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Quandle coverings and their Galois correspondence

Michael Eisermann (2014)

Fundamenta Mathematicae

This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p: (Q̃,q̃̃) → (Q,q). This in turn leads us to define the algebraic fundamental group π ₁ ( Q , q ) : = A u t ( p ) = g ∈ A d j ( Q ) ' | q g = q , where Adj(Q) is the adjoint group of Q. We then establish the Galois correspondence between connected coverings of (Q,q) and subgroups of π₁(Q,q). Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble Kervaire’s...

Quasitriangular Hom-Hopf algebras

Yuanyuan Chen, Zhongwei Wang, Liangyun Zhang (2014)

Colloquium Mathematicae

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 ̃ ( H ℳ ) . 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

Shiyin Zhao, Jing Wang, Hui-Xiang Chen (2014)

Czechoslovak Mathematical Journal

Let π be a group, and H be a semi-Hopf π -algebra. We first show that the category H ℳ of left π -modules over H is a monoidal category with a suitably defined tensor product and each element α in π induces a strict monoidal functor F α from H ℳ to itself. Then we introduce the concept of quasitriangular semi-Hopf π -algebra, and show that a semi-Hopf π -algebra H is quasitriangular if and only if the category H ℳ is a braided monoidal category and F α is a strict braided monoidal functor for any α ∈ π . Finally,...

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