Results 1 
5 of
5
GRAPHICAL METHODS FOR TANNAKA DUALITY OF WEAK BIALGEBRAS AND WEAK HOPF ALGEBRAS
"... Abstract. Tannaka duality describes the relationship between algebraic objects in ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. Tannaka duality describes the relationship between algebraic objects in
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 ..."
Abstract
 Add to MetaCart
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 VNcore in Set, while the free vector space on it is a particular type of VNcore in Vect, and partly from the properties of the paths which generate a (rowfinite) 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 kalgebras 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
 Add to MetaCart
We discuss an inversion property of the fusion map associated to many semibialgebras. Please note that a characterisation of Hopf kalgebras 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 VNcore in Set, while the free vector space on it is a particular type of VNcore in Vect, and partly from the properties of the paths which generate a (rowfinite) graph algebra[5]. The fusion map f = (1 ⊗ µ)(δ ⊗ 1) : A ⊗ A − → A ⊗ A
WHAT SEPARABLE FROBENIUS MONOIDAL FUNCTORS PRESERVE
, 904
"... Abstract. Separable Frobenius monoidal functors were de ned and studied under that name in [10], [11] and [4] and in a more general context in [3]. Our purpose here is to develop their theory in a very precise sense. We determine what kinds of equations in monoidal categories they preserve. For exam ..."
Abstract
 Add to MetaCart
Abstract. Separable Frobenius monoidal functors were de ned and studied under that name in [10], [11] and [4] and in a more general context in [3]. Our purpose here is to develop their theory in a very precise sense. We determine what kinds of equations in monoidal categories they preserve. For example we show they preserve lax (meaning not necessarily invertible) YangBaxter operators, weak YangBaxter operators in the sense of [1], and (in the braided case) weak bimonoids in the sense of [8]. In fact, we characterize which monoidal expressions are preserved (or rather, are stable under conjugation in a wellde ned sense). We show that every weak YangBaxter operator is the image of a genuine YangBaxter operator under a separable Frobenius monoidal functor. Prebimonoidal functors are also de ned and discussed. 1.
ON THE ITERATION OF WEAK WREATH PRODUCTS
"... Based on a study of the 2category of weak distributive laws, we describe a method of iterating Street’s weak wreath product construction. That is, for any 2category K and for any nonnegative integer n, we introduce 2categories Wdl (n) (K), of (n + 1)tuples of monads in K pairwise related by wea ..."
Abstract
 Add to MetaCart
Based on a study of the 2category of weak distributive laws, we describe a method of iterating Street’s weak wreath product construction. That is, for any 2category K and for any nonnegative integer n, we introduce 2categories Wdl (n) (K), of (n + 1)tuples of monads in K pairwise related by weak distributive laws obeying the YangBaxter equation. The first instance Wdl (0) (K) coincides with Mnd(K), the usual 2category of monads in K, and for other values of n, Wdl (n) (K) contains Mnd n+1 (K) as a full 2subcategory. For the local idempotent closure K of K, extending the multiplication of the 2monad Mnd, we equip these 2categories with n possible ‘weak wreath product ’ 2functors Wdl (n) (K) → Wdl (n−1) (K), such that all of their possible nfold composites Wdl (n) (K) → Wdl (0) (K) are equal; that is, such that the weak wreath product is ‘associative’. Whenever idempotent 2cells in K split, this leads to pseudofunctors Wdl (n) (K) → Wdl (n−1) (K) obeying the associativity property upto isomorphism. We present a practically important occurrence of an iterated weak wreath product: the algebra of observable quantities in an Ising type quantum spin chain where the spins take their values in a dual pair of finite weak Hopf algebras. We also construct a fully faithful embedding of Wdl (n) (K) into the 2category of commutative n + 1 dimensional cubes in Mnd(K) (hence into the 2category of commutative n + 1 dimensional cubes in K whenever K has EilenbergMoore objects and its idempotent 2cells split). Finally we give a sufficient and necessary condition on a monad in K to be isomorphic to an nary weak wreath product.