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A categorification of the square root of -1

Yin Tian (2016)

Fundamenta Mathematicae

We give a graphical calculus for a monoidal DG category ℐ whose Grothendieck group is isomorphic to the ring ℤ[√(-1)]. We construct a categorical action of ℐ which lifts the action of ℤ[√(-1)] on ℤ².

A construction of the Hom-Yetter-Drinfeld category

Haiying Li, Tianshui Ma (2014)

Colloquium Mathematicae

In continuation of our recent work about smash product Hom-Hopf algebras [Colloq. Math. 134 (2014)], we introduce the Hom-Yetter-Drinfeld category H H via the Radford biproduct Hom-Hopf algebra, and prove that Hom-Yetter-Drinfeld modules can provide solutions of the Hom-Yang-Baxter equation and H H is a pre-braided tensor category, where (H,β,S) is a Hom-Hopf algebra. Furthermore, we show that ( A H , α β ) is a Radford biproduct Hom-Hopf algebra if and only if (A,α) is a Hom-Hopf algebra in the category H H . Finally,...

A Maschke type theorem for relative Hom-Hopf modules

Shuangjian Guo, Xiu-Li Chen (2014)

Czechoslovak Mathematical Journal

Let ( H , α ) be a monoidal Hom-Hopf algebra and ( A , β ) a right ( H , α ) -Hom-comodule algebra. We first introduce the notion of a relative Hom-Hopf module and prove that the functor F from the category of relative Hom-Hopf modules to the category of right ( A , β ) -Hom-modules has a right adjoint. Furthermore, we prove a Maschke type theorem for the category of relative Hom-Hopf modules. In fact, we give necessary and sufficient conditions for the functor that forgets the ( H , α ) -coaction to be separable. This leads to a generalized...

Adjointness between theories and strict theories

Hans-Jürgen Vogel (2003)

Discussiones Mathematicae - General Algebra and Applications

The categorical concept of a theory for algebras of a given type was foundet by Lawvere in 1963 (see [8]). Hoehnke extended this concept to partial heterogenous algebras in 1976 (see [5]). A partial theory is a dhts-category such that the object class forms a free algebra of type (2,0,0) freely generated by a nonempty set J in the variety determined by the identities ox ≈ o and xo ≈ o, where o and i are the elements selected by the 0-ary operation symbols. If the object class of a dhts-category...

Bicovariant differential calculi and cross products on braided Hopf algebras

Yuri Bespalov, Bernhard Drabant (1997)

Banach Center Publications

In a braided monoidal category C we consider Hopf bimodules and crossed modules over a braided Hopf algebra H. We show that both categories are equivalent. It is discussed that the category of Hopf bimodule bialgebras coincides up to isomorphism with the category of bialgebra projections over H. Using these results we generalize the Radford-Majid criterion and show that bialgebra cross products over the Hopf algebra H are precisely described by H-crossed module bialgebras. In specific braided monoidal...

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