On endomorphism algebras of functors with non-compact domain

We shall use the term “VN-core” (in Vect k ) to mean a (usual) k-semibialgebra E together with a k-linear endomorphism S such that

The VN-core is called “antipodal” if S(xy) = SySx (and S(1) = 1) for all x, y ∈ E. This minimal type of structure is introduced here in order to avoid compactness assumptions on the generator A ⊂ C and, at the same time, retain the “fusion” operator, namely

satisfying the usual fusion equation [7]. Note that here the fusion operator always has a partial inverse (see [1]).

In §2 we establish sufficient conditions on a functor U in order that

be a VN-core in Vect k (following [2]). This core can be completed to a VN-core End ∨ U ⊕ k with a unit element. In §3 we give several examples of suitable functors U for the theory.

2 The construction of End ∨ U Let C = (C, ⊗, I) be a monoidal category and let U : C → Vect be a functor with both a semigroupal structure, denoted

and a cosemigroupal structure, denoted

We shall suppose also that there exists a small full subcategory A of C with the properties:

UeA y y r r r r r r r r r r r r and

UeA y y r r r r r r r r r r r r where e UA = 1 ⊗ ev in Vect, and r 3 i 3 = 1.

We now define the semibialgebra structure (End ∨ U, µ, δ) on

as in [2] §2, with the isomorphism of k-linear spaces

given (as in [2] §3) by the usual components

where d is the canonical map from a vector space to its double dual. Furthermore, each map

commutes, and

Then we obtain:

Proof. The von Neumann axiom

becomes the diagram (in which we have omitted “⊗”): where ( * ) is the exterior of the diagram

which commutes using (E2) and commutativity of

The first type of example is derived from the idea of a (contravariant) involution on a (small) comonoidal category D. This includes the doubles D = B op + B and D = B op ⊗ B with their respective “switch” maps (where B is a given small comonoidal Vect k -category), or any small comonoidal and compact-monoidal Vect k -category D (such as the category Mat k of finite matrices over k) with the tensor duals of objects now providing an antipode on the comonoidal aspect of the structure rather than on the monoidal part, or any * -algebra structure on a given k-bialgebra (e.g., a C * -bialgebra) with the * -operation providing the antipode.

In each case, an even functor from D to Vect is defined to be a (k-linear) functor F equipped with a (chosen) dinatural isomorphism

If we take the morphisms of even functors to be all the natural transformations between them then we obtain a category

Let A = E(D, Vect fd ) fs be the full subcategory of E consisting of the finitely valued functors of finite support. While this category is generally not compact, it has on it a natural antipode derived from those on D and Vect fd , namely

Of course, there are also examples where A is actually compact, such as those where D is a Hopf algebroid, in the sense of [4], with antipode (-) * = S, in which case each A from D to Vect has a symmetry structure on it. Now let C be the full subcategory of E consisting of the small coproducts in E of objects from A. This category C is easily seen to be monoidal under the pointwise convolution structure from D, and the inclusion A ⊂ C is U -dense for the functor

which is split semigroupal with U A finite dimensional for all A ∈ A. Moreover,

for all A ∈ A. The conditions of ( 5) are easily verified if we define maps

by commutativity of the diagrams

where the exterior of

is a genuine map in C when C is given the pointwise monoidal structure from D. This completes the details of the general example.

In the case where k = C and D has just one object D whose endomorphism algebra is a C * -bialgebra, we have a one-object comonoidal category D with a C-conjugate-linear antipode given by the * -operation. Then the convolution [D, Hilb fd ], where [D,

is a monoidal category, with a C-linear antipode given by

where H • denotes the conjugate-transpose of H ∈ Hilb fd . We now interpret an even functor F to be a functor equipped with a dinatural isomorphism

Let V = (V, ⊗, I) be a (small) braided monoidal category and let B be the klinearization of Semicoalg(V) with the monoidal structure induced from that on V. By analogy with [5], let X ⊂ B be a finite full subcategory of B with I ∈ X and X op promonoidal when

for x, y, z ∈ X . For example (cf. [5]), one could take X to be a (finite) set of non-isomorphic “basic” objects in some braided monoidal category V, where each x ∈ X has a coassociative diagonal map δ : x → x ⊗ x. However, we won’t need the category X to be discrete o