Gerstenhaber algebra with an additional operator Δ of degree 1 which is a derivation with respect to the bracket such that Δ^{2} = 0 and

{a, b} = (−1)^{|a|}Δ(ab) − (−1)^{|a|}Δ(a)b − aΔ(b).  (2)

Tha paper is organised as follows. Using relations between braided categories and braid groups we present braided (balanced) monoidal categories as an algebras over certain categorical operads (the section 2). Calculating K-theory of these operads we prove that algebras over K-theory operads are exactly Gerstenhaber (BV-) algebars (the section 3). As an application we prove a generalization of a theorem of Anderson-Moore and Vafa Anderson-Moore-Vafa [1, 18] saing that the order of the twist in a semi-simple balanced monoidal category with finitely many simple objects is finite (the section 4, Theorem 4.6).

Acknowledgment

The paper was prepared with the partial supports of the ARC grant A00000187 and the RFBR grant no. 01-01-01144. The author would like to thank A.Bondal and M.Batanin for very fruitful discussions. The paper was completed during authors stay at Max Planck Institute für Mathematik which the author would like to thank for hospitality and inspiring atmosphere.

2 Braided monoidal categories as algebras over an operad

2.1 braided and balanced categories

Throughout the paper we mean by a monoidal category a strict monoidal category, e.g. we assume that associativity constraint is identity. Although it seems restrictive, the coherence theorem (see [9], Theorem 1.2) says that any monoidal category is monoidal equivalent to a strict one.

A monoidal category G is braided [9] if it is equipped with a natural family of isomorphisms

c_{X,Y} : X ⊗ Y → Y ⊗ X, ∀X, Y ∈ G,

called commutativity constrains, which satisfy the equations:

c_{X,Y ⊗Z} = (Y ⊗ c_{X,Z})(c_{X,Y} ⊗ Z) c_{X⊗Y,Z} = (c_{X,Z} ⊗ Y)(X ⊗ c_{Y,Z}) (3)

A braided monoidal category G is balanced if there given a natural family of automorphisms τ_{X} : X → X such that for any X, Y ∈ G

τ_{X⊗Y} = c_{Y,X}c_{X,Y} (τ_{X} ⊗ τ_{Y}).  (4)

The name braided motivated by the relation with Artin braid groups B_{n} which algebraically can be given by generators and relations:

B_{n} =< s_{i}, i = 1, ..., n − 1 : s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1}, s_{i}s_{j} = s_{j}s_{i}, |i − j| ≥ 2 > .

Any element σ of the braid group B_{n} defines an isomorphism c(σ) : X_{1} ⊗ ... ⊗ X_{n} → X_{σ(1)} ⊗ ... ⊗ X_{σ(n)} natural in objects X_{1}, ..., X_{n} of the braided category G. This assignment has the following properties:

c(στ) = c(σ)c(τ), c(s_{i}) = I ⊗ c_{X_{i},X_{i+1}}) ⊗ I

which actually allow to define it uniquely. There are natural surjections from the braid groups B_{n} to symmetric groups S_{n} sending generators s_{i} to the transpositions (i, i + 1). The kernel P_{n} of the surjection B_{n} → S_{n} is called the pure braid group. Note that pure braids defines natural automorphisms in a braided category.

In the balanced case the above assignment extends from the braid groups to the semi-direct products (the balanced group Bal_{n}) of the the braid group B_{n} by the free abelian group Z^{n} with respect to the permutation action of B_{n} on Z^{n}. The generator t_{i} of Z^{n} corresponds to the natural automorphism I ⊗ τ_{X_{i}} ⊗ I of X_{1} ⊗ ... ⊗ X_{n}. Note that the surjection B_{n} → S_{n} lifts to the surjection Bal_{n} → S_{n} with the kernel P b_{n} = P_{b} × Z^{n} (the pure balanced group).

2.2 braided and balanced operads

Consider an operad structure on the collection P_{n}, n ≥ 0 of pure braid groups which sends the collection of braids (σ_{1}, ..., σ_{n}, σ) from P_{m_{1}} × ... × P_{m_{n}} × P_{n} to the braid μ(σ_{1}, ..., σ_{n}, σ) in P_{m_{1}+...+m_{n}} which is a product of replacement of strings of σ by σ_{1}, ..., σ_{n}. This operad is not symmetric, but conjugation actions of braid groups on its components are compatible with the operadic substitutions. The composition maps are group homomorphisms so we can think of this group-operad as a Cat-operad P of one-object groupoids with morphisms groups P_{∗}.

The braid-operad structure on the collection P_{n}, n ≥ 0 can be extended to P b_{n}, n ≥ 0. To define the operad extension is enough to specify the image of the pair (I, t) under μ : P b_{2} × P b_{1} → P b_{2}. We set it to be s_{1}^{2}t_{1}t_{2} in accordance with the equation 4. Again the operadic substitutions will be a group homomorphisms which give us a Cat-operad of one-object groupoids Pb with compatible braid groups actions. The incliusions P_{n} ⊂ P b_{n} induce a homomorphism of operads P → Pb.

We define an End-operad End(A) of a category A as a Cat-operad which n-th component is the category of functors Funct(A^{×n}, A) from the cartesian product of n copies of A to A. Direct product and composition of functors will give an operadic substitution in the usual way. We say that the category A is an algebra over the Cat-operad O if there given a homomorphism of Cat-operads O → End(A) which is a collection of functors O_{n} → Funct(A^{×n}, A) compatible with the operadic substitutions. In other words, the action is a collection of functors

O_{n} × A × ... × A → A,  (5)

compatible with operadic composition of O_{∗}.

The construction of the natural automorphisms c(σ) implies the following

Proposition 2.3 Braided strict monoidal categories are algebras over the Cat-operad P. Balanced strict monoidal categories are algebras over the Cat-operad Pb.

We can not speak about K-theory of the operads P, Pb for their components are not abelian (exact) categories. To do so we need to replace them by their abelian variants.

2.4 linearisation

Recall that a pseudo-abelian (or Cauchy complete) category is an additive category containing images of all its idempotent endomorphisms. Here we describe k-linear pseudo-abelian Cat-operads P_{k}, Pb_{k}. The components of the operads P_{k}, Pb_{k} are k-linear pseudo-abelian envelopes of the components of P, Pb. By k-linear abelian envelope of G we mean an initial object in the category of functors from G to k-linear pseudo-abelian categories. For a small category G with the set of morphisms Mor its k-linear pseudo-abelian envelope can be identified with the category of projective modules over the semigroup ring k[Mor]. In particular, the k-linear pseudo-abelian envelope of a one-object groupoid with the morphism group G is the category of projective modules over the group ring k[G].

We denote by P_{k} the collection of the categories of finitely generated projective modules over the group rings k[P_{∗}]. The operadic composition of pure braid groups induces a Cat-operad structure on the colection P_{k}. Analogously, the collection of the categories of finitely generated projective modules over the group rings k[P b_{∗}] forms an operad Pb_{k}. Note that the components of P_{k} and Pb_{k} are exact categories and the operadic compositions are exact functors.

Now we can formulate the linear analog of the proposition 2.3.

Proposition 2.5 Any k-linear abelian braided strict monoidal category is an algebra over the Cat-operad P_{k}. Any k-linear abelian balanced strict monoidal category is an algebra over the Cat-operad Pb_{k}.

Sckech of the proof: In a braided category G the natural automorphisms c(σ) defines the action of the pure braid group P_{n} on the product X_{1} ⊗ ... ⊗ X_{n} of any objects X_{1}, ..., X_{n} ∈ G. When G is abelian we can define the product M ⊗_{P_{n}} (X_{1} ⊗ ... ⊗ X_{n}) for any finitely generated projective k[P_{n}]-module M. Indeed, presenting M as the image of an idempotent p ∈ M_{m}(k[P_{n}]) we set M ⊗_{P_{n}} (X_{1} ⊗ ... ⊗ X_{n}) to be the image of c(p) ∈ End_{G}(X_{1} ⊗ ... ⊗ X_{n}). The setting does not depend (upto an isomorphism) on the choice of the idempotent p. The functors

(X_{1}, ..., X_{n}) ↦ M ⊗_{P_{n}} (X_{1} ⊗ ... ⊗ X_{n})  (6)

will give the action of the operad P_{k} on G. The balanced case can be treated similarly. □

Note that when the tensor product of the category G is bi-exact the functors (6) will be poly-exact.

3 K-theory of braided monoidal categories

3.1 K-theory of Cat-operad action

Remind that for a bi-exact functor A × B → C Waldhausen [19] assigned a homomorphism K_{∗}(A) ⊗ K_{∗}(B) → K_{∗}(C). We will need the multi-version of this which sets a homomorphism K_{∗}(A_{1}) ⊗ ... ⊗ K_{∗}(A_{n}) → K_{∗}(C) in correspondance to a poly-exact functor A_{1} × ... × A_{n} → C which can be acheived by applying the arguments of [19] to poly-exact functors. Applying it to the exact operadic compositions of a Cat-operad (with exact components) we see that its K-theory is naturally an operad of graded abelian groups. Moreover, when such a Cat-operad O acts on an exact category A with exact action functors (5) its K-theory acts on the K-theory K_{∗}(A) of its algebra. In particular, homomorphisms in K-theory induced by the functors (6) give the following.

Proposition 3.2 The operad K_{∗}(kP_{∗}) acts on K-theory of any k-linear abelian braided monoidal category with exact tensor product.

Similarly, the operad K_{∗}(kP b_{∗}) acts on K-theory of any k-linear abelian balanced monoidal category with exact tensor product.

To calculate the K-theory of pure braid groups we use the assembly map h_{∗}(BG, K(k)) → K_{∗}(kG) defined by J.-L. Loday [15]. Here h_{∗}(BG, K(k)) = π_{∗}(BG ∧ K(k)) is the generalized homology associated to the classifying space BG of a group G with coefficients in the K-theory spectrum K(k) of the ground ring k. The Atiyah-Hirzebruch spectral sequence

E_{p,q}(2) = H_{p}(G, K_{q}(k)) ⟹ h_{p+q}(BG, K(k))

gives a way of calculating generalized homology h_{∗}(BG, K(k)) in terms of group homology H_{∗}(G, K_{∗}(k)).

3.3 homology of braid groups

Cohomology of pure braid groups were calculated by V. I. Arnold in [2]. In particular, his result implies that homology of pure braid groups are free of finite rank as abelian groups. Thus the operad structure on the collection {P_{∗}} gives rise to the graded operad on homologies H_{∗}(P_{∗}). Operad composition for H_{∗}(P_{∗}) was described explicitly by F. Cohen [6]. As was noticed by P. Deligne, it follows from F. Cohen description that H_{∗}(P_{∗}) is the operad of Gerstenhaber algebras. It is generated by two elements: a generator of H_{0}(P_{2}) (binary operation of degree zero, the product) and a generator [s_{1}^{2}] of H_{1}(P_{2}) (binary operation of degree one, the bracket). It follows easily from the presentation for the Gerstenhaber operad, it has no non-trivial endomorphism. This fact allow us to prove the following

Lemma 3.4 For any multiplicative spectre E with π_{0}(E) = Z the Atiyah-Hirzebruch spectral sequence for pure braid group P_{n}

E_{p,q}(2) = H_{p}(P_{n}, π_{q}(E)) ⟹ h_{p+q}(BP_{n}, E)

degenerates and gives an isomorphism H_{∗}(P_{n}) ⊗ π_{∗}(E) → h_{∗}(BP_{n}, E).

Proof: We prove that all differentials of Atiyah-Hirzebruch spectral sequences for pure braid groups P_{∗} are zero. Since the cohomologies H_{∗}(P_{∗}) are free abelian, the Universal Coefficients Formula implies that H_{∗}(P_{∗}, π_{∗}(E)) is isomorphic to H_{∗}(P_{∗}) ⊗ π_{∗}(E). Again since H_{∗}(P_{∗}) is free abelian, differentials can be considered as elements of End(H_{∗}(P_{∗})) ⊗ End(π_{∗}(E)). By the naturality of differentials with respect to group homomorphisms between pure braid groups, their components in End(H_{∗}(P_{∗})) should preserve the operad structure on H_{∗}(P_{∗}). As we have seen there are no such maps in non-zero degree.

Finally, the inclusion of the free abelian groups H_{∗}(P_{n}) → H_{∗}(P_{n}) ⊗ π_{0}(E) → gr_{∗}(h_{∗}(BP_{n}, E)) lifts to a map H_{∗}(P_{n}) → h_{∗}(BP_{n}, E). By the multiplicative property of Atiyah-Hirzebruch spectral sequence, its product with the inclusion π_{∗}(E) → h_{∗}(BP_{n}, E) gives the isomorphism H_{∗}(P_{n}) ⊗ π_{∗}(E) → h_{∗}(BP_{n}, E). □

Lemma 3.5 The homology H_{∗}(P b_{∗}) of the pure balanced groups form an operad of Batalin-Vilkovisky algebras.

Proof: Remind that the pure balanced groups are the direct sums of pure braid groups and free abelian groups. Thus, by the Kunneth formula, their homology is a tensor product H_{∗}(P b_{∗}) = H_{∗}(P_{∗}) ⊗ H_{∗}(Z^{∗}). The operadic composition on it is given by the operad structure on P b_{∗}. In particular, for the image of the 1-class Δ = [t] ∈ H_{1}(P b_{1}) under μ : H_{∗}(P b_{2}) ⊗ H_{∗}(P b_{1}) → H_{∗}(P b_{2}) we have

μ([t]) = [s_{1}^{2}] + [t_{1}] + [t_{2}]

which is just the equation (2) for the Batalin-Vilkovisky differential. Note that the class [t] generates H_{1}(P b_{1}) = H_{1}(Z) and together with the bracket [s_{1}^{2}] and the product 1 ∈ H_{0}(P b_{2}) generates H_{∗}(P b_{∗}). □

The composition of the assembly map with an isomorphism from the lemma 3.4 is a homomorphism

H_{∗}(P_{∗}) ⊗ K_{∗}(k) → K_{∗}(k[P_{∗}]),

which is obviously compatible with the operad composition. Analogously, we have a homomorphism of operads

H_{∗}(P b_{∗}) ⊗ K_{∗}(k) → K_{∗}(k[P b_{∗}]),

Combining these with the proposition 3.2 we have proven the following

Theorem 3.6 For any braided monoidal abelian category with exact tensor product its K-theory is a Gerstenhaber algebra. Similarly, for any balanced monoidal abelian category with exact tensor product its K-theory is a Batalin-Vilkovisky algebra. The Gerstenhaber bracket and the BV-differential are trivial in the case of symmetric monoidal abelian category.

3.7 explicit formulas

Here we express explicitly the zero degree components of the bracket { , } : K_{0}(G)⊗K_{0}(G) → K_{1}(G) and the BV-differential Δ : K_{0}(G) → K_{1}(G). Define by [X, x] ∈ K_{1}(G) the image of the class of the object (X, x) ∈ Aut(G) by the map K_{0}(Aut(G)) → K_{1}(G). Then

{X, Y } = [X ⊗ Y, σ_{X,Y}] and Δ(X) = [X, τ_{X}].

3.8 semi-simple braided categories

Let now G be a semi-simple category. Let S be the set of isomorphism classes of simple objects. We identify elements of S with some of their representatives in G. Note that the functor

×_{T ∈S}Hom_{G}(T, ) : G → ×_{T ∈S}End_{G}(T) − mod

is an equivalence between G and the direct product of categories of finitely generated modules over simple algebras. This equivalence gives the isomorphism in K-theory

K_{∗}(G) → ⊕_{T ∈S}K_{∗}(End_{G}(T)).  (7)

The Grothendieck ring K_{0}(E) of a simple algebra E is isomorphic to the ring of integers. The isomorphism is given by the dimension function which is defined for finitely generated modules. The Whitehead group K_{1}(E) coincides with the quotient by the commutant of the group of invertible elements in E. The map K_{1}(E) → E^{∗}/[E^{∗}, E^{∗}] is called the generalized determinant. Combining it with the formulas for the bracket and BV-differential from 3.7 we get the following version for the semi-simple case:

{X, Y } = ⊕_{T ∈S} det(Hom_{G}(T, σ_{X,Y})),  (8)
Δ(X) = ⊕_{T ∈S} det(Hom_{G}(T, τ_{X})),  (9)

where det : GL_{n}(End_{G}(T)) → Aut_{G}(T)/[Aut_{G}(T), Aut_{G}(T)] is the generalized determinant.

In the case when the category G is k-linear over algebraically closed field k, the endomorphism algebras of simple objects coincides with k and we can think of det in the formulas (8),(9) as of ordinary determinant. Moreover, the inverse to the isomorphism (7) which in this case has the form K_{∗}(G) ≃ ⊕_{T ∈S}K_{∗}(k) is given by the multiplication K_{0}(G)⊗K_{∗}(k) → K_{∗}(G). Since this map is obviously a ring homomorphism and the bracket and the BV-differential are always zero on K_{∗}(k), the formulas (8),(9) define the bracket and BV-differential on the whole K_{∗}(G).

4 Finite semi-simple braided rigid categories; a theorem of Anderson-Moore and Vafa

4.1 bilinear form on Grothendieck ring

From now on by a k-linear category we mean an additive category whose Hom-sets are finite dimensional vector spaces over k.

Let G be a k-linear semi-simple catgeory. The Grothendieck ring K_{0}(G) of a semi-simple k-linear monoidal category G has a bilinear form:

([X], [Y]) = dim_{k}Hom_{G}(X, Y), ∀X, Y ∈ G.

Classes of simple objects form an orthogonal basis, proving that ( , ) is symmetric and positively defined. The later condition means that (x, x) = 0 for x ∈ K_{0}(G) if and only if x = 0.

4.2 rigid categories

Remind that the monoidal category G is (left and right) closed if for any pair of objects X, Y ∈ G there exist the terminals among the pairs Z, z : Z ⊗X → Y and W, w : X ⊗ W → Y, correspondingly. The objects of these terminal pairs are called the left and the right inner Hom-objects (Hom^{l}(X, Y) and Hom^{r}(X, Y)) while the morphisms

Hom^{l}(X, Y) ⊗ X → Y, X ⊗ Hom^{r}(X, Y) → Y

are the evaluation maps. In the case of rigid (autonomous) category the natural maps

Hom^{l}(X ⊗ Y, Z) → Hom^{l}(X, Hom^{l}(Y, Z))
Hom^{r}(X ⊗ Y, Z) → Hom^{r}(Y, Hom^{r}(X, Z)),  (10)

defined by the terminality, are isomorphisms. In the case when Y = 1, the objects X^{∗} = Hom^{l}(X, 1) and X^{∘} = Hom^{r}(X, 1) are called left and right duals to X. The inner Hom-objects can be expressed in terms of duals since the natural maps

Y ⊗ X^{∗} → Hom^{l}(X, Y), X^{∘} ⊗ Y → Hom^{r}(X, Y)  (11)

are bijections. We are interested in Hom-objects because of their relations with ordinary Hom-spaces: the natural maps

Hom_{G}(1, Hom^{l}(X, Y)) ← Hom_{G}(X, Y) → Hom_{G}(1, Hom^{r}(X, Y))

are isomorphisms. This, together with (10),(11), gives the following multiplicative properties for the bilinear form on the Grothendieck ring of a semi-simple rigid category:

(xy, z) = (x, zy^{∗}) = (y, x^{∘}z),  (12)

where ( )^{∗}, ( )^{∘} : K_{0}(G) → K_{0}(G) are the (mutually inverse) operators given by the assignments [X]^{∗} = [X^{∗}], [X]^{∘} = [X^{∘}].

The next result is a variant of the theorem due to E.Artin (Theorem 1.5.3 from [12]).

Lemma 4.3 Let G be a k-linear semi-simple autonomous category with finitely many simple objects. Then its rational Grothendieck algebra K_{0}(G)⊗ Q is semi-simple.

Proof: We prove that any ideal of K_{0}(G)⊗Q is a direct summand (as a subalgebra) which, by induction, means that the Grothendieck algebra is a direct sum of simple subalgebras. Let I ⊂ K_{0}(G) ⊗ Q be a two-sided ideal. Then, by the formulas 12, the compliment

I^{⊥} = {x ∈ K_{0}(G) ⊗ Q : (I, x) = 0}

is also a two-sided ideal. By the positive definiteness of the form ( , ), the intersection I ∩ I^{⊥} is trivial and the inclusion I ⊕ I^{⊥} → K_{0}(G) ⊗ Q is an isomorphism. Finally, the products II^{⊥}, I^{⊥}I are trivial as subspaces of I ∩ I^{⊥} thus turning the additive decomposition K_{0}(G) ⊗ Q = I ⊕ I^{⊥} in to a decomposition of algebras. □

Remark 4.4

Using the first equation of (12) twice and the symmetry of the form ( , ) we can see that

(x, x) = (1, xx^{∗}) = (xx^{∗}, 1) = (x, x^{∗∗})

for any x. For a simple x, it will give the equality x = x^{∗∗}. In particular, left and right duals in semi-simple monoidal category coincide. So, in Lemma 4.3, we can assume only one-side rigidness instead of two-side.

4.5 eigenvalues of double braiding and twist

Now we can prove a generalization of a theorem of Anderson-Moore and Vafa [1, 18] on the rationality of h-phases of RCFT which in categorical terms sais that the twist in modular category has finite order. Remind that modular category is a special kind of balanced monoidal semi-simple category with finitely many simple objects.

Theorem 4.6 Let G be a k-linear semi-simple rigid category with finitely many simple objects. Suppose that

1. G is braided with the braiding c, then for any X, Y, Z ∈ G the determinant det(Hom_{G}(Z, c_{X,Y} c_{Y,X}) is a root of unity; 2.G is balanced with the twist τ, then for any X ∈ G the determinant det(Hom_{G}(Z, τ_{X}) is a root of unity.

Proof: By the lemma 4.3 we know that for such G the rational Grothendieck ring K_{0}(G) ⊗ Q is a semi-simple commutative Q-algebra. In particular, any its derivation is zero. Now the formula (1) shows that the Gertenhaber bracket is a bi-derivation. Thus the value of the bracket is torsion: {K_{0}(G), K_{0}(G)} ⊂ K_{1}(G)_{tors}, which together with (8) gives the first statement. Since the bracket is zero on K_{0}(G)⊗Q the formula (2) for the BV-differencial becomes a derivation condition and implies the triviality of Δ on K_{0}(G)⊗Q. So the image of the BV-differential is also torsion: Δ(K_{0}(G)) ⊂ K_{1}(G)_{tors}, which combined with (9) gives the second part of the theorem. □

Corollary 4.7 Let G be a semi-simple rigid balanced category with finitely many simple objects. Let c and τ be the commutativity constraint and the twist correspondingly. Then there is an integer n such that (c_{Y,X}c_{X,Y})^{n} = I_{X,Y} and τ_{X}^{n} = I_{X} for all X, Y ∈ G.

Proof: Since τ_{X} = det(Hom_{G}(X, τ_{X}) for a simple X ∈ G, the part 2 of the theorem 4.6 implies that τ_{X}^{n} = I_{X} for all simple X and, thus for all X. This implies (c_{Y,X}c_{X,Y})^{n} = I_{X,Y}, since c_{Y,X}c_{X,Y} = (τ_{X} ⊗τ_{Y})τ_{X⊗Y}^{−1} and τ_{X} ⊗τ_{Y} commutes with tau_{X⊗Y}. □

The next example shows that the finiteness condition for the number of simple objects is important.

Example 4.8 Grothendieck ring of quantum sl(2).

Let x denote the class of fundamental (2-dimensional) representation X of sl_{q}(2). Then the Grothendieck ring R(sl_{q}(2)) of the catgeory of finite dimensional representations of sl_{q}(2) is a polynomial ring Z[x] in it. Since the quantum R-matrix c_{X,X} has two eigenvalues −1 and q and the the eigenspace of the first is the trivial representation we have the formula for the bracket

{x, x} = (x^{2} − 1) ⊗ q^{2}.

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