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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,...

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...

Classifying bicrossed products of two Sweedler's Hopf algebras

Costel-Gabriel Bontea (2014)

Czechoslovak Mathematical Journal

We continue the study started recently by Agore, Bontea and Militaru in “Classifying bicrossed products of Hopf algebras” (2014), by describing and classifying all Hopf algebras E that factorize through two Sweedler’s Hopf algebras. Equivalently, we classify all bicrossed products H 4 H 4 . There are three steps in our approach. First, we explicitly describe the set of all matched pairs ( H 4 , H 4 , , ) by proving that, with the exception of the trivial pair, this set is parameterized by the ground field k . Then, for...

Cobraided smash product Hom-Hopf algebras

Tianshui Ma, Haiying Li, Tao Yang (2014)

Colloquium Mathematicae

Let (A,α) and (B,β) be two Hom-Hopf algebras. We construct a new class of Hom-Hopf algebras: R-smash products ( A R B , α β ) . Moreover, necessary and sufficient conditions for ( A R B , α β ) to be a cobraided Hom-Hopf algebra are given.

Galois H-objects with a normal basis in closed categories. A cohomological interpretation.

José N. Alonso Alvarez, José Manuel Fernández Vilaboa (1993)

Publicacions Matemàtiques

In this paper, for a cocommutative Hopf algebra H in a symmetric closed category C with basic object K, we get an isomorphism between the group of isomorphism classes of Galois H-objects with a normal basis and the second cohomology group H2(H,K) of H with coefficients in K. Using this result, we obtain a direct sum decomposition for the Brauer group of H-module Azumaya monoids with inner action:BMinn(C,H) ≅ B(C) ⊕ H2(H,K)In particular, if C is the symmetric closed category of C-modules with K a...

More examples of invariance under twisting

Florin Panaite (2012)

Czechoslovak Mathematical Journal

The so-called “invariance under twisting” for twisted tensor products of algebras is a result stating that, if we start with a twisted tensor product, under certain circumstances we can “deform” the twisting map and we obtain a new twisted tensor product, isomorphic to the given one. It was proved before that a number of independent and previously unrelated results from Hopf algebra theory are particular cases of this theorem. In this article we show that some more results from literature are particular...

On generalized partial twisted smash products

Shuangjian Guo (2014)

Czechoslovak Mathematical Journal

We first introduce the notion of a right generalized partial smash product and explore some properties of such partial smash product, and consider some examples. Furthermore, we introduce the notion of a generalized partial twisted smash product and discuss a necessary condition under which such partial smash product forms a Hopf algebra. Based on these notions and properties, we construct a Morita context for partial coactions of a co-Frobenius Hopf algebra.

On Lie algebras in braided categories

Bodo Pareigis (1997)

Banach Center Publications

The category of group-graded modules over an abelian group G is a monoidal category. For any bicharacter of G this category becomes a braided monoidal category. We define the notion of a Lie algebra in this category generalizing the concepts of Lie super and Lie color algebras. Our Lie algebras have n -ary multiplications between various graded components. They possess universal enveloping algebras that are Hopf algebras in the given category. Their biproducts with the group ring are noncommutative...

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