Quantizations of braided derivations. III: Modules with action by a group.
Quantum symmetries for exceptional modular invariants associated with conformal embeddings.
Quasitriangular Hom-Hopf algebras
A twisted generalization of quasitriangular Hopf algebras called quasitriangular Hom-Hopf algebras is introduced. We characterize these algebras in terms of certain morphisms. We also give their equivalent description via a braided monoidal category . Finally, we study the twisting structure of quasitriangular Hom-Hopf algebras by conjugation with Hom-2-cocycles.
Quasitriangular Hopf group algebras and braided monoidal categories
Let be a group, and be a semi-Hopf -algebra. We first show that the category of left -modules over is a monoidal category with a suitably defined tensor product and each element in induces a strict monoidal functor from to itself. Then we introduce the concept of quasitriangular semi-Hopf -algebra, and show that a semi-Hopf -algebra is quasitriangular if and only if the category is a braided monoidal category and is a strict braided monoidal functor for any . Finally,...
Quiver bialgebras and monoidal categories
We study bialgebra structures on quiver coalgebras and monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed...
Quotients of Tannakian categories.
Relating quantum and braided Lie algebras
We outline our recent results on bicovariant differential calculi on co-quasitriangular Hopf algebras, in particular that if is a quantum tangent space (quantum Lie algebra) for a CQT Hopf algebra A, then the space is a braided Lie algebra in the category of A-comodules. An important consequence of this is that the universal enveloping algebra is a bialgebra in the category of A-comodules.
Relative normalizing extensions.
Reshetikhin-Turaev invariants of Seifert 3-manifolds and a rational surgery formula.
Rewriting on cyclic structures : equivalence between the operational and the categorical description
Rewriting on cyclic structures: Equivalence between the operational and the categorical description
We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2-theories. We show that this presentation is equivalent to the well-accepted operational definition proposed by Barendregt et al. – but for the case of circular redexes , for which we propose (and justify formally) a different treatment. The categorical framework allows us to model in a concise way also automatic garbage collection and rules for sharing/unsharing and...
Roots of unity as a Lie algebra.
Semidirect products of categorical groups. Obstruction theory.
Separable and Frobenius monoidal Hom-algebras
As generalizations of separable and Frobenius algebras, separable and Frobenius monoidal Hom-algebras are introduced. They are all related to the Hom-Frobenius-separability equation (HFS-equation). We characterize these two Hom-algebraic structures by the same central element and different normalizing conditions, and the structure of these two types of monoidal Hom-algebras is studied. The Nakayama automorphisms of Frobenius monoidal Hom-algebras are considered.
Separable functors for the category of Doi Hom-Hopf modules
Let be the category of Doi Hom-Hopf modules, be the category of A-Hom-modules, and F be the forgetful functor from to . The aim of this paper is to give a necessary and suffcient condition for F to be separable. This leads to a generalized notion of integral. Finally, applications of our results are given. In particular, we prove a Maschke type theorem for Doi Hom-Hopf modules.
Serre Theorem for involutory Hopf algebras
We call a monoidal category C a Serre category if for any C, D ∈ C such that C ⊗ D is semisimple, C and D are semisimple objects in C. Let H be an involutory Hopf algebra, M, N two H-(co)modules such that M ⊗ N is (co)semisimple as a H-(co)module. If N (resp. M) is a finitely generated projective k-module with invertible Hattory-Stallings rank in k then M (resp. N) is (co)semisimple as a H-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel’d...
Simplicial -fold monoidal categories model all loop spaces
Skew-monoidal structures on simple rings with several objects.
Smash (co)products and skew pairings.
Let τ be an invertible skew pairing on (B,H) where B and H are Hopf algebras in a symmetric monoidal category C with (co)equalizers. Assume that H is quasitriangular. Then we obtain a new algebra structure such that B is a Hopf algebra in the braided category HHγD and there exists a Hopf algebra isomorphism w: B ∞ H → B [×]τ H in C, where B ∞ H is a Hopf algebra with (co)algebra structure the smash (co)product and B [×]τ H is the Hopf algebra defined by Doi and Takeuchi.
Some algebraic applications of graded categorical group theory.