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**1 - 2**of**2**### Note on the Fusion Map

, 2009

"... We note an inversion property of the fusion map associated to many semibialgebras. 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 (µ3 = µ ..."

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We note an inversion property of the fusion map associated to many semibialgebras. 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 (µ3 = µ(1⊗µ) = µ(µ⊗1) : A⊗A⊗A − → A) and a coassociative comultiplication: δ: A − → A ⊗ A (δ3 = (1 ⊗ δ)δ = (δ ⊗ 1)δ: A − → A ⊗ A ⊗ A) such that: δµ = (µ ⊗ µ)(1 ⊗ c ⊗ 1)(δ ⊗ δ) : A ⊗ A − → A ⊗ A It is also equipped with an endomorphism in C such that: S: A − → A µ3(1 ⊗ S ⊗ 1)δ3 = 1: A − → A The name “Von Neumann core ” stems partly from the notion of a Von Neumann regular semigroup, which is then 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 f = (1 ⊗ µ)(δ ⊗ 1) : A ⊗ A − → A ⊗ A then satisfies the fusion equation by the semibialgebra axiom of A (see [6]), and if we set: g = (1 ⊗ µ)(1 ⊗ S ⊗ 1)(δ ⊗ 1) as a tentative “inverse ” to f, then we get the following (partial) results:

### Note on the Fusion Map and Hopf Algebras

, 2009

"... We discuss an inversion property of the fusion map associated to many semibialgebras. Please note that a characterisation of Hopf k-algebras has been added at the end of this version. Let C = (C, ⊗, I, c) be a symmetric (or just braided) monoidal category. A Von Neumann “core ” in C is firstly a sem ..."

Abstract
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We discuss an inversion property of the fusion map associated to many semibialgebras. Please note that a characterisation of Hopf k-algebras has been added at the end of this version. 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 (µ3 = µ(1⊗µ) = µ(µ⊗1) : A⊗A⊗A − → A) and a coassociative comultiplication: δ: A − → A ⊗ A (δ3 = (1 ⊗ δ)δ = (δ ⊗ 1)δ: A − → A ⊗ A ⊗ A) such that: δµ = (µ ⊗ µ)(1 ⊗ c ⊗ 1)(δ ⊗ δ) : A ⊗ A − → A ⊗ A It is also equipped with an endomorphism S: A − → A in C such that: µ3(1 ⊗ S ⊗ 1)δ3 = 1: A − → A 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 f = (1 ⊗ µ)(δ ⊗ 1) : A ⊗ A − → A ⊗ A