Drinfeld Double for Braided Infinitesimal Hopf Algebras

Wang, Shengxiang; Wang, Shuanhong

doi:10.1080/00927872.2013.766796

Cited by 8 publications

(7 citation statements)

References 10 publications

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“…Moreover, the notions of coboundary and quasitriangular infinitesimal bialgebra, infinitesimal Hopf algebra, and the basic theory were established by Aguiar in [1,2]. Further research can be found in [4,27].…”

Section: Introductionmentioning

confidence: 99%

An associative analogy of Lie H-pseudobialgebra

Liu,

Guo

2020

Preprint

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The purpose of this paper is to study infinitesimal H-pseudobialgebra, which is an associative analogy of Lie H-pseudobialgebra. We first define the infinitesimal H-pseudobialgebra and investigate some properties of this new algebraic structure. Then we consider the coboundary infinitesimal H-pseudobialgebra, which is the subclass of infinitesimal H-pseudobialgebra and we obtain the associative Yang-Baxter equation over an associative H-pseudoalgebra. Finally, we found the connection between the (coboundary) infinitesimal H-pseudobialgebra and the (coboundary) Lie H-pseudobialgebra. Meanwhile, the relationship between the associative Yang-Baxter equation and the classical Yang-Baxter equation (over an H-pseudoalgebra) is established. Contents1. Introduction 1 2. Associative H-pseudoalgebras and their representations 3 3. Infinitesimal H-pseudobialgebras 8 4. Coboundary infinitesimal H-pseudobialgebras 12 5. From (coboundary) infinitesimal H-pseudobialgebras to (coboundary) Lie Hpseudobialgebras 17 6. From associative pseudo-Yang-Baxter equation to the classical type 22 References 23

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Section: Introductionmentioning

confidence: 99%

An associative analogy of Lie H-pseudobialgebra

Liu,

Guo

2020

Preprint

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“…Furthermore, infinitesimal bialgebras are also closely related to associative Yang-Baxter equations, Drinfeld's doubles, pre-Lie algebras and Drinfeld's Lie bialgebras [2]. Recently, Wang [39] generalized Aguiar's results by studying the Drinfeld's double for braided infinitesimal Hopf algebras in Yetter-Drinfeld categories. Another different version of infinitesimal bialgebras and infinitesimal Hopf algebras was defined by Loday and Ronco [35] and further studied by Foissy [15,16], in the sense that (2) ∆(ab) = a · ∆(b) + ∆(a) · b − a ⊗ b for a, b ∈ A.…”

Section: Introductionmentioning

confidence: 99%

Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles

et al. 2019

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In this paper, we define a new coproduct on the space of decorated planar rooted forests to equip it with a weighted infinitesimal unitary bialgebraic structure. We introduce the concept of Ω-cocycle infinitesimal bialgebras of weight λ and then prove that the space of decorated planar rooted forests H RT (X, Ω), together with a set of grafting operations {B + ω | ω ∈ Ω}, is the free Ωcocycle infinitesimal unitary bialgebra of weight λ on a set X, involving a weighted version of a Hochschild 1-cocycle condition. As an application, we equip a free cocycle infinitesimal unitary bialgebraic structure on the undecorated planar rooted forests, which is the object studied in the well-known (noncommutative) Connes-Kreimer Hopf algebra.

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“…The basic theory of infinitesimal bialgebras and infinitesimal Hopf algebras was developed by Aguiar [1,3,4,5], has proven useful not only in combinatorices [4,18], but in other areas of mathematics as well, such as associative Yang-Baxter equations [1,5], Drinfeld's doubles [1,39] and pre-Lie algebras [1]. The second version of infinitesimal bialgebras was defined by Loday and Ronco [31] and brought new life on rooted trees by Foissy [21,22] in the sense that…”

Section: Introductionmentioning

confidence: 99%

Weighted infinitesimal unitary bialgebras, pre-Lie, matrix algebras, and polynomial algebras

Zhang

Zheng

Luo

2019

Communications in Algebra

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Motivated by comatrix coalgebras, we introduce the concept of a Newtonian comatrix coalgebra. We construct an infinitesimal unitary bialgebra on matrix algebras, via the construction of a suitable coproduct. As a consequence, a Newtonian comatrix coalgebra is established. Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on matrix algebras. By the close relationship between pre-Lie algebras and infinitesimal unitary bialgebras, we erect a pre-Lie algebra and a new Lie algebra on matrix algebras. Finally, a weighted infinitesimal unitary bialgebra on non-commutative polynomial algebras is also given.

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Drinfeld Double for Braided Infinitesimal Hopf Algebras

Abstract: In this paper, we mainly construct the Drinfeld double for braided infinitesimal Hopf algebras in Yetter-Drinfeld categories as a generalization of Aguiar's result.

Cited by 8 publications

References 10 publications

An associative analogy of Lie H-pseudobialgebra

An associative analogy of Lie H-pseudobialgebra

Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles

Weighted infinitesimal unitary bialgebras, pre-Lie, matrix algebras, and polynomial algebras

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