Narrow Results

We define notions of pivotal and ribbon objects in a monoidal category. These constructions give pivotal or ribbon monoidal categories from a monoidal category which is not necessarily with duals. We apply this construction to the braided monoidal category of Yetter–Drinfeld modules over a Hopf algebra. This gives rise to the notion of ribbon Yetter–Drinfeld modules over a Hopf algebra, which form ribbon categories. This gives an invariant of tangles.

In his book on noncommutative geometry, Connes constructed a differential graded algebra out of a spectral triple. Lack of monoidality of this construction is investigated. We identify a suitable monoidal subcategory of the category of spectral triples and show that when restricted to this subcategory the construction of Connes is monoidal. Richness of this subcategory is exhibited by establishing a faithful endofunctor to this subcategory.

This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

In this paper, we consider Hom-(co)modules associated to a Hom-(co)associative algebra and define the notion of Hom-triple. We introduce the definitions of cleft extension and Galois extension with normal basis in this setting and we show that, as in the classical case, these notions are equivalent in the Hom setting.

We study the algebraic structures of the virtual singular braid monoid, $VS{B}_n$, and the virtual singular pure braid monoid, $VS{P}_n$. The monoid $VS{B}_n$ is the splittable extension of $VS{P}_n$ by the symmetric group $S_n$. We also construct a representation of $VS{B}_n$.

We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor category. We also show that all objects in the tensor category have the structure of left-left crossed bimodules over the Hopf algebra object. For 4D factorizable topological quantum filed theories, we provide by analogous methods a construction of a Hopf algebra category.

We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

Let B be a bialgebra, and A be a left B-comodule algebra in a braided monoidal category , and assume that A is also a coalgebra, with a not-necessarily associative or unital left B-action. Then we can define a right A-action on the tensor product of two relative Hopf modules, and this defines a monoidal structure on the category of relative Hopf modules if and only if A is a bialgebra in the category of left Yetter–Drinfeld modules over B. Some examples are given.

We define the concept of weak pseudotwistor for an algebra $(A,\mu )$ in a monoidal category ${𝒞}$, as a morphism $T:A\otimes A\to A\otimes A$ in ${𝒞}$, satisfying some axioms ensuring that $(A,\mu \circ T)$ is also an algebra in ${𝒞}$. This concept generalizes the previous proposal called pseudotwistor and covers a number of examples of twisted algebras that cannot be covered by pseudotwistors, mainly examples provided by Rota–Baxter operators and some of their relatives (such as Leroux’s TD-operators and Reynolds operators). By using weak pseudotwistors, we introduce an equivalence relation (called “twist equivalence”) for algebras in a given monoidal category.

We give two alternate presentations of the Frobenius Heisenberg category, ${\mathscr{H}}ei{s}_{F,k}$, defined by Savage, when the Frobenius algebra $F=F_1\oplus \cdots \oplus F_n$ decomposes as a direct sum of Frobenius subalgebras. In these alternate presentations, the morphism spaces of ${\mathscr{H}}ei{s}_{F,k}$ are given in terms of planar diagrams consisting of strands “colored” by integers $i=1,\dots ,n$, where a strand of color $i$ carries tokens labelled by elements of $F_i.$ In addition, we prove that when $F$ decomposes this way, the tensor product of Frobenius Heisenberg categories, ${\mathscr{H}}ei{s}_{F_1,k}\otimes \cdots \otimes {\mathscr{H}}ei{s}_{F_n,k},$ is equivalent to a certain subcategory of the Karoubi envelope of ${\mathscr{H}}ei{s}_{F,k}$ that we call the partial Karoubi envelope of ${\mathscr{H}}ei{s}_{F,k}$.

In this paper we introduce the notion of weak operators and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strict monoidal category. We prove that the class of such objects constitutes a non-strict monoidal category. It is also shown that this category is not trivial, that is to say, it admits objects generated by the adjoint action (coaction) associated to the weak braided Hopf algebra.