Hopf箭图与Hopf代数

本文研究了简单无向Hopf箭图的图论性质以及路代数与路余代数的关系.利用简单无向Hopf箭图与简单图的关系.证明了路代数是路余代数的对偶的直和部分.     

用箭图构造非分次的双Frobenius代数

利用箭图构造了一类非分次的双Frobenius代数, 并根据结构常数分类出一种给定类型的双Frobenius代数.     

一类Hopf路余代数

设F是域,G是3阶循环群,Q是群G的箭图.借助于模范畴的等价性,给出了Hopf路余代数FQc的所有结构分类,并给出了FQc的子Hopf代数FG[FQc1]的结构.     

一类Hopf路余代数

设F是域,G是3阶循环群,Q是群G的箭图.借助于模范畴的等价性,给出了Hopf路余代数FQ~c的所有结构分类,并给出了FQ~c的子Hopf代数FG[FQ_1~c]的结构.     

循环矩阵构成的Hopf代数

Cn(R)表示交换环R上n阶循环矩阵的全体.根据通常矩阵的乘法·和加法,Cn(R)同构于一个箭图代数.考虑矩阵的Hadamard积o,Cn(R)也为结合代数,在Cn(R)上定义了新的余乘△,余单位ε和对极S使得(Cn(R),o,μ,Δ,ε,S)做成Hopf代数.     

二面体群上Hopf路余代数的结构分类

从Hopf quiver出发,借助于右kZu(C)-模的直积范畴ПC∈K(G) MkZu(C)与kG-Hopf双模范畴kG kG M kG kG之间的同构,当G是二面体群D3时,给出了Hopf路余代数kQc的同构分类及其子Hopf代数kG[kQ1]结构.     

二面体群上Hopf路余代数的结构分类

从Hopf quiver出发,借助于右kZ_u(c)-模的直积范畴■ Mkz_(u(C))与kG-Hopf双模范畴kG/kG M kG/kG之间的同构,当G是二面体群D_3时,给出了Hopf路余代数kQ~c的同构分类及其子Hopf代数kG[kQ_1]结构.     

二面体群D_2上Hopf路余代数的结构分类

本文研究了Hopf代数的构造问题.利用模范畴和箭图,获得了当G是二面体群D2这一4阶交换群时的Hopf路余代数kQc的同构分类及其子Hopf代数kG[kQ1]的结构,推广了当G是2阶循环群时的相应结论.     

双扭HOPF代数的对偶

本文研究了一个双扭Hopf代数的分次对偶空间以及两个双扭Hopf代数的分次对偶关系.利用代数和余代数分次对偶空间的性质,得出一个局部有限的双扭(χ1,χ2)-Hopf代数的分次对偶空间是一个双扭(χ1T,χ2)-Hopf代数,并判定两个双扭Hopf代数的分次对偶可以简化为判定它们作为双扭双代数是分次对偶的.     

Smash积代数和量子模范畴中的Hopf代数的新对偶

本文引入模代数的一种新对偶,它推广了代数的有限对偶概念.并证明通过这种新对偶,模代数的对偶为余模余代数,从而形成Smash余积,而且证明了Smash积的对偶是Smash余积,即有(A#H)0≌HA0×H0余代数同构.最后证明量子模范畴中的Hopf代数通过这种新对偶是自对偶的.     

Quiver Hopf algebras

In this paper we study subHopfalgebras of graded Hopf algebra structures on a path coalgebra kQ^c. We prove that a Hopf structure on some subHopfquivers can be lifted to a Hopf structure on the whole Hopf quiver. If Q is a Schurian Hopf quiver, then we classified all simple-pointed subHopfalgebras of a graded Hopf structure on kQ^c. We also prove a dual Gabriel theorem for pointed Hopf algebras.     

Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

Graded Quantum Groups and Quasitriangular Hopf Group-Coalgebras

ABSTRACT

Starting from a Hopf algebra endowed with an action of a group π by Hopf automorphisms, we construct (by a “twisted” double method) a quasitriangular Hopf π-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular Hopf π-coalgebras for any finite group π and for infinite groups π such as GL _n (𝕂). In particular, we define the graded quantum groups, which are Hopf π-coalgebras for π = ?[[h]] ^l and generalize the Drinfeld-Jimbo quantum enveloping algebras.     

The Hopf Structure of Some Dual Operator Algebras

We study the Hopf structure of a class of dual operator algebras corresponding to certain semigroups. This class of algebras arises in dilation theory, and includes the noncommutative analytic Toeplitz algebra and the multiplier algebra of the Drury–Arveson space, which correspond to the free semigroup and the free commutative semigroup respectively. The preduals of the algebras in this class naturally form Hopf (convolution) algebras. The original algebras and their preduals form (non-self-adjoint) dual Hopf algebras in the sense of Effros and Ruan. We study these algebras from this perspective, and obtain a number of results about their structure.     

弱准三角Hopf代数和解量子Yang-Baxter方程

出于解量子Yang-Baxter方程的需要,本文定义了弱准三角Hopf代数,并且发现了一类构造弱准三角Hopf代数的方法,文中称之为杨-积,它可以提供量子Yang-Baxter方程的解.     

Actions of Pointed Hopf Algebras on Quantum Torus

Actions of finite dimensional pointed Hopf algebras on generic quantum torus are classified.

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Sunto Vengono classificate le azioni delle algebre di Hopf puntate di dimensione finita sul toro quantico generico.

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Research partially supported by grants RFBR 03-01-00167, NSh-1910.2003.1 and INTAS00-566     

Exceptional Sequences in Hall Algebras and Quantum Groups

Let be a finite-dimensional hereditary algebra over a finite field k, () and () be, respectively, the Hall algebra and the composition algebra of , be the isomorphism classes of finite dimensional -modules and I the isomorphism classes of simple -modules. We define and , in , to be the right and left derivations of () respectively. By using these derivations and the action of the braid group on the set of exceptional sequences of -mod, we provide an effective algorithm of calculating the root vectors of real Schur roots. This means that we get an inductive method to express u as the combinations of elements u_i in the Hall algebra, where i I and in is any exceptional -module. Because of the canonical isomorphism between the Drinfeld–Jimbo quantum group and the generic composition algebra, our algorithm is applicable directly to quantum groups. In particular, all the root vectors are obtained in this way in the finite type cases.     

Actions of Hopf Algebras on General Quantum Mal'tsev Power Series and Quantum Planes

A classification of automorphisms, derivations, and Hopf algebra actions of the quantum plane and its completions is presented. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 18, Algebra, 2004.     

Cluster Tilted Algebras with a Cyclically Oriented Quiver

In association with a finite dimensional algebra A of global dimension two, we consider the endomorphism algebra of A, viewed as an object in the triangulated hull of the orbit category of the bounded derived category, in the sense of Amiot. We characterize the algebras A of global dimension two such that its endomorphism algebra is isomorphic to a cluster-tilted algebra with a cyclically oriented quiver. Furthermore, in the case that the cluster tilted algebra with a cyclically oriented quiver is of Dynkin or extended Dynkin type then A is derived equivalent to a hereditary algebra of the same type.     

On Approximation of Groups, Group Actions, and Hopf Algebras

We give new examples and criteria in the theory of approximation of groups by finite groups. Bibliography: 17 titles.