Sweedler四维代数及其子代数上的带权无穷小双代数和预李代数

带权无穷小双代数是非齐次结合经典杨巴方程的代数抽象,在数学和数学物理领域有诸多重要的应用.给出了Sweedler四维代数及其子代数上的带权无穷小双代数的分类.作为应用,得到了Sweedler四维代数上的预李代数,进而诱导出它们的李代数结构.     

非交换群上一类代数的不可约表示

在一类无限维非交换Hopf代数上,借助其Hopf理想,构造出商Hopf代数,讨论了此商代数上的有限维不可约模,得出此非平凡不可约模的维数一定是2.     

自由交换Nijenhuis代数上的左余单位Hopf代数结构

本文基于自由交换Rota-Baxter代数上的Hopf代数结构,探讨自由交换Nijenhuis代数上的Hopf代数相关结构;借助于上闭链(cocycle)条件证明左余单位双代数(即不满足右余单位性)上的自由交换Nijenhuis代数具有左余单位双代数结构.本文获得更具一般性的结论,连通分次左余单位双代数是左余单位右对极Hopf代数,即其对极只是右侧的.由此证明连通左余单位双代数上的自由交换Nijenhuis代数是连通且分次的,从而,它是左余单位右对极Hopf代数.     

Heisenberg-Virasoro代数的q-形变的量子群结构

构造了水平为零的扭的Heisenberg-Virasoro代数的一个q-形变Hvirq,证明它是一个quasi-hom-李代数.给出该代数的一个非平凡的量子群结构,即它是一个非交换且余交换的Hopf代数.     

Weyl型单代数

对于任意特征的域F上具有单位元的交换结合代数A和它的交换导子的子空间D的多项式代数F［Ｄ］,在张量空间A［Ｄ］=AÄF［Ｄ］中定义了Weyl型结合代数和Lie代数.证明了A［Ｄ］作为Lie代数(模去中心)或结合代数是单的充要条件是A为D-单的并且A［Ｄ］在A上的作用是忠实的.[KG*2]由此可构造出许多单代数.     

Doi-Hopf模以及其上的平坦联络

对于任意Hopf代数,在其上建立了一种新的具有非交换微分几何形式的代数结构,并论证了Doi-hopf模与具有平坦联络的模之间是同构的,其中,联络对应于我们所给的微分算子.     

乘子Hopf代数的 L-R Smash余积

设 $A$ 是正则乘子Hopf代数, $R$ 是 $A$-\!\!双余模代数, 首先定义了 $A$-\!\!双余模双代数, 并利用它构造了 L-R Smash 积的对偶形式, 即 $R\otimes A$ 上一种非平凡的乘子Hopf代数结构, 称之为 L-R Smash余积. 然后给出了 L-R Smash余积上的积分和$*$-\!\!结构.     

弱Hopf代数作用与冲积

本文研究了弱Hopf代数上的冲积并讨论了它约性质．设H是弱Hopf代数,A是左H-摸代数．我们给出了冲积A#H是弱双代数的一个充分条件以及A#H是A可分扩张的一个判定条件．另外,利用积分理论研究了Hopf模代数的有限性条件．     

Hopf π-子模

设H为有限型Hopfπ-代数,研究Hopfπ-代数H上的Hopfπ-模与Hopf π-余代数H *上的Hopfπ-余模之间的对偶关系,得出了Hopfπ-子模与Hopfπ-子 余模之间的充分必要条件,推广了Hopf代数中的相关结论.     

相对Hopf模和广义cleft扩张

本文研究了π-cleft扩张的余代数分解问题及对偶的η-cocleft扩张的代数分解问题.利用相对Hopf模结构定理,给出了一个条件,使得具有余代数结构的π-cleft扩张有较好的余代数分解形式.作为特例,可以重新得到Radford给出的具有投射的双代数的双积分解定理.     

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.     

Bidendriform bialgebras, trees, and free quasi-symmetric functions

We introduce bidendriform bialgebras, which are bialgebras such that both product and coproduct can be split into two parts satisfying good compatibilities. For example, the Malvenuto-Reutenauer Hopf algebra and the non-commutative Connes-Kreimer Hopf algebras of planar decorated rooted trees are bidendriform bialgebras. We prove that all connected bidendriform bialgebras are generated by their primitive elements as a dendriform algebra (bidendriform Milnor-Moore theorem) and then is isomorphic to a Connes-Kreimer Hopf algebra. As a corollary, the Hopf algebra of Malvenuto-Reutenauer is isomorphic to the Connes-Kreimer Hopf algebra of planar rooted trees decorated by a certain set. We deduce that the Lie algebra of its primitive elements is free in characteristic zero (G. Duchamp, F. Hivert and J.-Y. Thibon conjecture).     

The infinitesimal Hopf algebra and the poset of planar forests

We introduce an infinitesimal Hopf algebra of planar trees, generalising the construction of the non-commutative Connes-Kreimer Hopf algebra. A non-degenerate pairing and a dual basis are defined, and a combinatorial interpretation of the pairing in terms of orders on the vertices of planar forests is given. Moreover, the coproduct and the pairing can also be described with the help of a partial order on the set of planar forests, making it isomorphic to the Tamari poset. As a corollary, the dual basis can be computed with a Möbius inversion.     

Hom-弱Hopf代数上的Hom-smash余积

本文研究了在Hom-Hopf代数上引入Hom-弱Hopf代数的问题.利用建立弱左H-Hom-余模双代数的方法,获得了Hom-smash余积的代数结构,并证明了Hom-smash余积是Hom-余代数和Hom-弱Hopf代数,推广了由Molnar定义的smash余积Hopf代数.     

On Galois Extension of Hopf Algebras

Let H be a cosemisimple Hopf algebra over a field k, and π ： A→ H be a surjective cocentral bialgebra homomorphism of bialgebras. The authors prove that if A is Galois over its coinvariants B=LH Ker π and B is a sub-Hopf algebra of A, then A is itself a Hopf algebra. This generalizes a result of Cegarra [3] on group-graded algebras.     

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.     

On bimeasurings

We introduce and study bimeasurings from pairs of bialgebras to algebras. It is shown that the universal bimeasuring bialgebra construction, which arises from Sweedler's universal measuring coalgebra construction and generalizes the finite dual, gives rise to a contravariant functor on the category of bialgebras adjoint to itself. An interpretation of bimeasurings as algebras in the category of Hopf modules is considered.     

On the operad of bigraft algebras

In this paper, we study the notion of a bigraft algebra, generalizing the notions of left and right graft algebras. We construct the free bigraft algebra on one generator in terms of certain planar rooted trees with decorated edges, and therefore describe explicitly the bigraft operad. We then compute its Koszul dual and show that the bigraft operad is Koszul. Moreover, we endow the free bigraft algebra on one generator with a universal Hopf algebra structure and a pairing. Finally, we prove an analogue of the Poincaré–Birkhoff–Witt and Cartier–Milnor–Moore theorems. For this, we define the notion of infinitesimal bigraft bialgebras and we prove the existence of a new good triple of operads.     

Hopf modules,Frobenius functors and (one-sided) Hopf algebras

We investigate the property of being Frobenius for some functors strictly related with Hopf modules over a bialgebra and how this property reflects on the latter. In particular, we characterize one-sided Hopf algebras with anti-(co)multiplicative one-sided antipode as those for which the free Hopf module functor is Frobenius. As a by-product, this leads us to relate the property of being an FH-algebra (in the sense of Pareigis) for a given bialgebra with the property of being Frobenius for certain naturally associated functors.     

(ω,σ)-Smash积积和和(ν,α)-Smash余余积积

设B,H是两个Hopf代数,构造了(ω,σ)-Smash积Bω#σH和(ν,α)-Smash余积Bν■αH,并给出了Bω#σH是Hopf代数和Bν■αH是双代数的充要条件,证明了许多已知的积和余积是它们的特殊情况.