Note on the Fusion Map

Let C = (C, ⊗, I, c) be a symmetric (or just braided) monoidal category. A Von Neumann "core" in C is firstly a semibialgebra in C, that is, an object A in C with an associative multiplication: µ : A ⊗ A -→ A (µ 3 = µ(1⊗µ) = µ(µ⊗1) : A⊗A⊗A -→ A) and a coassociative comultiplication:

The name “Von Neumann core” stems partly from the notion of a Von Neumann regular semigroup, which is then precisely a VN-core in Set, while the free vector space on it is a particular type of VN-core in Vect, and partly from the properties of the paths which generate a (row-finite) graph algebra [5].

The fusion map

then satisfies the fusion equation by the semibialgebra axiom of A (see [6]), and if we set:

as a tentative “inverse” to f , then we get the following (partial) results:

Proof. Define the (left) Fourier transform l(α) of a map α : A -→ B to be the composite

, where ⋆ is the convolution α ⋆ β = µ(α ⊗ β)δ of two maps α and β from A to B. Thus:

Proposition 2. gf g = g if S 2 = 1 and S is an antihomomorphism either of algebras or of coalgebras.

The proof is straightforward.

Recall that a V N -core is called “unital” [1] if it satisfies the (stronger) axiom

where A is assumed to have the unit η : I -→ A. (A unital VN-core in C = Set is precisely a group).

[1] gf = 1 for any unital VN-core.

Note that, in general, if for a map f there exists a map g with f gf = f , then we can always find a map h with f hf = f and hf h = h provided idempotents split in C.

A semibialgebra is called a very weak bialgebra in [1] if it also has both a unit η : I -→ A (µ(1 ⊗ η) = µ(η ⊗ 1) = 1) and a counit ǫ : A -→ I ((1 ⊗ ǫ)δ = (ǫ ⊗ 1)δ = 1). A very weak bialgebra A is then called a very weak Hopf algebra if it is equipped with a map S : A -→ A satisfying the axioms:

Hence S⋆1⋆S = S so that gf g = g and, as a consequence of the semibialgebra axiom, we have 1 ⋆ t = 1 (see [4]) whence 1 ⋆ S ⋆ 1 = 1 so that f gf = f (using S ⋆ 1 = t by the first axiom).

Example: Suppose that (A, µ, δ, η, ǫ) is a bialgebra for which δ is not known to be coassociative, and suppose that A is also equipped with a map S : A -→ A (not necessarily an antihomomorphism), and invertible elements λ : I -→ A and ρ : I -→ A such that the standard Drinfel’d axioms hold, namely:

Proposition 4. This is a quasi-VN-bialgebra in the sense that both the equations

Note that the two standard Drinfel’d conditions were still satisfied in the definition of a weak quasi-Hopf algebra (in the sense of Haring-Oldenburg et al. [2]).

We note also that any VN-core A in Vect k can be completed to a (VN-) bialgebra A ⊕ k in a fairly obvious way.

Moreover, there are Tannaka-type reconstruction results for VN-bialgebras, based on the notion of a partially compact monoidal category, which is simply a k-linear monoidal category (A, ⊗, I) equipped with an antipode functor S and two k-linear natural transformations:

such that ene = e and nen = n. The reconstruction results use the facts that the finite-dimensional left k-representations of a VN-bialgebra (A, µ, δ, η, ǫ, S) satisfying S(xy) = SySx and S1 = 1 form such a partially compact monoidal category, and, in general, the self-dual representations of any VN-bialgebra form a partially compact monoidal category.

Enquires (etc.) regarding this article can be made to the author through Micah McCurdy (Macquarie University), who kindly typed the manuscript.

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