International Conference on

Secondary calculus and cohomological Physics Moscow, August 1997

ON SOME HOCHSCHILD COHOMOLOGY CLASSES OF FUSION ALGEBRAS

A. A. Davydov

Abstract. The obstructions for an arbitrary fusion algebra to be a fusion algebra of some semisimple monoidal category are constructed. Those obstructions lie in groups which are closely related to the Hochschild cohomology of fusion algebras with coefficients in the K-theory of the ground (algebraically closed) field. The special attention is devoted to the case of fusion algebra of invariants of finite group action on the group ring of abelian group.

Contents

1 Introduction 1

2 Fusion rings 2

3 Semisimple monoidal categories 2

4 A_{n}-spaces 5

5 Cohomological obstruction for monoidality 9

6 Calculation of the first obstruction 11

1. Introduction

Recent activity in the theory of fusion rings (a special kind of semiring) was inspired by the development of quantum field theory [13, 8, 14], although this notion appeared before in different branches of mathematics, for example in the representation theory (semiring of irreducible representations). Most fusion rings considered so far can be represented as fusion rings of simple objects of some semisimple monoidal category [3] (so-called monoidal fusion rings). As we shall see, not all fusion rings admit such a representation.

The article is devoted to the construction of obstructions for an arbitrary fusion ring to be the monoidal. These obstrucions are elements of the Hochschild cohomology of the fusion ring with coefficients in the algebraic K-theory of the ground field. The construction is based on the notion of A_{n}-space (homotopy associative space) [10, 6, 7, 2]. The obstruction to monoidality is a special case of the obstruction to the extension of A_{n}-structure to A_{n+1}.

The first obstruction to monoidality, which is an element of the fourth Hochschild cohomology can be expressed explicitly by means of structural constants of a given fusion ring (section 5). In section 6 the triviality of the first obstruction in case of two element fusion ring with identity is analysed.

Acknowledgement

The author would like to thank J. Stasheff for stimulating attention, fruitful discussions and corrections.

2. Fusion rings

Bt definition, a fusion ring is a set S with a collection of non-negative integers {m_{x_{1},x_{2}}^{x}, x, x_{1}, x_{2} ∈ S} (structural constants) which satisfy the (associativity) condititon

m_{x_{1},x_{2},x_{3}}^{x} = ∑_{t∈S} m_{x_{1},t}^{x}m_{x_{2},x_{3}}^{t} = ∑_{s∈S} m_{x_{1},x_{2}}^{s}m_{s,x_{3}}^{x}, ∀x, x_{1}, x_{2}, x_{3} ∈ S.

An element e of the fusion algebra S is an identity if m_{t,e}^{s} = m_{e,t}^{s} = δ_{s,t} for all s, t ∈ S. A morphism of a fusion ring S to a fusion ring S′ is a collection {n_{t}^{s}, s ∈ S, t ∈ S′} of

non-negative integers which satisfy the following condition:

∑_{s∈S} m_{s_{1},s_{2}}^{s}n_{s}^{t} = ∑_{t_{1},t_{2}∈S′} m′_{t_{1},t_{2}}^{t}n_{s_{1}}^{t_{1}}n_{s_{2}}^{t_{2}}

for any s_{1}, s_{2} ∈ S and t ∈ S′.

The enveloping ring A(S) of a fusion algebra S is the free Z-module with the basis {[s], s ∈ S} labeled by the elements of S and with the multiplications [i][j] = ∑_{s∈S} m_{i,j}^{s}[s]. A morphism of fusion rings defines a homomorphism of their enveloping rings f : A(S) → A(S′) where f([s]) = ∑_{t∈S′} n_{s}^{t}[t].

3. Semisimple monoidal categories

This section is devoted to the most important case of fusion algebras, namely fusion algebras corresponding to semisimple monoidal categories (monoidal fusion algebras).

Let G be a semisimple monoidal k-linear category over the field k with the set S of isomorphism classes of simple objects. The collection of dimensions m_{y,z}^{x} = dim_{k}Hom_{G}(X, Y ⊗ Z) form a fusion ring structure on the set S. Here X, Y and Z are some representatives of the classes x, y, z ∈ S. This fusion ring structure (together with the map from S to the set of simple k-algebras, x ↦ End_{G}(X)) contains all the information about the tensor product in the category G. But as we shall see, it is not sufficient to reconstruct associativity constraint ϕ of the category. To define this constraint in terms of simple objects it is necessary to fix representative of any class from S. If we do this we can construct a semisimple monoidal category in which isomorphic simple objects coincides and which is equivalent to the category G. We call such a category a model M(S) of the semisimple monoidal category G with the set S of simple objects. Objects of the model are maps from the set S to the set of finite dimensional vector spaces with only finitely many nonzero values (or formal finite sums ⊕_{x∈S}V_{x}x of vector spaces). The morphism space Hom(⊕_{x∈S}V_{x}x, ⊕_{y∈S}U_{y}y) coincides with ⊕_{x∈S}Hom_{k}(V_{x}, U_{x}) ⊗ End_{G}(X). By definition, the tensor product of two simple objects x ⊗ y is the sum ⊕_{z∈S}H_{x,y}^{z}, where H_{x,y}^{z} = Hom_{G}(Z, X ⊗ Y). The associativity constraint for the model can be given by the collection of isomorphisms of vector spaces

Φ_{x_{1},x_{2},x_{3}}^{x} : ⊕_{u∈S}H_{x_{1},u}^{x} ⊗ H_{x_{2},x_{3}}^{u} → ⊕_{v∈S}H_{x_{1},x_{2}}^{v} ⊗ H_{v,x_{3}}^{x},  (1)

which are defined by the associativity constraint ϕ of the category G

Φ_{x_{1},x_{2},x_{3}}^{x} =
Hom_{G}(X, ϕ_{X_{1},X_{2},X_{3}}) : Hom_{G}(X, X_{1} ⊗ (X_{2} ⊗ X_{3}))
→ Hom_{G}(X, (X_{1} ⊗ X_{2}) ⊗ X_{3}).

The pentagon axiom for the constraint ϕ consists of commutativity of the pentagon diagram [3] for any x, x_{2}, x_{3}, x_{3}, x_{4} ∈ S. This condition can be written in a form of the equation:

(⊕_{b∈S}(Φ_{b,x_{3},x_{4}}^{x})_{23})(⊕_{a∈S}(Φ_{x_{1},x_{2},a}^{t})_{12}) =
(⊕_{f∈S}(Φ_{x_{1},x_{2},x_{3}}^{f})_{12})(⊕_{d∈S}(Φ_{x_{1},d,x_{4}}^{t})_{13})(⊕_{c∈S}(Φ_{x_{2},x_{3},x_{4}}^{c})_{23}),

which is denoted by A_{x_{1},x_{2},x_{3},x_{4}}^{x}.

Below, in case of simplicity we consider k-linear semisimple categories for which End(X) = k for any simple object X. For example, it is true for algebraically closed field k.

If we start from arbitrary fusion ring S we also can construct the semisimple category G(S) with a tensor product whose simple objects are parametrized by the elements of S. Since the vector spaces

⊕_{u∈S} H_{x_{1},u}^{x} ⊗ H_{x_{2},x_{3}}^{u} and ⊕_{v∈S} H_{x_{1},x_{2}}^{v} ⊗ H_{v,x_{3}}^{x}  (2)

have the same dimension m_{x_{1},x_{2},x_{3}}^{x}, they are isomorphic, i.e. the tensor product is quasiassociative. It is associative if it is possible to choose the isomorphisms between vector spaces (2) satisfying the pentagon axiom.

In case of the fusion ring consisting of one element x with the fusion rule x ∗ x = nx (the structural constant m_{x,x}^{x} equals n) the quasiassociative structure is an automorphism Φ of vector space H ⊗ H. Here H = H_{x,x}^{x} is an n-dimensional vector space. The pentagon axiom for this structure is equivalent to the following (pentagon) equation

Φ_{12}Φ_{13}Φ_{23} = Φ_{23}Φ_{12}.  (3)

Here Φ_{ij} denotes the automorphism of H^{⊗3} acting on i-th and j-th components.

With any structure of Hopf algebra on the vector space H we can associate a solution of the equation (3):

Φ(g ⊗ h) = ∑_{(g)} g_{(0)} ⊗ h_{(0)} ∀g, h ∈ H.

General solutions of the equation (3) are similar to these ones. The proof of this fact will be published elsewhere.

The pair consisting of the identity e and the element x with fusion rule x∗x = ne provides an example of a fusion ring whose category does not admit a monoidal structure (there is no quasiassociative structure satisfying the pentagon axiom). To simplify the verification of this fact let us prove the following lemma.

Lemma 1 Let S be a fusion ring with identity e and Φ be an associativity constraint for a corresponding semisimple k-linear category. Then there are identifications ρ_{x} : H_{x,e}^{x} → k and λ_{x} : H_{e,x}^{x} → k (for any x) such that Φ_{x,y,e}^{z}, Φ_{x,e,y}^{z} and Φ_{e,x,y}^{z} coincide with the compositions:

(I ⊗ ρ_{z}^{−1})(I ⊗ ρ_{y}) : H_{x,y}^{z} ⊗ H_{y,e}^{y} → H_{x,y}^{z} → H_{x,y}^{z} ⊗ H_{z,e}^{z},  (4)
(ρ_{x}^{−1} ⊗ I)(I ⊗ λ_{y}) : H_{x,y}^{z} ⊗ H_{e,y}^{y} → H_{x,y}^{z} → H_{x,e}^{x} ⊗ H_{z,e}^{z},  (5)
(λ_{x}^{−1} ⊗ I)(λ_{z} ⊗ I) : H_{e,z}^{z} ⊗ H_{x,y}^{z} → H_{x,y}^{z} → H_{e,x}^{x} ⊗ H_{z,e}^{z}.  (6)

Proof:

It follows from the equations A_{x,e,e,e}^{x}, A_{e,x,e,e}^{x}, A_{e,e,x,e}^{x} and A_{e,e,e,x}^{x} that one can choose the isomorphisms ρ_{x} : H_{x,e}^{x} → k and λ_{x} : H_{e,x}^{x} → k such that Φ_{x,e,e}^{x}, Φ_{e,x,e}^{x} and Φ_{e,e,x}^{x} have the form 5,4 and 6 respectively.

Conversely, equations which involve three identity elements imply the statement of the lemma. □

Let us return to the fusion ring {e, x} with fusion rule x ∗ x = ne. Identifying H_{x,e}^{x} and H_{e,x}^{x} with k we can see that unique nontrivial component ϕ_{x,x,x}^{x} : H_{x,e}^{x} ⊗ H_{x,x}^{e} → H_{x,x}^{e} ⊗ H_{e,x}^{x} of a monoidal structure for the category G(S) (if it exists) corresponds to automorphism Φ of n-dimensional vector space H = H_{x,x}^{e}. The pentagon condition A_{x,x,x,x}^{e} is equivalent to the equation Φ^{2}⊗I = t, where I is the identity automorphism of the space H and t : H^{⊗2} → H^{⊗2} is the permutation of tensor factors (the isomorphisms (Φ_{x,x,e}^{e})_{12} and (Φ_{e,x,x}^{e})_{23} are identical by the lemma, (Φ_{x,x,x}^{x})_{23} and (Φ_{x,x,x}^{x})_{12} correspond to I ⊗ A and A ⊗ I respectively and (Φ_{x,e,x}^{e})_{13} to t). It is easy to see that this equation has solutions iff n = 1.

This example motivates the following definition. A fusion ring is called monoidal if it coincides with the fusion ring of simple objects of some semisimple monoidal category.

Any monoidal k-linear functor [3] F : G → G′ between semisimple monoidal k-linear categories provides the morphism of fusion algebras S and S′ of simple objects. This morphism is given by the collection of dimensions n_{y}^{x} = dim_{k}Hom_{G′}(X, F (Y)). Here X ∈ G′ and Y ∈ G are some representatives of the classes x ∈ S′,y ∈ S. This morphism structure contains all information about the functor F (without its monoidal structure). Let us denote by H_{y}^{x} the n_{y}^{x}-dimensional vector space Hom_{G′}(X, F (Y)).

The monoidal constraint for this functor can be given by the collection of isomorphisms of vector spaces

Ψ_{y,z}^{x} : ⊕_{u∈S}H_{y,z}^{u} ⊗ H_{u}^{x} → ⊕_{v,w∈S′}H_{y}^{v} ⊗ H_{z}^{w} ⊗ H_{v,w}^{x},

which are defined by the monoidal constraint ψ of the functor F:

Ψ_{y,z}^{x} = Hom_{G′}(X, ϕ_{Y,Z}) : Hom_{G′}(X, F (Y ⊗ Z)) → Hom_{G′}(X, F (Y) ⊗ F (Z)).

The compatibility axiom for the constraint ϕ states the commutativity of hexagon diagrams [3] for any x ∈ S′, y, z, w ∈ S. This condition can be written in a form of the equation:

(⊕_{t_{1},t_{2},t_{3}∈S′}(Φ_{t_{1},t_{2},t_{3}}^{t})_{13})(⊕_{y∈S′}(Ψ_{s_{1},s_{2}}^{y})_{12})(⊕_{x∈S}t_{12}(Ψ_{s,x}^{t})_{23})t_{12}) =
t_{34}(⊕_{w∈S′}(Ψ_{s_{1},s_{2}}^{w})_{12})(⊕_{z∈S}(Ψ_{z,s_{3}}^{t})_{23})(⊕_{s∈S}(Φ_{s_{1},s_{2},s_{3}}^{s})_{12}).

The condition of an existence of a monoidal functor between semisimple monoidal categories M(S), M(S′) which induce a given morphism of fusion rings S, S′ is called monoidality of the morphism (with respect to the given monoidal structures on the categories M(S), M(S′)).

The simplest example of a nonmonoidal morphism of fusion rings is provided by the identity map of the one-element fusion ring S = {x, x ∗ x = nx} with respect to monoidal structures on the category M(S) defined by two non-isomorphic n-dimensional Hopf algebras.

4. A_{n}-spaces

Let us denote by T^{k} the set of planar trees with k ends. This set is a disjoint union ⨆_{i=0,...,k−2} T_{i}^{k} of subsets T_{i}^{k}, which consists of trees with k − i − 1 vertices. Junction of two neighboring vertices defines the correspondence R^{k} ⊂ T^{k} × T^{k}. Namely,

R^{k} = {(t, t′)} ∈ T^{k} × T^{k},

where t′ is a result of the contraction of an internal edge of t.

This correspondence also decomposes into a disjoint union ⨆_{i=0,...,k−3} R_{i}^{k}, where R_{i}^{k} is contained in T_{i}^{k} × T_{i+1}^{k}. The unique element of T_{k−2}^{k} is denoted by t(k).

This data permits us to define cell complexes BT^{k} (Stasheff’s complexes K_{k}, [10]): elements of the set T_{i}^{k} parametrizes the i-dimensional cells of BT^{k}, which are glued by means of the correspondence R.

The operation of gluing trees (roots of the trees t_{1}, ..., t_{k} are glued to the ends of the tree t ∈ T^{k}) defines the maps

c^{k,l_{1},...,l_{k}} : T_{i}^{k} × T_{i_{1}}^{l_{1}} × ... × T_{i_{k}}^{l_{k}} → T_{i_{1}+...+l_{k}+i}^{l_{1}+...+l_{k}}.

These maps induce a (non-symmetric) operad structure [6]:

BT^{k} × BT^{l_{1}} × ... × BT^{l_{k}} → BT^{l_{1}+...+l_{k}}.

Let us denote by ∂_{k}^{i}(t) ∈ T^{k+1} the result of gluing the root of the tree t(2) to i-th end of the tree t for t ∈ T^{k} and i = 1, ..., k. For i = 0, k + 1 the expression ∂_{k}^{i}(t) denotes the result of gluing the root of the tree t to the second and first ends of the tree t(2) respectively.

Let us shoose the orientations of the complexes BT^{k} such that the maps ∂_{k}^{1} : BT^{k} → BT^{k+1} preserve the orientations.

It should be noted that for any space X the collection of spaces Hom(X^{k}, X) also forms an operad.

An A_{n}-space structure [10, 2, 6] is a collection of continuous maps BT^{k} → Hom(X^{k}, X) for k ≤ n that are compatible with compositions.

It is easy to see that the cells which correspond to the trees from the image of the composition map

c^{k,l_{1},...,l_{k}}(T_{i}^{k} × T_{i_{1}}^{l_{1}} × ... × T_{i_{k}}^{l_{k}})

are contractible in Hom(X^{l_{1}+...+l_{k}}, X) if i_{p}, i_{q} > 0 for some p, q.

For example, an A_{3}-space structure on X consists of a product μ : X × X → X (the image of pt = BT^{2} → Hom(X^{2}, X)) and a homotopy between μ(I ∧ μ) and μ(μ ∧ I) (the image of I = BT^{3} → Hom(X^{3}, X)).

An A_{n+i}-structure on the space X for which the restriction on the k-cells for k ≤ n coincides with the given A_{n}-structure is called an extension. Two A_{n}-extensions of a given A_{n−1}-structure on the space X define a map X^{n} → Ω^{n−2}X. Indeed, these A_{n}-structures differ only by the maps I^{n−2} = BT^{n} → Hom(X^{n}, X) of the n − 2-cell. In particular, these maps coincide on the boundary S^{n−3} = I^{n−2}. Hence we can glue them together along the boundary and create a map S^{n−2} = I^{n−2} ⨆_{∂I^{n−2}} I^{n−2} → Hom(X^{n}, X) or X^{n} → Ω^{n−2}X.

Now let us define the cohomology of A_{3}-spaces with coefficients in some bimodule.

An A_{3}-bimodule over an A_{3}-space X is a space Y together with the continuous maps

ν : Y ∧ X → Y, υ : X ∧ Y → Y

and homotopies

ν(I ∧ μ) → ν(ν ∧ I), υ(I ∧ ν) → ν(υ ∧ I), υ(I ∧ υ) → υ(μ ∧ I).

For example, the space of maps to the A_{3}-space is a A_{3}-bimodule over this space.

Denote by [X, Y] the set of homotopy classes of continuous maps from X to Y. The topological cobar complex of a A_{3}-space X with coefficients in a A_{3}-bimodule space Y is a cosimplicial complex of sets

C_{∗}(X, Y), C_{n}(X, Y) = [X^{n}, Y]

with the coface maps ∂_{n}^{i} : C_{n−1}(X, Y) → C_{n}(X, Y) defined as follows:

∂_{n}^{i}(f) = \{ \begin{matrix}υ(I × f),  i = 0 \\ f(I × ... × μ × ... × I),  1 ≤ i ≤ n \\ ν(f × I),  i = n + 1 \end{matrix}

If Y is a loop space (with a loop structure which is not connected with the given A_{3}-structure) then C_{·}(X, Y) is a complex of groups and these groups are abelian if Y is a double loop space. In the second case the cohomology of the cochain complex associated with C_{∗}(X, Y) is called the topological cohomology (H^{∗}(X, Y)) of the A_{3}-space X with coefficient in the A_{3}-bimodule Y.

Let us note that an A_{n}-space structure on X (and the operad structures on BT^{∗} and Hom(X^{∗}, X)) allows to define maps BT_{≤n−2}^{l} → Hom(X^{l}, X) for any l. Here BT_{≤n−2}^{l} denotes the union of i-dimensional cells in BT^{l} with i ≤ n − 2.

For example, the map S^{n−2} = ∂BT^{n+1} = BT_{≤n−1}^{n+1} → Hom(X^{n+1}, X) defines an element in Hom(X^{n+1}, Ω^{n−2}X).

Theorem 1 Let X be an A_{n}-space for n ≥ 3 (and independently a loop space if n = 3). Then the class in [X^{n+1}, Ω^{n−2}X] of the map BT_{≤n−2}^{n+1} → Hom(X^{n+1}, X) is a cocycle. Its class α(X) in cohomology H^{n+1}(X, Ω^{n−2}X) does not depend on the choice of the n−2-dimensional component of an A_{n}-structure on the space X and it is trivial iff there is a n−2-deformation of the given A_{n}-structure which can be extended to an A_{n+1}-structure.

Proof:

Let us consider the map BT_{≤n−2}^{n+2} → Hom(X^{n+2}, X). The complex BT_{≤n−2}^{n+2} is a union of n − 2-spheres (boundaries of n − 1-cells), which correspond to planar trees with n + 2 ends and two vertices. Hence this map provides an equation on the classes of these spheres in [X^{n+2}, Ω^{n−2}X]. The cells labeled by trees, whose vertices have valences (the number of incident, e.g., incoming and outcoming edges) greater then three, are contractible in Hom(X^{n+2}, X). So (potentially) non-trivial components correspond to the trees that can be presented as glueing of trees t(n + 1) and t(2), e.g., trees of the form ∂_{n+1}^{i}(t) for i = 0, ..., n + 2. Their classes in [X^{n+2}, Ω^{n−1}X] are equal to ∂_{n+1}^{i}(α(X)) respectively. Hence the map BT_{≤n−2}^{n+2} → Hom(X^{n+2}, X) defines the equation ∂(α(X)) = 0 in [X^{n+2}, Ω^{n−2}X] which means that α(X) is a n + 1-cocycle.

Another choice of the n-component of an A_{n}-structure provides the cocycle which differs from the previous one by coboundary ∂(f) of the map f which is defined by two A_{n}-extensions of the given A_{n−1}-structure.

Triviality of the cohomology class of the cocycle α means that we can choose the ncomponent of A_{n}-structure such that it can be extendable to an A_{n+1}-structure. □

To define A_{n}-maps between A_{n}-spaces we need the following notion. A tricolored tree is a planar tree which vertexes are labeled by one of three ordered indices 0 < ϵ < 1. The index of vertix which lies over are not less then index of given vertix; moreover the index of the vertix which lies over ϵ-labeled vertix is 1.

As in the non-colored case the set CT^{k} of tricolored trees with k ends can be presented as the disjoint union ⨆_{i=0,...,k−1} CT_{i}^{k} of subsets, which consist of trees with k − i − 1 + n_{ϵ} vertices. Here n_{ϵ} is the number of ϵ-labeled vertices.

We also define the color version CR of the correspondence R by the condition that (t, t′) ∈ CR if the tree t′ can be obtained from the tree t by contraction of an edge between equally labeled vertices or between vertices one of which is labeled by ϵ.

This allows us to define cell complexes BCT^{k} whose i-cells are parametrized by elements of CT_{i}^{k} and are glued by means of the correspondence CR.

Two collection of inclusions T_{i}^{k} → CT_{i}^{k} which label all vertices by 0 and 1 respectively provides two collections of compositions:

u^{k,l_{1},...,l_{k}} : T_{i}^{k} × CT_{i_{1}}^{l_{1}} × ... × CT_{i_{k}}^{l_{k}} → CT_{i_{1}+...+l_{k}+i}^{l_{1}+...+l_{k}},

and

d^{k,l_{1},...,l_{k}} : CT_{i}^{k} × T_{i_{1}}^{l_{1}} × ... × T_{i_{k}}^{l_{k}} → CT_{i_{1}+...+l_{k}+i}^{l_{1}+...+l_{k}},

which satisfy the usual operad axioms and define two collections of cell complex maps:

u^{k,l_{1},...,l_{k}} : BT^{k} × BCT^{l_{1}} × ... × BCT^{l_{k}} → BCT^{l_{1}+...+l_{k}},

and

d^{k,l_{1},...,l_{k}} : BCT^{k} × BT^{l_{1}} × ... × BT^{l_{k}} → BCT^{l_{1}+...+l_{k}}.

An A_{n}-map between A_{n}-spaces X and Y is a collection of continuous maps BCT^{k} → Hom(X^{k}, Y) for k ≤ n − 1, which are compatible with the A_{n}-structures.

For example, an A_{3}-map between A_{2}-spaces X, Y is a continuous map f : X → Y, which is the image of pt = BCT^{1} → Hom(X, Y) and a homotopy α : fμ → μ(f × f), which is defined by the map I = BCT^{2} → Hom(X^{2}, Y) (fμ is the image of the 0-cell which corresponds to the 1-labeled tree t(2), μ(f × f) is the image of the 0-cell, corresponding to the 0-labeled tree t(2) and homotopy α is the image of the 1-cell, corresponding to the ϵ-labeled tree t(2)). We also call this structure an A_{3}-structure on the map f. Similarly, A_{n}-structure on the map f : X → Y between A_{n}-spaces is an A_{n}-map structure such that the image of pt = BCT^{1} → Hom(X, Y) coincides with f.

Note that A_{3} map between A_{3}-spaces X, Y allows to define the structure of A_{3}-bimodule over an A_{3}-space X on a space Y. In particular, if Y is a loop, the space we have defined is the topological cohomology space H^{∗}(X, Ω^{∗}Y).

Theorem 2 Let (X, μ_{X}), (Y, μ_{Y}) be A_{n}-spaces (and Y also is independently a loop spaces if n = 3) and let f : X → Y be a A_{n}-map. Then the images f_{∗}(α(X)), f^{∗}(α(Y)) of their canonical classes in H^{n+1}(X, Ω^{n−2}Y) coincide.

If the A_{n}-structures on X, Y can be extended to an A_{n+1}-structure, then the class α(f) in the cohomology H^{n}(X, Ω^{n−2}Y) is defined. It does not depend of the choice of n-dimesional component of A_{n}-structure on f and it is trivial iff this structure can be extended to an A_{n+1}-structure.

Proof:

Let us note that an A_{n}-map structure on f (and composition operations on BCT^{∗} and Hom(X^{∗}, Y)) allows us to define maps BCT_{≤n−2}^{l} → Hom(X^{l}, Y) for any l. Here BCT_{≤n−2}^{l} denotes the union of i-dimensional cells in BCT^{l} with i ≤ n − 2.

Let us consider the element α(f) in [X^{n}, Ω^{n−2}Y] which corresponds to the map of n − 2sphere ∂BCT^{n} = BCT_{≤n−2}^{n} → Hom(X^{n}, Y). This element satisfies the equation which is defined by the map BCT_{≤n−2}^{n+1} → Hom(X^{n+1}, Y). The cell complex BCT_{≤n−2}^{n+1} is a union of boundaries of n − 1-cells, which correspond to tricolored trees with n + 1 ends and n_{ϵ} + 1 vertices (it means that all except one vertices of the tree are ϵ-labeled). Nondegenerated components correspond to the trees with n_{ϵ} ≤ 1. The case n_{ϵ} = 0 consists of two n − 2spheres (two tricolored trees of the form t(n + 1): one is 1-labeled and another 0-labeled) whose classes in [X^{n+1}, Ω^{n−2}Y] are f_{∗}(α(X)) and f^{∗}(α(Y)) respectively. In case of n_{ϵ} = 1 we have a trees of the form ∂_{n}^{i}(t(n)) for i = 0, ..., n+1. Nondegenerate colored trees correspond to the case where the tree t(n) is ϵ-labeled. Their classes in [X^{n+2}, Ω^{n}X] are equal to ∂_{n+1}^{i}(α(X)) respectively. Hence the map BT_{≤n−2}^{n+2} → Hom(X^{n+2}, X) defines the equation ∂(α(X)) = 0 in [X^{n+2}, Ω^{n−2}X]. □

Let us note that for an A_{4}-space (X, μ_{X}) and A_{4}-bimodule (Y, μ_{Y}) the collection of abelian groups H^{p}(X, Ω^{q}Y) is equipped with a differential d^{p,q} : H^{p}(X, Ω^{q}Y) → H^{p+2}(X, Ω^{q+1}Y). Indeed, let χ be a representative for some cocycle x ∈ C^{p}(X, Ω^{q}Y). We can choose the homotopy τ between the map ∂(χ) : X^{p+1} → Ω^{q}Y and the trivial map 1_{p+1} (the map into a fixed point). Since ∂(1_{p+1}) and ∂^{2}(χ) can be canonically identified with the trivial map 1_{p+2} : X^{p+2} → Ω^{q}Y the homotopy ∂(τ) can be regarded as an autohomotopy of 1_{p+2} or as a map K^{p+2} → Ω^{q+1}Y. It is easy to see that the class of this map in [X^{p+2}, Ω^{q+1}Y] is a cocycle and the class of this cocycle in H^{p+2}(X, Ω^{q+1}Y) does not depend of the choice of χ and τ and defines a homomorphism d : H^{p}(X, Ω^{q}Y) → H^{p+2}(X, Ω^{q+1}Y).

Direct checking shows that the canonical class of an A_{n}-space (A_{n}-map) lies in the kernel of the differential d and the image of the canonical class in E_{3}^{p,q} = ker(d^{p,q})/im(d^{p−2,q−1}) does not depend on the choice of not only the n-component of the A_{n}-structure (theorems 1,2) by also on the choice of the n − 1-component.

Moreover, for an A_{n}-space (X, μ_{X}) and A_{n}-bimodule (Y, μ_{Y}) there is a ”restricted spectral sequence” consisting of abelian groups E_{k}^{p,q} (k ≤ n) and differentials

d_{k}^{p,q} : E_{k}^{p,q} → E_{k}^{p+k,q+k−1}

(k ≤ n−1) such that E_{k}^{p,q} = ker(d_{k−1}^{p,q})/im(d_{k−1}^{p−k,q−k+1}). This restricted spectral sequence can be constructed using filtrations on the groups [X^{∗}, Y] which are defined as follows.

An A_{k}-cocycle of an A_{n}-space X with coefficients in an A_{n}-bimodule Y (k ≤ n) is the class in [X^{m}, Y] of a map f : X^{m} → Y with the homotopies τ_{1} : ∂(f) → 1, τ_{i} : ∂(τ_{i−1}) → 1 for 2 ≤ i ≤ k.

It is useful for applications to change the group of values of canonical classes. Since an A_{3}-space (bimodule) structure on a loop space induces a graded ring (bimodule) structure on its homotopy groups, the natural map from C_{∗}(X, Y) to the Hochschild complex C_{∗}(π_{·}(X), π_{·}(Y)) of the ring π_{∗}(X) with coefficients in the bimodule π_{∗}(Y) induces the homomorphism of cohomology

H^{∗}(X, Y) ⟶ HH_{∗}(π_{∗}(X), π_{∗}(Y)).

So we have the class α(X) ∈ HH_{n+3}(π_{0}(X), π_{n}(X)) for any A_{n}-space X.

5. Cohomological obstruction for monoidality

In [9] Quillen had associated to an abelian (exact) category A a topological space BQA such that exact functors between categories define continuous maps between the corresponding spaces and isomorphisms of functors define homotopies between the corresponding maps. In other words, Quillen’s space is a 2-functor from the 2-category of abelian (exact) categories (with isomorphisms of functors as 2-morphisms) to the 2-category of topological spaces. Waldhausen [15] proved that the 2-functor K = ΩBQ is permutative (in some sense) with the natural product operations. Namely, he constructed a continuous map K(A)∧K(B) → K(C) for any biexact functor A×B → C. He also proved that K(A) is an infinite loop space for any abelian category A. The homotopy groups K_{∗}(A) = π_{∗}(A) of the Waldhausen space K(A) are called the algebraic K-theory of the category A. The 2-categorical nature of the functor K implies that the Waldhausen space K(G) is an A_{∞}-space for any abelian monoidal category G with respect to the product corresponding to the monoidal product in G. In particular, the space K(M_{k}(S)) is A_{∞}-space for any monoidal (over the field k) fusion algebra S.

The structure of the space K(M_{k}(S)) motivates the following construction. If S is a fusion ring and K is a homotopy associative space (which is also a loop space), then we can construct a new space K(S) = ⋁_{s∈S} K with the product defined by the fusion rule of S. Namely, the component of μ_{K(S)} from the i and j-labeled components of K(S) to the klabeled has the form μ_{K(S)}_{i,j}^{k} = m_{i,j}^{k}μ_{K} where the multiplication by the non-negative integer m_{i,j}^{k} is defined by means of internal loop-space structure.

We say that the fusion ring S is of A_{n}-type with respect to an A_{∞}-space K if the natural A_{3}-structure on K(S) is extendeble to an A_{n}-structure. The morphism of fusion rings S, S′ is of A_{n}-type with respect to A_{∞}-space K if the corresponding map K(S) → K(S′) has an A_{n}-structure.

The canonical class α(S) of the fusion ring S with respect to the A_{∞}-space K is a class of the homotopy associative space K(S) in the Hochschild cohomology HH^{∗}(A(S), A(S) ⊗ π_{∗}(K)). The following theorem is a direct corollary of the definitions.

Theorem 3 For the monoidal (over the field k) fusion ring S the canonical cohomology class α_{∗}(S) ∈ HH^{∗}(A(S), A(S) ⊗ K_{∗}(k)) is trivial.

Properties of canonical classes of A_{n}-spaces proved in the previous section (theorem 2) imply the following theorem.

Theorem 4 Let the fusion rings S, S′ be of A_{n}-type with respect to A_{∞}-space K. Let f : A(S) → A(S′) be the homomorphism of enveloping rings which corresponds to the morphism of fusion rings S and S′ of A_{n}-type. Then the canonical classes of these fusion rings satisfy the following condition:

f_{∗}(α(X)) = f^{∗}(α(Y)) ∈ HH^{n+3}(A(S), A(S′) ⊗ π_{n}(K)).

If the fusion rings S, S′ are of A_{n+1}-type then the class α(f) in cohomology HH^{n+2}(A(S), A(S′) ⊗ π_{n}(K)) is defined. It is trivial if the map f admits a A_{n+1}-structure.

For example, any fusion ring is of A_{3}-type with respect to arbitrary A_{∞}-space K. Hence the class α(S) ∈ HH^{4}(A(S), A(S′) ⊗ π_{1}(K)) is defined. Any morphism of fusion rings is also of A_{3}-type; hence the images f_{∗}(α(X)), f^{∗}(α(Y)) in HH^{4}(A(S), A(S′) ⊗ π_{1}(K)) of the first classes of the fusion rings S, S′ (connected by the morphism f) coincide.

The first class admits a more direct description with respect to the space K(M(k)) = BGL(k)^{+} for the field k. Namely, it coincides with the class of the Hochcshild cocycle A ∈ Z^{4}(A(S), A(S) ⊗ k^{∗}), which is defined by the following formula

A(x_{1}, x_{2}, x_{3}, x_{4}) = ∑_{x∈S} det(A_{x_{1},x_{2},x_{3},x_{4}}^{x}),

where k^{∗} is the group of invertible elements of the field k and det(A_{x_{1},x_{2},x_{3},x_{4}}^{x}) is the determinant of the linear automorphism, which is a (clockwise) circumference of the diagram A_{x_{1},x_{2},x_{3},x_{4}}^{x}).

It follows from the description that first class of fusion ring S in HH^{4}(A(S), A(S) ⊗ k^{∗}) is the image of the class in HH^{4}(A(S), A(S) ⊗ π_{1}(BΣ_{∞}^{+})) where the space BΣ_{∞}^{+} is Quillen’s plus-construction applyed to the infinite symmetric group (an analog of the Waldhausen space K(A) for the category of finite sets). Barrat-Priddy-Quillen theorem [1, 11, 12] states that this space is homotopically equivalent to the spheric spectrum. In particular, its homotopy groups (stable homotopy groups of spheres) are torsion and

π_{1}(BΣ_{∞}^{+}) = Z/2Z, π_{2}(BΣ_{∞}^{+}) = Z/2Z, π_{3}(BΣ_{∞}^{+}) = Z/24Z.

6. Calculation of the first obstruction

In this section we give the combinatorial description of the cocycle representing the first obstruction and calculate its class in case of two element fusion ring with identity.

To write the first obstruction in an explicit form, choose a linear ordering > on the set S and linearly ordered sets

X_{y,z}^{x} = {f_{y,z}^{x}(i), i = 1, ..., m_{y,z}^{x}}

for any x, y, z ∈ S. This data allows us to define the ‘well’ order on the sets

⨆_{u∈S} X_{y,u}^{x} × X_{z,w}^{u}, ⨆_{v∈S} X_{y,z}^{v} × X_{v,w}^{x}

and to define the map Φ_{y,z,w}^{x} such sets as an (unique) order preserving the bijection. More precisely, elements of the first set

f_{y,z,w}^{x}(u, i, j) = f_{y,u}^{x}(i) × f_{z,w}^{u}(j)

are parametrized by the collections

(u, i, j), u ∈ S, i = 1, ..., m_{y,u}^{x}, j = 1, ..., m_{z,w}^{u},

which are ordered lexicographically

(u, i, j) > (u′, i′, j′) ⟺ \{ \begin{matrix}u > u′ \\ u = u′, & i > i′ \\ u = u′, & i = i′, & j > j′ \end{matrix} .

The elements of the second set

g_{y,z,w}^{x}(v, s, t) = f_{y,z}^{v}(t) × f_{v,w}^{x}(s)

are parametrized by the collections

(v, s, t), v ∈ S, s = 1, ..., m_{y,z}^{v}, j = 1, ..., m_{v,w}^{x},

are ordered analogously:

(v, s, t) > (v′, s′, t′) ⟺ \{ \begin{matrix}v > v′ \\ v = v′, & s > s′ \\ v = v′, & s = s′, & t > t′ \end{matrix} .

The following lemma is useful for the calculation of the first obstruction.

Lemma 2 1. The sign of the unique order preserving bijection between lexicographically ordered sets X × Y and Y × X (the permutation of factors) equals (−1)^{\binom{|X|}{2}}^{\binom{|Y |}{2}}.

2. The sign of the unique order preserving bijection between lexicographically ordered disjoint unions ⨆_{a,b∈S} X(a, b) and ⨆_{b,a∈S} X(a, b) equals

(−1)^{∑_{a_{1}>a_{2},b_{1}<b_{2}} |X(a_{1},b_{1})||X(a_{2},b_{2})|}.

Proof:

The sign the automorphism of the set X sending one well order >_{1} to another >_{2} is determined by the evenness of the number of inversions of orders, e.g. the quantity of pairs x, y ∈ X such that x >_{1} y and x <_{2} y.

1. The inversion of two lexicographic orders on X × Y is a pair (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ X × Y such that x_{1} > x_{2} and y_{1} < y_{2}. The quantity of these pairs is \binom{|X|}{2}\binom{|Y |}{2}.

2. The inversion of two lexicographic orders on the disjoint union ⨆_{a,b∈S} X(a, b) is a pair x ∈ X(a_{1}, b_{1}), y ∈ X(a_{2}, b_{2}) such that a_{1} > a_{2} and b_{1} < b_{2}. The quantity of these pairs is ∑_{a_{1}>a_{2},b_{1}<b_{2}} |X(a_{1}, b_{1})||X(a_{2}, b_{2})|. □

To write down the first obstruction, let us fix an order on the set of vertices of any planar binary tree.

First of all, we can regard any planar tree as a partial order on the set of its vertices V. Thus the set V decomposes into the disjoint union of subsets of vertices. Since the tree is planar we can draw all vertices from these subsets lying on the same horizontal line. We can define the well order on V ordering them left to right.

The order on the set of vertices of the planar binary tree T with n ends allows to define an (lexicographic) order on the set

X_{x_{1},...,x_{n}}^{y}(T) = ⨆_{f:E→S} ×_{v∈V} X_{f(e_{2}),f(e_{3})}^{f(e_{1})},

where the sum is over all markings of the tree T, e.g., the functions from the set of edges E to the fusion ring S, such that the values on the ends are x_{1}, ..., x_{n}; e_{1} ∈ E is the unique edge with the end v and e_{2}, e_{3} are edges which begins at v.

In particular, we can define the orders on the (five) sets corresponding to the planar binary trees with four ends which parametrize the vertices of the pentagon diagram A_{x_{1},x_{2},x_{3},x_{4}}^{x}:

⨆_{a,b∈S} X_{x_{1},a}^{x} × X_{x_{2},b}^{a} × X_{x_{3},x_{4}}^{b}, ⨆_{c,b∈S} X_{c,b}^{x} × X_{x_{2},x_{2}}^{c} × X_{x_{3},x_{4}}^{b},

⨆_{a,e∈S} X_{x_{1},a}^{x} × X_{e,x_{4}}^{a} × X_{x_{2},x_{3}}^{e}, ⨆_{d,e∈S} X_{d,x_{4}}^{x} × X_{x_{1},e}^{d} × X_{x_{2},x_{3}}^{e},

⨆_{c,d∈S} X_{d,x_{4}}^{x} × X_{c,x_{3}}^{d} × X_{x_{1},x_{2}}^{c}.

These orders define the correspondence between the (order preserving) map and any arrow of the diagram A_{x_{1},x_{2},x_{3},x_{4}}^{x}. On the other hand, we can define a map that corresponds to any arrow of the diagram A_{x_{1},x_{2},x_{3},x_{4}}^{x} by means of the maps Φ. These pairs of maps does not coincide. Using lemma 2 we can calculate the signs of the differences:

S_{x_{1},x_{2},x_{3},x_{4}}^{x} =

∑_{a>a′,b<b′} m_{x_{1},a}^{x}m_{x_{2},b}^{a}m_{x_{3},x_{4}}^{b}m_{x_{1},a′}^{x}m_{x_{2},b′}^{a′}m_{x_{3},x_{4}}^{b′}+

∑_{c>c′,b<b′} m_{c,b}^{x}m_{x_{1},x_{2}}^{c}m_{x_{3},x_{4}}^{b}m_{c′,b′}^{x}m_{x_{1},x_{2}}^{c′}m_{x_{3},x_{4}}^{b′}+

∑_{a>a′,e<e′} m_{x_{1},a}^{x}m_{e,x_{4}}^{a}m_{x_{2},x_{3}}^{e}m_{x_{1},a′}^{x}m_{e′,x_{4}}^{a′}m_{x_{2},x_{3}}^{e′}+

∑_{d>d′,e<e′} m_{d,x_{4}}^{x}m_{x_{1},e}^{d}m_{x_{2},x_{3}}^{e}m_{d′,x_{4}}^{x}m_{x_{1},e′}^{d′}m_{x_{2},x_{3}}^{e′}+

∑_{d>d′,c<c′} m_{d,x_{4}}^{x}m_{c,x_{3}}^{d}m_{x_{1},x_{2}}^{c}m_{d′,x_{4}}^{x}m_{c′,x_{3}}^{d′}m_{x_{1},x_{2}}^{c′}+

∑_{c,b} m_{c,b}^{x}\binom{m_{x_{1},x_{2}}^{c}}{2}\binom{m_{x_{3},x_{4}}^{b}}{2}.

The cocycle representing the class of the first obstruction can be written as follows

α(x_{1}, x_{2}, x_{3}, x_{4}) = ∑_{x∈S} x ⊗ α_{x_{1},x_{2},x_{3},x_{4}}^{x},

where α_{x_{1},x_{2},x_{3},x_{4}}^{x} = (−1)^{S_{x_{1},x_{2},x_{3},x_{4}}^{x}}.

Example.

Let us consider the case of two elements fusion ring S = {e, x} with the fusion rule x ∗ x = mx + n. Let us note that the first obstruction lies in H^{4}(A(S), A(S) ⊗ Z/2Z) ≅ H^{4}(A(S), A(S)/2A(S)).

Lemma 3 The fourth Hochcshild cohomology group H^{4}(A(S), A(S)/2A(S)) of the above algebra A(S) are isomorphic to

A(S)/(2, m)A(S) ≅ \{ \begin{matrix}0, & m ≡ 1(2) \\ Z/2Z ⊕ Z/2Z, & m ≡ 0(2) \end{matrix} .

The isomorphism sends the class of the cocyle α of the standard complex to the class of its value α(x, x, x, x) ∈ A(S)/(2, m)A(S).

Proof: We use the presentation of Hochcshild cohomology groups H^{k}(A(S), M) with coefficients in the A(S)-bimodule M as the groups of extensions Ext_{A(S)−A(S)}^{k}(A(S), M) in the category of A(S)-bimodules. Since A(S) = Z[x, x^{2} = mx + n] we can identify free A(S)-bimodule of rank one A(S) ⊗_{Z} A(S) with Z[x_{1}, x_{2}, x_{i}^{2} = mx_{i} + n]. The A(S)-bimodule A(S) has a 2-periodic resolution 14 a. a. davydov

A(S) ←^{μ} A(S) ⊗ A(S) ←^{x_{1}−x_{2}} A(S) ⊗ A(S) ←^{x_{1}+x_{2}−m}

A(S) ⊗ A(S) ←^{x_{1}−x_{2}} A(S) ⊗ A(S) ←^{x_{1}+x_{2}−m} ...

Applying the functor Hom_{A(S)−A(S)}(?, M) to this resolution we can calculate the cohomology H^{∗}(A(S), M)

H^{2i}(A(S), M) ≡ {y ∈ M, xy = yx}/{xz + zx − mz, z ∈ M},

H^{2i+1}(A(S), M) ≡ {y ∈ M, xy + yx = my}/{xz − zx, z ∈ M}.

Since in our case the bimodule M = A(S)/2A(S) is symmetric we have

H^{4}(A(S), A(S)/2A(S)) ≡ A(S)/(2, m)A(S).

□ The unique nontrivial factor of the expression α_{x,x,x,x}^{e} is the last one which equals

(−1)^{\binom{n}{2}+n\binom{m}{2}}. The last factor of the expression α_{x,x,x,x}^{x} is equal to (−1)^{m\binom{m}{2}} and all the other coincide with (−1)^{nm}. Hence

α_{x,x,x,x}^{x} = (−1)^{m\binom{m}{2}+nm}.

It follows from the above description that the first obstruction of the fusion ring S is nontrivial iff

m ≡ 0(2), n ≡ 2, 3(4), or m ≡ 2(4), n ≡ 1(4).

References

[1] Barratt M.G., Priddy S.B., On the homotopy of non-connected monoids and their associated groups, Comment. Math. Helvet., 47(1972), 1-14.

[2] Bordman J.M., Vogt P.M., Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math., Springer, 347, 1973.

[3] Deligne P., Milne J.S., Tannakian categories, Lecture Notes in Math., Springer, 900(1982), 101-228.

[4] Kashiwara M., Cristalizing the q-analoue of universal enveloping algebras, Commun. Math. Phys., 112 (1990), 249-260.

[5] Lusztig G., Canonical bases arasing from quantized universal enveloping algebras, Journal of Amer. Math. Soc.,

[6] May J.P., The geometry of iterated loop spaces, Lecture Notes in Math., Springer, 271(1972).

[7] May J.P., A_{∞}-spaces and algebraic K-theory, Lecture Notes in Math., Springer, 658, 1978.

[8] Moore G., Seiberg S., Polynomial equations for rational conformal field theories, Phys. Lett., B 212 (1988), 451-460.

[9] Quillen D., Higher algebraic K-theory, Lecture Notes in Math., Springer, 341, 1973.

[10] Stasheff J.P., Homotopy associativity of H-spaces I,II, Trans. Amer. Math. Soc., 108(1963), 275-292, 293-312.

[11] Segal G., Configuration-spaces and iterated loop-spaces, Inventiones math. 21(1973), 213-221.

[12] Segal G., Categories and cohomology theories, Topology, v.13(1974), 293-312.

[13] Verlinde E., Fusion rules and modular transformations in 2d conformal field theory, Nucl. Phys., B 300 [FS 22], 1988, 360-376.

on some hochschild cohomology classes 15

[14] Witten E., Qyantum field theory and the Jones polinomials, IASSNS-HEP-88/Preprint.

[15] Waldhausen F., Algebraic K-theory of spaces, Lecture Notes in Math., Springer, 1126, 1985.