关于μ~H中的Lie双代数和余Poisson-Hopf代数(英文)

本文研究余三角 Ｈｏｐｆ代数余模范畴中的 Ｌｉｅ双代数和余 Ｐｏｉｓｓｏｎ－Ｈｏｐｆ代数．我们主要讨论余三角Ｈｏｐｆ代数余模范畴中的Ｌｉｅ双代数和余Ｐｏｉｓｓｏｎ－Ｈｏｐｆ代数之间的关系．     

三角Hopf代数表示范畴上的Lie余代数

本文在三角Ｈｏｐｆ代数表示范畴上系统地研究了Ｌｉｅ余代数,在此范畴上 的Ｌｉｅ余代数与Ｈｏｐｆ代数之间建立了重要的联系．主要给出了Ｌｉｅ余代数的余包络 余代数的结构．所得结果自然是关于Ｌｉｅ代数的对偶结果,推广了 Ｓｗｅｅｄｌｅｒ Ｍ． Ｅ．, Ｇｕｒｅｖｉｃｈ Ｄ．Ｉ．, Ｍｉｃｈａｅｌｉｓ Ｗ．和 Ｍａｉｉｄ Ｓ．等人的结果．     

三角Hopf代数表示范畴上的Lie余代数

本文在三角Ｈｏｐｆ代数表示范畴上系统地研究了Ｌｉｅ余代数,在此范畴上 的Ｌｉｅ余代数与Ｈｏｐｆ代数之间建立了重要的联系．主要给出了Ｌｉｅ余代数的余包络 余代数的结构．所得结果自然是关于Ｌｉｅ代数的对偶结果,推广了 Ｓｗｅｅｄｌｅｒ Ｍ． Ｅ．, Ｇｕｒｅｖｉｃｈ Ｄ．Ｉ．, Ｍｉｃｈａｅｌｉｓ Ｗ．和 Ｍａｉｉｄ Ｓ．等人的结果．     

三角Hopf代数表示范畴上的代数结构

Ｙｕ．Ⅰ．Ｍａｎｉｎ［５］在范畴上引入各种代数结构，但没有进行深入的研究．本文在三角Ｈｏｐｆ代数的表示范畴上进行系统的研究，在此范畴上的Ｌｉｅ代数与Ｈｏｐｆ代数之间建立了重要的联系，主要结果有：（１）三角Ｈｏｐｆ代数表示范畴上Ｌｉｅ代数的包络代数是此范畴上的Ｈｏｐｆ代数；（２）三角Ｈｏｐｆ代数表示范畴上Ｌｉｅ双代数结构可唯一扩张为其包络代数的余Ｐｏｉｓｓｏｎ－Ｈｏｐｆ代数结构．因而推广了Ｍ．Ｅ．Ｓｗｅｅｄｌｅｒ的经典结果与Ｖ．Ｇ．Ｄｒｉｎｆｅｌｄ的一个重要定理．     

关于伪紧余代数、余挠对和弦余代数

本文利用箭图和拓扑伪紧空间研究了K-余代数及其表示.定义了域K上的伪紧K-余代数,研究了伪紧K-余代数和K-代数范畴之间的关系,研究了余挠对和余模逼近,描述了余倾斜余挠对.通过有限维的支撑子余代数和基本的路余代数研究了弦余代数.     

交叉余积上的Hopf代数结构

本文给出了Ｓｍａｓｈ积代数结构和交叉余积余代数结构构成双代数的一个充分必要条件．另外,还给出了这一新的双代数成为Ｈｏｐｆ代数的一个充分条件．     

有限W-代数与横截Poisson结构

证明了物理学中的有限Ｗ－代数是半单Ｌｉｅ代数上的横截Ｐｏｉｓｓｏｎ结构．     

H－弱余模余代数和交叉余积

引进了交叉积的对偶交叉余积，证明了：余Ｃｌｅｆｔ模余代数的结构定理（作为余代数）；如果为Ｈｏｐｆ代数余可裂正合序列，那么作为Ｈｏｐｆ代数，由此有强增广余代数Ｃ的结构定理（作为双代数）；如果为Ｈｏｐｆ代数可裂正合序列，那么作为Ｈｏｐｆ代数并简单地讨论了Ｃ×αＨ的余半单性．     

弱Hopf群余代数Kegel定理

本文研究了余三角弱Hopfπ-余代数H的左弱π-H-余模代数.通过构造左弱π-H-余模代数的导出π-σ-李代数,得到了弱Hopf π-余代数Kegel定理,推广了文献[4]的结果.     

半完备余代数上余模范畴的黏合

本文主要研究半完备余代数上余模范畴的黏合,证明黏合中的范畴是余模范畴当且仅当它是由半完备余代数的余幂等子余代数诱导的黏合,进一步还将此结果应用到Morita-Takeuchi关系余代数和余模复形范畴上.     

三角Hopf余模范畴上的张量余代数

在三角Ｈｏｐｆ代数余模范畴上研究张量余代数．主要给出三角Ｈｏｐｆ代数余模范畴上的张量余代数的结构．     

Engel’s Theorem for Generalized Lie Algebras

We prove analogues of the classical Engel’s Theorem for Lie algebras in the category of comodules over a cotriangular Hopf algebra, generalizing the known result for Lie coloralgebras.     

Monoidal Hom–Hopf Algebras

Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Hom-algebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras, and Lie algebras.     

Twisted Comodule Algebras and the Generalized Quantum Double

In this article we show that the notion of twisted comodule algebras is a generalization of Drinfeld's cocycle twisting construction of bialgebras or Hopf algebras, and that several other twisting constructions, such as Ferrer's twisting product and Lu's one-sided twisting, are also special examples of twisted comodule algebras. Some general properties on twisted comodule algebras are investigated, and applications to the generalized quantum double are given.     

Categorical Constructions for Hopf Algebras

We prove that both the embedding of the category of Hopf algebras into that of bialgebras and the forgetful functor from the category of Hopf algebras to the category of algebras have right adjoints; in other words, every bialgebra has a Hopf coreflection, and on every algebra there exists a cofree Hopf algebra. In this way, we give an affirmative answer to a forty-years old problem posed by Sweedler. On the route, the coequalizers and the coproducts in the category of Hopf algebras are explicitly described.     

Quantization of Lie bialgebras, II

This paper is a continuation of [EK]. We show that the quantization procedure of [EK] is given by universal acyclic formulas and defines a functor from the category of Lie bialgebras to the category of quantized universal enveloping algebras. We also show that this functor defines an equivalence between the category of Lie bialgebras over k [[h]] and the category of quantized universal enveloping (QUE) algebras.     

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

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.     

Generalized Lie Algebras and Cocycle Twists

We use the results of Etingof and Gelaki on the classification of (co)triangular Hopf algebras to extend Scheunert's “discoloration” technique to Lie algebras in the category of (co)modules. As an application, we prove a PBW-type theorem for such Lie algebras. We also discuss the relationship between Lie algebras in the category of (co)modules and symmetric braided Lie algebras introduced by Gurevich. Finally, we construct examples of symmetric braided Lie algebras that are essentially different from Lie coloralgebras.     

Hopf algebra deformations of binary polyhedral groups

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on cosemisimple Hopf algebras with a two-dimensional irreducible comodule in the finite-dimensional context. Finally, we give some applications to the classification of certain classes of semisimple Hopf algebras.