L-R-Smash biproducts, double biproducts and a braided category of Yetter-Drinfeld-Long bimodules

Communicated by T. H. LenaganWe study some examples of braided categories and quasitriangular Hopf algebras and decide which of them is pseudosymmetric, respectively pseudotriangular. We show also that there exists a universal pseudosymmetric braided category.Pseudosymmetric categories are a special class of braided categories and have been introduced in [10]. The motivation was the study of certain categorical structures called twines, strong twines and pure-braided structures (introduced in [1, 9, 13]). A braiding on a strict monoidal category is called pseudosymmetric if it satisfies a sort of modified braid relation; any symmetric braiding is pseudosymmetric. One of the most intriguing results obtained in [10] was that the category of Yetter-Drinfeld modules over a Hopf algebra H is pseudosymmetric if and only if H is commutative and cocommutative. We proved in [8] that pseudosymmetric categories can be used to construct representations for the group PS n = Bn [Pn,Pn] , the quotient of the braid group by the commutator subgroup of the pure braid group. There exists also a Hopf algebraic analogue of pseudosymmetric braidings: a quasitriangular structure on a Hopf algebra is called pseudotriangular if it satisfies a sort of modified quantum Yang-Baxter equation.In this paper we tie some lose ends from [8,10]. We study more examples of braided categories and quasitriangular Hopf algebras and decide when they are pseudosymmetric, respectively pseudotriangular. Namely, we prove that the canonical braiding of the category LR(H) of Yetter-Drinfeld-Long bimodules over a Hopf algebra H (introduced in [11]) is pseudosymmetric if and only if H is commutative and cocommutative. We show that any quasitriangular structure on the 4ν-dimensional Radford's Hopf algebra H ν (introduced in [12]) is pseudotriangular. We analyze the positive quasitriangular structures R(ξ, η) on a Hopf algebra with positive bases H(G; G + , G − ) (as defined in [6,7]), where ξ, η are group homomorphisms from G + to G − , and we present a list of necessary and sufficient conditions for R(ξ, η) to be pseudotriangular. If R(ξ, η) is normal (i.e. if ξ is trivial) these conditions reduce to the single relation η(uv) = η(vu) for all u, v ∈ G + .In Sec. 5 we recall the pseudosymmetric braided category PS introduced in [8] and we show that it is a universal pseudosymmetric category. More precisely, we prove that it satisfies two universality properties similar to the ones satisfied by the universal braid category B (see [5]).

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“…Moreover, it has a (canonical) braiding given by We recall now the braided monoidal category LR(H) defined in [11].…”

Section: Yetter-drinfeld-long Bimodulesmentioning

confidence: 99%

More Examples of Pseudosymmetric Braided Categories

Panaite

Staic

2013

J. Algebra Appl.

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“…Panaite and F. V. Oystaeyen in [3] introduced the notion of L-R smash biproduct, with the L-R smash product and L-R smash coproduct introduced in [2] as multiplication, respectively comultiplication. When an object A which is both an algebra and a coalgebra and a bialgebra H form a L-R-admissible pair (H, A), A♮H becomes a bialgebra with smash product and smash coproduct, and the Radford biproduct is a special case.…”

Section: Introductionmentioning

confidence: 99%

“…When an object A which is both an algebra and a coalgebra and a bialgebra H form a L-R-admissible pair (H, A), A♮H becomes a bialgebra with smash product and smash coproduct, and the Radford biproduct is a special case. It turns out that A is in fact a bialgebra in the category LR(H) of Yetter-Drinfeld-Long bimodules (introduced in [3]) with some compatible condition.…”

Section: Introductionmentioning

confidence: 99%

Yetter-Drinfeld-Long bimodules are modules

Wang

2017

Czech.Math.J.

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“…The Radford biproduct plays an important role in the lifting method for the classification of finite dimensional pointed Hopf algebras [3]. Some related results about Radford biproduct were recently given in [6,10,12,13,17,21].…”

Section: Introductionmentioning

confidence: 99%

On Radford Biproduct

2015

Communications in Algebra

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Let H be a bialgebra, A an algebra, and a left H-comodule coalgebra. Let f H ⊗ H −→ A ⊗ H and R H ⊗ A −→ A ⊗ H be two linear maps. Let B be a right H-module algebra and also a right H-comodule coalgebra. In this paper, necessary and sufficient conditions are given for the one-sided Brzeziński's crossed product algebra A# f R H#B and the two-sided smash coproduct coalgebra A × H × B to form a bialgebra, which we call the Brzeziński's double biproduct. The celebrated Radford biproduct [18], Majid's double biproduct [13], Agore and Militaru's unified product [2], and the Wang-Jiao-Zhao's crossed product [21] are all derived as special cases. On the other hand, we construct a class of Radford biproducts by a twisting method.

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L-R-Smash biproducts, double biproducts and a braided category of Yetter-Drinfeld-Long bimodules

Cited by 13 publications

References 15 publications

More Examples of Pseudosymmetric Braided Categories

More Examples of Pseudosymmetric Braided Categories

Yetter-Drinfeld-Long bimodules are modules

On Radford Biproduct

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