二重Tame型遗传代数的Ringel-Hall代数和Hopf超箭图
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摘要
1989年,Ringel定义了任意有限性环上的Hall代数(见文[Rin2]),其主要目的是为了探讨箭图表示与李代数和量子群之间的关系.随后,许多数学工作者一直试图利用有限域上有限维代数的Hall代数来实现李代数和量子群,已产生许多有重要意义的结果,见文[Rin3, Rin4, Rin6, Gr2, PX1-PX3, X, DX].Hall代数中由单模同构类生成的子代数称为合成代数,合成代数在实现李代数和量子群的过程中起到重要作用,引起大家的广泛研究,见文[Rin3, Rin7, Gr2, X, GP, GZ, ZP2-Zp5, ZZ].另外,Hall多项式在计算相应李代数和量子群的结构常数时起到非常关键的作用,因此Hall多项式是否存在也是备受关注的问题,见文[Rin2, Rin6, Gu, P, Zs1].有限维遗传代数的二重(duplicated)代数的倾斜模与cluster范畴中的倾斜对象一一对应,引起我们对二重代数的研究兴趣,见文[ABST1, Zs5].
Hopf代数在上世纪六十年代就引起了很多数学家的研究兴趣.尤其近年来量子群的兴起,特别是量子群与统计力学中的Yang-Baxter方程之间深刻的联系,Hopf代数已发展成为与物理等多个学科有密切联系的代数学分支.用箭图的方式来研究Hopf代数始于上世纪九十年代,Cibils,Rosso,Green, Solberg, Van Oystaeyen和章璞等都有许多重要工作(见文[Ci,CR1,CR2,GSo,VZ]),在推动Hopf代数的发展上起到不可忽视的作用.
本学位论文主要做了下面两方面工作：
一.研究了二重tame型遗传代数的Ringel-Hall代数的结构,并得到二重tame型遗传代数导出的李子代数；
二.建立了Hopf超箭图理论,并用箭图理论研究了Hopf超代数和拟Hopf超代数这两类重要的Hopf结构.
本论文共分四部分：
第一章是引言部分,我们介绍与本文有关的研究发展概况,较全面阐述论文的工作背景和思路.
第二章,我们研究了二重tame型遗传代数的Ringel-Hall代数和合成代数结构,证明了二重tame型遗传代数的一些Hall多项式存在,并得到由二重tame型遗传代数导出的李子代数.主要结果如下：
定理2.2.4设A为tame型遗传代数,A为A的二重代数,M为不可分解A-模,则u[M]∈(?)(A)当且仅当M为例外A-模.
定理2.2.7设A为tame型遗传代数,A为A的二重代数, M为非单且不可分解A-模,则(?)(A)中的元素u[M]可以写成单A-模同构类的多重斜换位子.
定理2.3.10设A为tame型遗传代数,A为A的二重代数,X和Y为不可分解A-模,则对任意的A-模M,Hall多项式g XY M存在.
第三章,建立了Hopf超箭图理论,作为应用给出了分次Hopf超代数的分类理论和一些重要的结构性定理.主要结果如下：
定理3.2.2设(Q,p)为超箭图,则路超余代数(kQ,p)具有分次Hopf超代数结构当且仅当(Q,p)为由群和与之对应的一个超分歧数据确定的Hopf超箭图.
推论3.2.3设G为群,C为群G的共轭类集.对任意C∈C,记ZC为C中某个元的中心化子.设R=(R0,R1)为G的一个超分歧数据,其中和记相应的Hopf超箭图(Q(G,R0,R1),p)为(Q,p).则路超余代数(kQ,p)具有Q0(?)G的分次Hopf超代数结构与表示组{(VC,0)C∈C,(VC,1)C∈C}一一对应.其中VC,0和VC,1分别为维数是RC,0和RC,1的kZC-模,对任意C∈C.
命题3.3.1设H为点化Hopf超代数,grH为由H的余根滤链诱导的Hopf超代数,则存在唯一的Hopf超箭图(Q,p)以及路超余代数(kQ,p)上的分次Hopf超代数,使得grH同构于该分次Hopf超代数的一个包含kQ0(?)kQ1的子Hopf超代数.
第四章,用建立的Hopf超箭图理论来研究拟Hopf超代数.主要结果有：
定理4.2.1设(Q,p)为超箭图,则路超余代数(kQ,p)具有分次对偶拟Hopf超代数结构当且仅当(Q,p)为Hopf超箭图.
命题4.2.3设G为群,(kG,Φ,S,α,β)为对偶拟Hopf超代数.取群G的一个超分歧数据R=R0 R0.设(Q,p)=(Q(G,R0,R1),p)为Hopf超箭图.则路超余代数(kQ,p)具有分次对偶拟Hopf超代数结构,其中以kQ0(?)(kG,Φ,S,α,β)作为子对偶拟Hopf超代数,当且仅当kQ1具有kG-quasi-Hopf超双模结构.并且,路超余代数(kQ,p)上的分次对偶拟Hopf超代数结构与kQ1上的kG-quasi-Hopf超双模结构一一对应.
命题4.2.4设H为点化对偶拟Hopf超代数,grH为由H的余根滤链诱导的对偶拟Hopf超代数,则存在唯一的Hopf超箭图(Q,p)以及路超余代数(kQ,p)上的分次对偶拟Hopf超代数,使得grH同构于该分次对偶拟Hopf超代数的一个包含kQ0(?)kQ1的子对偶拟Hopf超代数.
推论4.2.5设(Q,p)为超箭图,则路超余代数(kQ,p)具有对偶拟Hopf超代数结构(不一定分次)当且仅当(Q,p)是Hopf超箭图.
In order to study relations between representations of quivers and Lie algebras and quantum groups, Ringel defined Hall algebras over finitary rings in 1989(see [Rin2]). Later, many working mathematicians have been trying to use Hall algebras of finite dimensional algebras over a finite field to realize Lie algebras and quantum groups, and many important and inspiring results have been obtained, see [Rin3, Rin4, Rin6, Gr2, PX1-PX3, X, DX]. The subalgebra of Hall algebra generated by the isoclasses of simple modules is called composition algebra, playing an important role in realizing Lie algebras and quantum groups, see [Rin3, Rin7, Gr2, X, GP, GZ, Zp2-Zp5, ZZ]. Also, Hall polynomials provide some convenience in calculating the structural coefficients of the corresponding Lie algebras and quantum groups. So it is interesting and important to investigate the existence of Hall polynomials, see [Rin2, Rin6, Gu, P, Zs1].Tilting modules of duplicated algebra of a finite dimensional hereditary algebra are in one-to-one correspondence with cluster tilting objects of the corresponding cluster category. This motivates further interest on this kind of algebras, see [ABST1, Zs5].
Hopf algebra has attracted much research interest since the sixties of last century. In recent years with the development of quantum groups, and the strong relations between quantum groups and Yang-Baxter equations in statistics mechanics, Hopf algebra has been found to have many other connections and applications in physics. Since the nineties of last century, many people have started to study Hopf algebra using quivers. Cibils, Rosso, Green, Solberg, Van Oystaeyen and Zhang Pu have obtained many interesting and important results since then, see [Ci, CR1, CR2, GSo, VZ].
In this dissertation, we investigate the structure of Ringel-Hall algebras of dupli-cated tame hereditary algebras, obtain some Lie subalgebras induced by duplicated tame hereditary algebras; and we also develop a super version of the Hopf quiver the-ory, investigate Hopf superalgebras and quasi-Hopf superalgebras. This dissertation includes four parts altogether.
In chapter 1 of this dissertation, we give an introduction, including important re-sults needed and recent developments related to this dissertation, and make a systemic exposition of our main results.
In chapter 2, we investigate the structure of Ringel-Hall algebras and composition algebras of duplicated tame hereditary algebras, and we obtain some Lie subalgebras induced by duplicated tame hereditary algebras.The main results are as follows.
Theorem 2.2.4 Let A be a tame hereditary algebra over k and A be the duplicated algebra of A. And let M be an indecomposable A-module. Then u[M]∈(?)(A) if and only if M is an exceptional A-module.
Theorem 2.2.7 Let A be a tame hereditary algebra over k and A be the duplicated algebra of A. And let M be a non-simple indecomposable A-module. Then the element u[M]∈(?)(A) can be written as an iterated skew commutator of the isoclasses of simple A-modules.
Theorem 2.3.10 Let A be a tame hereditary algebra over k and A be the duplicated algebra of A. Let X and Y be indecomposable A-modules. Then for any A-module M, there exists the Hall polynomial gXYM.
In chapter 3,we develop a super version of the Hopf quiver theory, the nontion of Hopf superquivers is introduced. As an application, we give the classification of graded Hopf superalgebras and some structure theorems. The main results are as follows.
Theorem 3.2.2 Let (Q, p) be a superquiver. Then the path supercoalgebra (kQ, p) admits a graded Hopf superalgebra structure if and only if (Q, p) is a Hopf superquiver of some group with respect to a super ramification datum.
Corollary 3.2.3 Let G be a group, C its set of conjugacy classes and for each C∈C, let ZC denote the centralizer of one of its elements. Let R=(R0, R1)be a super ramification datum of G with R0=∑C∈CRC,0C. and R1=∑C∈CRC,1C, Denote the associated Hopf superquiver (Q(G,R0,R1),p) as (Q,p).Then the set of graded Hopf superalgebra structures on (kQ, p) with Q0≌G as groups is in one-to-one correspondence with the set of pairs of collections{(VC,0)C∈C, (VC,1)C∈C} in which VC,0 (resp. VC,1) is a kZC-module of dimension RC,0 (resp. RC,1) for all C∈C.
Proposition 3.3.1 Let H be a pointed Hopf superalgebra and gr H its graded version induced by the coradical filtration. Then there is a unique Hopf superquiver (Q, p) and a graded Hopf superalgebra structure on the path supercoalgebra (kQ, p) such that gr H can be embedded into it as a sub Hopf superalgebra which contains kQ0⊕kQ1.
In chapter 4, we will investigate quasi-Hopf superalgebras using Hopf superquivers. The main results are as follows.
Theorem 4.2.1 Let (Q, p) be a superquiver. Then the path supercoalgebra (kQ,p) admits a graded dual quasi-Hopf superalgebra structure if and only if (Q, p) is a Hopf superquiver.
Proposition 4.2.3 Let G be a group and (kG,Φ, S,α,β) a dual quasi-Hopf superalgebra. Let (Q,p)= (Q(G,R0,R1),p) be the Hopf superquiver associated to a ramification datum R=R0⊕R1 of G. Then the path supercoalgebra (kQ,p) admits a graded dual quasi-Hopf superalgebra structure with kQ0≌(kG,Φ, S,α,β) as a sub dual quasi-Hopf superalgebra if and only if kQ1 admits a kG-quasi-Hopf superbimodule structure. Moreover, the set of such graded dual quasi-Hopf superalgebra structures on the path supercoalgebra (kQ, p) is in one-to-one correspondence with the set of kG-quasi-Hopf superbimodule structures on kQ1.
Proposition 4.2.4 Let H be a pointed dual quasi-Hopf superalgebra and gr H its graded version induced by the coradical filtration. Then there is a unique Hopf superquiver (Q, p) and a graded dual quasi-Hopf superalgebra structure on the path supercoalgebra (kQ, p) such that gr H can be embedded into it as a sub dual quasi-Hopf superalgebra which contains kQ0⊕kQi.
Corollary 4.2.5 Let (Q, p) be a superquiver. Then the path supercoalgebra (kQ,p) admits a dual quasi-Hopf superalgebra structure (not necessarily graded) if and only if (Q, p) is a Hopf superquiver.
Hopf代数在上世纪六十年代就引起了很多数学家的研究兴趣.尤其近年来量子群的兴起,特别是量子群与统计力学中的Yang-Baxter方程之间深刻的联系,Hopf代数已发展成为与物理等多个学科有密切联系的代数学分支.用箭图的方式来研究Hopf代数始于上世纪九十年代,Cibils,Rosso,Green, Solberg, Van Oystaeyen和章璞等都有许多重要工作(见文[Ci,CR1,CR2,GSo,VZ]),在推动Hopf代数的发展上起到不可忽视的作用.
本学位论文主要做了下面两方面工作：
一.研究了二重tame型遗传代数的Ringel-Hall代数的结构,并得到二重tame型遗传代数导出的李子代数；
二.建立了Hopf超箭图理论,并用箭图理论研究了Hopf超代数和拟Hopf超代数这两类重要的Hopf结构.
本论文共分四部分：
第一章是引言部分,我们介绍与本文有关的研究发展概况,较全面阐述论文的工作背景和思路.
第二章,我们研究了二重tame型遗传代数的Ringel-Hall代数和合成代数结构,证明了二重tame型遗传代数的一些Hall多项式存在,并得到由二重tame型遗传代数导出的李子代数.主要结果如下：
定理2.2.4设A为tame型遗传代数,A为A的二重代数,M为不可分解A-模,则u[M]∈(?)(A)当且仅当M为例外A-模.
定理2.2.7设A为tame型遗传代数,A为A的二重代数, M为非单且不可分解A-模,则(?)(A)中的元素u[M]可以写成单A-模同构类的多重斜换位子.
定理2.3.10设A为tame型遗传代数,A为A的二重代数,X和Y为不可分解A-模,则对任意的A-模M,Hall多项式g XY M存在.
第三章,建立了Hopf超箭图理论,作为应用给出了分次Hopf超代数的分类理论和一些重要的结构性定理.主要结果如下：
定理3.2.2设(Q,p)为超箭图,则路超余代数(kQ,p)具有分次Hopf超代数结构当且仅当(Q,p)为由群和与之对应的一个超分歧数据确定的Hopf超箭图.
推论3.2.3设G为群,C为群G的共轭类集.对任意C∈C,记ZC为C中某个元的中心化子.设R=(R0,R1)为G的一个超分歧数据,其中和记相应的Hopf超箭图(Q(G,R0,R1),p)为(Q,p).则路超余代数(kQ,p)具有Q0(?)G的分次Hopf超代数结构与表示组{(VC,0)C∈C,(VC,1)C∈C}一一对应.其中VC,0和VC,1分别为维数是RC,0和RC,1的kZC-模,对任意C∈C.
命题3.3.1设H为点化Hopf超代数,grH为由H的余根滤链诱导的Hopf超代数,则存在唯一的Hopf超箭图(Q,p)以及路超余代数(kQ,p)上的分次Hopf超代数,使得grH同构于该分次Hopf超代数的一个包含kQ0(?)kQ1的子Hopf超代数.
第四章,用建立的Hopf超箭图理论来研究拟Hopf超代数.主要结果有：
定理4.2.1设(Q,p)为超箭图,则路超余代数(kQ,p)具有分次对偶拟Hopf超代数结构当且仅当(Q,p)为Hopf超箭图.
命题4.2.3设G为群,(kG,Φ,S,α,β)为对偶拟Hopf超代数.取群G的一个超分歧数据R=R0 R0.设(Q,p)=(Q(G,R0,R1),p)为Hopf超箭图.则路超余代数(kQ,p)具有分次对偶拟Hopf超代数结构,其中以kQ0(?)(kG,Φ,S,α,β)作为子对偶拟Hopf超代数,当且仅当kQ1具有kG-quasi-Hopf超双模结构.并且,路超余代数(kQ,p)上的分次对偶拟Hopf超代数结构与kQ1上的kG-quasi-Hopf超双模结构一一对应.
命题4.2.4设H为点化对偶拟Hopf超代数,grH为由H的余根滤链诱导的对偶拟Hopf超代数,则存在唯一的Hopf超箭图(Q,p)以及路超余代数(kQ,p)上的分次对偶拟Hopf超代数,使得grH同构于该分次对偶拟Hopf超代数的一个包含kQ0(?)kQ1的子对偶拟Hopf超代数.
推论4.2.5设(Q,p)为超箭图,则路超余代数(kQ,p)具有对偶拟Hopf超代数结构(不一定分次)当且仅当(Q,p)是Hopf超箭图.
In order to study relations between representations of quivers and Lie algebras and quantum groups, Ringel defined Hall algebras over finitary rings in 1989(see [Rin2]). Later, many working mathematicians have been trying to use Hall algebras of finite dimensional algebras over a finite field to realize Lie algebras and quantum groups, and many important and inspiring results have been obtained, see [Rin3, Rin4, Rin6, Gr2, PX1-PX3, X, DX]. The subalgebra of Hall algebra generated by the isoclasses of simple modules is called composition algebra, playing an important role in realizing Lie algebras and quantum groups, see [Rin3, Rin7, Gr2, X, GP, GZ, Zp2-Zp5, ZZ]. Also, Hall polynomials provide some convenience in calculating the structural coefficients of the corresponding Lie algebras and quantum groups. So it is interesting and important to investigate the existence of Hall polynomials, see [Rin2, Rin6, Gu, P, Zs1].Tilting modules of duplicated algebra of a finite dimensional hereditary algebra are in one-to-one correspondence with cluster tilting objects of the corresponding cluster category. This motivates further interest on this kind of algebras, see [ABST1, Zs5].
Hopf algebra has attracted much research interest since the sixties of last century. In recent years with the development of quantum groups, and the strong relations between quantum groups and Yang-Baxter equations in statistics mechanics, Hopf algebra has been found to have many other connections and applications in physics. Since the nineties of last century, many people have started to study Hopf algebra using quivers. Cibils, Rosso, Green, Solberg, Van Oystaeyen and Zhang Pu have obtained many interesting and important results since then, see [Ci, CR1, CR2, GSo, VZ].
In this dissertation, we investigate the structure of Ringel-Hall algebras of dupli-cated tame hereditary algebras, obtain some Lie subalgebras induced by duplicated tame hereditary algebras; and we also develop a super version of the Hopf quiver the-ory, investigate Hopf superalgebras and quasi-Hopf superalgebras. This dissertation includes four parts altogether.
In chapter 1 of this dissertation, we give an introduction, including important re-sults needed and recent developments related to this dissertation, and make a systemic exposition of our main results.
In chapter 2, we investigate the structure of Ringel-Hall algebras and composition algebras of duplicated tame hereditary algebras, and we obtain some Lie subalgebras induced by duplicated tame hereditary algebras.The main results are as follows.
Theorem 2.2.4 Let A be a tame hereditary algebra over k and A be the duplicated algebra of A. And let M be an indecomposable A-module. Then u[M]∈(?)(A) if and only if M is an exceptional A-module.
Theorem 2.2.7 Let A be a tame hereditary algebra over k and A be the duplicated algebra of A. And let M be a non-simple indecomposable A-module. Then the element u[M]∈(?)(A) can be written as an iterated skew commutator of the isoclasses of simple A-modules.
Theorem 2.3.10 Let A be a tame hereditary algebra over k and A be the duplicated algebra of A. Let X and Y be indecomposable A-modules. Then for any A-module M, there exists the Hall polynomial gXYM.
In chapter 3,we develop a super version of the Hopf quiver theory, the nontion of Hopf superquivers is introduced. As an application, we give the classification of graded Hopf superalgebras and some structure theorems. The main results are as follows.
Theorem 3.2.2 Let (Q, p) be a superquiver. Then the path supercoalgebra (kQ, p) admits a graded Hopf superalgebra structure if and only if (Q, p) is a Hopf superquiver of some group with respect to a super ramification datum.
Corollary 3.2.3 Let G be a group, C its set of conjugacy classes and for each C∈C, let ZC denote the centralizer of one of its elements. Let R=(R0, R1)be a super ramification datum of G with R0=∑C∈CRC,0C. and R1=∑C∈CRC,1C, Denote the associated Hopf superquiver (Q(G,R0,R1),p) as (Q,p).Then the set of graded Hopf superalgebra structures on (kQ, p) with Q0≌G as groups is in one-to-one correspondence with the set of pairs of collections{(VC,0)C∈C, (VC,1)C∈C} in which VC,0 (resp. VC,1) is a kZC-module of dimension RC,0 (resp. RC,1) for all C∈C.
Proposition 3.3.1 Let H be a pointed Hopf superalgebra and gr H its graded version induced by the coradical filtration. Then there is a unique Hopf superquiver (Q, p) and a graded Hopf superalgebra structure on the path supercoalgebra (kQ, p) such that gr H can be embedded into it as a sub Hopf superalgebra which contains kQ0⊕kQ1.
In chapter 4, we will investigate quasi-Hopf superalgebras using Hopf superquivers. The main results are as follows.
Theorem 4.2.1 Let (Q, p) be a superquiver. Then the path supercoalgebra (kQ,p) admits a graded dual quasi-Hopf superalgebra structure if and only if (Q, p) is a Hopf superquiver.
Proposition 4.2.3 Let G be a group and (kG,Φ, S,α,β) a dual quasi-Hopf superalgebra. Let (Q,p)= (Q(G,R0,R1),p) be the Hopf superquiver associated to a ramification datum R=R0⊕R1 of G. Then the path supercoalgebra (kQ,p) admits a graded dual quasi-Hopf superalgebra structure with kQ0≌(kG,Φ, S,α,β) as a sub dual quasi-Hopf superalgebra if and only if kQ1 admits a kG-quasi-Hopf superbimodule structure. Moreover, the set of such graded dual quasi-Hopf superalgebra structures on the path supercoalgebra (kQ, p) is in one-to-one correspondence with the set of kG-quasi-Hopf superbimodule structures on kQ1.
Proposition 4.2.4 Let H be a pointed dual quasi-Hopf superalgebra and gr H its graded version induced by the coradical filtration. Then there is a unique Hopf superquiver (Q, p) and a graded dual quasi-Hopf superalgebra structure on the path supercoalgebra (kQ, p) such that gr H can be embedded into it as a sub dual quasi-Hopf superalgebra which contains kQ0⊕kQi.
Corollary 4.2.5 Let (Q, p) be a superquiver. Then the path supercoalgebra (kQ,p) admits a dual quasi-Hopf superalgebra structure (not necessarily graded) if and only if (Q, p) is a Hopf superquiver.
引文
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[CR1]C. Cibils, M. Rosso, Algebres des chemins quantiques, Adv. Math.,125 (1997) 171-199.
[CR2]C. Cibils, M. Rosso, Hopf quivers, J. Algebra,254 (2002) 241-251.
[DH]H. Dong, H.-L. Huang, Hopf superquivers, submitted.
[Di]J. Dieudonne, Introduction to the theory of formal groups, Marcel Dekker, New York,1973.
[DR]V. Dlab, C.M. Ringel, Indecomposable representations of graphs and algebras. Mem.Amer.Math.Soc,173 (1976)
[Drl]V.G. Drinfeld, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol.1,2 (Berkeley, Calif.,1986),798-820, Amer. Math. Soc, Providence, RI,1987.
[Dr2]V.G. Drinfeld, Quasi-Hopf algebras, Leningrad Math. J.1 (1990) 1419-1457.
[DX]B. Deng, J. Xiao, On double Ringel-Hall algebras, J. Algebra,251(2002),110-149.
[DZ]H. Dong, S. Zhang, Ringel-Hall algebras of duplicated tame hereditary algebras. Preprint, arXiv:math. RT/1001.1188v1,2010.
[EG1]P. Etingof and S. Gelaki, On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic, Internat. Math. Res. Notices,7(1999) 387-394.
[EG2]P. Etingof and S. Gelaki, The classification of triangular semisimple and cosemisim-ple Hopf algebras and over an algebraically closed field, Internat. Math. Res. Notices,5(2000) 223-234.
[EG3]P. Etingof, S. Gelaki, Finite dimensional quasi-Hopf algebras with radical of codimension 2, Math. Res. Lett.11 (2004) 685-696.
[EG4]P. Etingof, S. Gelaki, On radically graded finite-dimensional quasi-Hopf alge-bras, Mosc. Math. J.5(2) (2005) 371-378.
[EG5]P. Etingof, S. Gelaki, Liftings of graded quasi-Hopf algebras with radical of prime codimension, J. Pure Appl. Algebra,205(2) (2006) 310-322.
[ENO]P. Etingof, D. Nikshych, V. Ostrik, On fusion categories, Ann.Math.,162 (2005)581-642.
[EO]P. Etingof, V. Ostrik, Finite tensor categories, Moscow Math. J.4 (2004) 627-654,782-783.
[Ga1]P. Gabriel, Unzerlegbare Darstellungen Ⅰ, Manuscripta Math.,6(1972)71-103.
[Ga2]P. Gabriel, Indecomposable representations Ⅱ, Symposia Mathematica, Vol. Ⅺ (Convegno di Algebra Gommutativa, INDAM, Rome,1971),81-104, Academic Press, London,1973.
[Ga3]P. Gabriel, Finite representation type is open, in:Representations of Algebras, Lecture Notes in Math., vol.488, Springer Verlag,1975, pp.132-155.
[Ge1]S. Gelaki, Basic quasi-Hopf algebras of dimension n3, J. Pure Appl. Algebra, 198(1-3) (2005) 165-174.
[Ger]M. Gerstenhaer, On the deformation of rings and algebras, Ann. Math.,79(2) (1964) 59-103.
[GP]J. Guo, L. Peng, Universal PBW-basis of Hall-Ringel algebras and Hall polyno-mials. J. Algebra,198(1997),339-351.
[Gr1]J.A. Green, Locally finite representations, J. Algebra,41 (1976) 137-171.
[Gr2]J.A. Green, Hall algebras,hereditary algebras and quantum groups.Invent. Math.,120 (1995)361-377.
[Gr3]E.L. Green, Constructing quantum groups and Hopf algebras from coverings, J. Algebra,176 (1995) 12-33.
[GSc]M. Gerstenhaber, S.D. Schack, Bialgebra cohomology, deformations, and quan-tum groups, Proc. Nat. Acad. Sci.,87 (1990) 478-481.
[GSo]E.L. Green,(?). Solberg, Basic Hopf algebras and quantum groups, Math. Z., 229 (1998) 45-76.
[Gu]J. Guo, The Hall polynomials of a cyclic serial algebra, Comm. Algebra,23 (1995) 743-751.
[GZ]E.L. Green, P. Zhang, Rigids as iterated skew commutators of simples. Algebr Represent Theory,9 (2006) 539-555.
[GZI1]M.D. Gould, Y.Z. Zhang, P.S. Isaac, Casimir invariants from quasi-Hopf (su-per)algebras, J. Math. Phys.41, no.1,547(2000).
[GZI2]M.D. Gould, Y.Z. Zhang, P.S. Isaac, On quasi-Hopf superalgebras, Comm. Math. Phys.,224 (2001)341-372.
[H1]H.-L. Huang, A simple proof for a theorem of Cartier,2008,submitted.
[H2]H.-L. Huang, Quiver approaches to quasi-Hopf algebras, J. Math. Phys.,50 (2009) 043501,9pp.
[H3]H.-L. Huang, From projective representations to quasi-quantum groups. Preprint, arXiv:math. RT/0903.1472,2009.
[HCZ]H.-L. Huang, H.-X. Chen, P. Zhang, Generalized Taft algebras, Algebra Colloq. 11(03) (2004) 313-320.
[HN]F. Hausser, F. Nill, Integral theory for quasi-Hopf algebras, math.QA/9904164.
[HRS]D. Happel, I. Reiten, S.O. Smal(?), Tilting in abelian categories and quasitilted algebras, Mem.Amer.Math.Soc,575,1996.
[HZ]Y. Han, D. Zhao, Superspecies and their representations, J. Algebra,321 (2009) 3668-3680.
[Kac]V. Kac, Lie superalgebras, Adv. Math.26 (1977) 8-96.
[Kar]G. Karpilovsky, Projecive Representations of Finite Groups, Marcel Dekker, New York,1985.
[Kas]C. Kassel, Quantum Groups, Graduate Texts in Math.155, Springer-Verlag, New York,1995.
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[KR]P.P. Kulish, N.Yu. Reshetikhin, Universal R-matrix of the quantum superalgebra osp(2|1), Lett. Math. Phys.18(2) (1989) 143-149.
[La]T.Y. Lam, First Course in Noncommutative Rings. Graduate Texts in Mathe-matics, Vol.131, Springer-Verlag, New York,1991.
[Lu]G. Lusztig, Intoduction to quantum groups. Birkhauser. Boston.1993.
[LZ]H. Lv, S. Zhang, Global dimensions of endomorphism algebras of generator-cogenerators over m-replicated algebras. Accepted for publication in Comm. Al-gebra.
[Ma1]S. Majid, Tannaka-Krein theorem for quasi-Hopf algebras and other results, Contemp. Math.134 (1992) 219-232.
[Ma2]S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge,1995.
[MM]J.W. Milnor, J.C. Moore, On the structure of Hopf algebras, Ann. Math.,81 (1965) 211-264.
[Mo1]S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conf. Series in Math.82, Amer. Math. Soc, Providence, RI,1993.
[Mo2]S. Montgomery, Indecomposable coalgebras, simple comodules and pointed Hopf algebras, Proceedings of the AMS,123 (1995) 2343-2351.
[P]L. Peng, Some Hall polynomials for representation-finite trivial extension algebras. J. Algebra,197 (1997) 1-13.
[PX1]L. Peng, J. Xiao, A realization of affine Lie algebras of type An-1 via the derived categories of cycle quivers, Canadian Math. Soc. Conference Proc,8 (1996) 539-554.
[PX2]L. Peng, J. Xiao, Root categories and simple Lie algebras, J. Algebra,198 (1997) 19-56.
[PX3]L. Peng, J. Xiao, Triangulated categories and Kac-Moody algebras. Invent. math.,140 (2000)563-603.
[R]M. Rosso, Quantum groups and quantum shuffles, Invent.Math.,133(1998) 399-416.
[Rie]C.H. Riedtmann, Lie algebras generated by indecomposables. J. Algebra,170(1994) 526-546.
[Rin1]C M. Ringel, Tame algebras and integral quadratic forms. Springer Lecture Notes in Math., (1099),1984
[Rin2]C.M. Ringel, Hall algebra. In:Topics in Algebras. Banach Centre Publ. Warszawa.1990,26,433-447.
[Rin3]C.M. Ringel, Hall algebras and quantum groups. Invent. Math.,101 (1990) 583-592.
[Rin4]C.M. Ringel, Hall polynomials for the representation-finite hereditary algebras. Adv. Math.,84 (1990) 137-178.
[Rin5]C.M. Ringel, Green's theorem on Hall algebras. Canad. Math. Soc. Conf. Proc. Vol 19,185-245.
[Rin6]C.M. Ringel, Lie algebra arising in representation theory, London Math Soc, 168, Cambridge University Press,1992,284-291.
[Rin7]C.M. Ringel, The composition algebra of a cyclic quiver. Proc.London Math. Soc,1993,66(3):507-537.
[Rin8]C.M. Ringel, PBW-basis of quantum group. J.rein angew. Math.,470(1996) 51-88.
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[SS1]S. Shnider, S. Sternberg, Quantum Groups:From Coalgebras to Drinfeld Alge-bras, International Press Inc.,Boston,1993.
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[X]J. Xiao, Drinfeld double and Ringel-Green theory of Hall algebras, J. Algebra,190 (1997) 100-144.
[ZG]Y.Z. Zhang, M.D. Gould, Quasi-Hopf superalgebras and elliptic quantum super-groups, J. Math. Phys.40, no.10,5264(1999).
[Zp1]P. Zhang, Composition algebras of affine type. J. Algebra,206 (1998)505-540.
[Zp2]P. Zhang, Representations as elements in affine composition algebras, Trans. Amer. Math. Soc,353 (2000)1221-1249.
[Zp3]P. Zhang, Indecomposables in the composition algebra of the Kronecker algebra. Comm. Algebra,27 (1999) 4633-4639.
[Zp4]P. Zhang, PBW-basis for the composition algebra of the Kronecker algebra. J. Reine Angew. Math.,527 (2000) 97-116.
[Zp5]P. Zhang, Triangular decomposition of the composition algebra of the Kronecker algebra. J. Algebra,184(1996)159-174.
[Zr]R.B. Zhang, Three-manifold invariants arising from Uq(osp(1|2)),Modern Phys. Lett. A 9 (1994) 1453-1465.
[Zs1]S. Zhang, The Hall polynomials for tame quiver algebras. J. Algebra,239 (2001) 606-614.
[Zs2]S. Zhang, Triangular decomposition of tame non-simply-laced composition alge-bras. J. Algebra,241 (2001) 548-577.
[Zs3]S. Zhang, Degenerate Ringel-Hall algebras of finite connected valued Auslander-Reiten quivers. Comm. Algebra,29(2) (2001)477-492.
[Zs4]S. Zhang, Lie algebra determined by tame hereditary algebras (In Chinese). Chinese Ann. Math.,21(5)(A) (2000) 609-612.
[Zs5]S. Zhang, Tilting mutation and duplicated algebras. Comm. Algebra,37 (2009) 3516-3524.
[ZZ]P. Zhang, S. Zhang, Indecomposables as elements in affine composition algebras. J. Algebra,210 (1998) 614-629.
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[Ci]C. Cibils, A quiver quantum group, Comm. Math. Phys.,157 (1993) 459-477.
[CM]W. Chin, S. Montgomery, Basic coalgebras, Modular interfaces (Riverside, CA, 1995),41-47, AMS/IP Stud. Adv. Math.4, Amer. Math. Soc, Providence, RI, 1997.
[CR1]C. Cibils, M. Rosso, Algebres des chemins quantiques, Adv. Math.,125 (1997) 171-199.
[CR2]C. Cibils, M. Rosso, Hopf quivers, J. Algebra,254 (2002) 241-251.
[DH]H. Dong, H.-L. Huang, Hopf superquivers, submitted.
[Di]J. Dieudonne, Introduction to the theory of formal groups, Marcel Dekker, New York,1973.
[DR]V. Dlab, C.M. Ringel, Indecomposable representations of graphs and algebras. Mem.Amer.Math.Soc,173 (1976)
[Drl]V.G. Drinfeld, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol.1,2 (Berkeley, Calif.,1986),798-820, Amer. Math. Soc, Providence, RI,1987.
[Dr2]V.G. Drinfeld, Quasi-Hopf algebras, Leningrad Math. J.1 (1990) 1419-1457.
[DX]B. Deng, J. Xiao, On double Ringel-Hall algebras, J. Algebra,251(2002),110-149.
[DZ]H. Dong, S. Zhang, Ringel-Hall algebras of duplicated tame hereditary algebras. Preprint, arXiv:math. RT/1001.1188v1,2010.
[EG1]P. Etingof and S. Gelaki, On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic, Internat. Math. Res. Notices,7(1999) 387-394.
[EG2]P. Etingof and S. Gelaki, The classification of triangular semisimple and cosemisim-ple Hopf algebras and over an algebraically closed field, Internat. Math. Res. Notices,5(2000) 223-234.
[EG3]P. Etingof, S. Gelaki, Finite dimensional quasi-Hopf algebras with radical of codimension 2, Math. Res. Lett.11 (2004) 685-696.
[EG4]P. Etingof, S. Gelaki, On radically graded finite-dimensional quasi-Hopf alge-bras, Mosc. Math. J.5(2) (2005) 371-378.
[EG5]P. Etingof, S. Gelaki, Liftings of graded quasi-Hopf algebras with radical of prime codimension, J. Pure Appl. Algebra,205(2) (2006) 310-322.
[ENO]P. Etingof, D. Nikshych, V. Ostrik, On fusion categories, Ann.Math.,162 (2005)581-642.
[EO]P. Etingof, V. Ostrik, Finite tensor categories, Moscow Math. J.4 (2004) 627-654,782-783.
[Ga1]P. Gabriel, Unzerlegbare Darstellungen Ⅰ, Manuscripta Math.,6(1972)71-103.
[Ga2]P. Gabriel, Indecomposable representations Ⅱ, Symposia Mathematica, Vol. Ⅺ (Convegno di Algebra Gommutativa, INDAM, Rome,1971),81-104, Academic Press, London,1973.
[Ga3]P. Gabriel, Finite representation type is open, in:Representations of Algebras, Lecture Notes in Math., vol.488, Springer Verlag,1975, pp.132-155.
[Ge1]S. Gelaki, Basic quasi-Hopf algebras of dimension n3, J. Pure Appl. Algebra, 198(1-3) (2005) 165-174.
[Ger]M. Gerstenhaer, On the deformation of rings and algebras, Ann. Math.,79(2) (1964) 59-103.
[GP]J. Guo, L. Peng, Universal PBW-basis of Hall-Ringel algebras and Hall polyno-mials. J. Algebra,198(1997),339-351.
[Gr1]J.A. Green, Locally finite representations, J. Algebra,41 (1976) 137-171.
[Gr2]J.A. Green, Hall algebras,hereditary algebras and quantum groups.Invent. Math.,120 (1995)361-377.
[Gr3]E.L. Green, Constructing quantum groups and Hopf algebras from coverings, J. Algebra,176 (1995) 12-33.
[GSc]M. Gerstenhaber, S.D. Schack, Bialgebra cohomology, deformations, and quan-tum groups, Proc. Nat. Acad. Sci.,87 (1990) 478-481.
[GSo]E.L. Green,(?). Solberg, Basic Hopf algebras and quantum groups, Math. Z., 229 (1998) 45-76.
[Gu]J. Guo, The Hall polynomials of a cyclic serial algebra, Comm. Algebra,23 (1995) 743-751.
[GZ]E.L. Green, P. Zhang, Rigids as iterated skew commutators of simples. Algebr Represent Theory,9 (2006) 539-555.
[GZI1]M.D. Gould, Y.Z. Zhang, P.S. Isaac, Casimir invariants from quasi-Hopf (su-per)algebras, J. Math. Phys.41, no.1,547(2000).
[GZI2]M.D. Gould, Y.Z. Zhang, P.S. Isaac, On quasi-Hopf superalgebras, Comm. Math. Phys.,224 (2001)341-372.
[H1]H.-L. Huang, A simple proof for a theorem of Cartier,2008,submitted.
[H2]H.-L. Huang, Quiver approaches to quasi-Hopf algebras, J. Math. Phys.,50 (2009) 043501,9pp.
[H3]H.-L. Huang, From projective representations to quasi-quantum groups. Preprint, arXiv:math. RT/0903.1472,2009.
[HCZ]H.-L. Huang, H.-X. Chen, P. Zhang, Generalized Taft algebras, Algebra Colloq. 11(03) (2004) 313-320.
[HN]F. Hausser, F. Nill, Integral theory for quasi-Hopf algebras, math.QA/9904164.
[HRS]D. Happel, I. Reiten, S.O. Smal(?), Tilting in abelian categories and quasitilted algebras, Mem.Amer.Math.Soc,575,1996.
[HZ]Y. Han, D. Zhao, Superspecies and their representations, J. Algebra,321 (2009) 3668-3680.
[Kac]V. Kac, Lie superalgebras, Adv. Math.26 (1977) 8-96.
[Kar]G. Karpilovsky, Projecive Representations of Finite Groups, Marcel Dekker, New York,1985.
[Kas]C. Kassel, Quantum Groups, Graduate Texts in Math.155, Springer-Verlag, New York,1995.
[Ko]B. Kostant, Graded manifolds, graded Lie theory, and prequantization, in:Differ-ential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn,1975), pp.177-306. Lecture Notes in Math., Vol.570, Springer, Berlin, 1977.
[KR]P.P. Kulish, N.Yu. Reshetikhin, Universal R-matrix of the quantum superalgebra osp(2|1), Lett. Math. Phys.18(2) (1989) 143-149.
[La]T.Y. Lam, First Course in Noncommutative Rings. Graduate Texts in Mathe-matics, Vol.131, Springer-Verlag, New York,1991.
[Lu]G. Lusztig, Intoduction to quantum groups. Birkhauser. Boston.1993.
[LZ]H. Lv, S. Zhang, Global dimensions of endomorphism algebras of generator-cogenerators over m-replicated algebras. Accepted for publication in Comm. Al-gebra.
[Ma1]S. Majid, Tannaka-Krein theorem for quasi-Hopf algebras and other results, Contemp. Math.134 (1992) 219-232.
[Ma2]S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge,1995.
[MM]J.W. Milnor, J.C. Moore, On the structure of Hopf algebras, Ann. Math.,81 (1965) 211-264.
[Mo1]S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conf. Series in Math.82, Amer. Math. Soc, Providence, RI,1993.
[Mo2]S. Montgomery, Indecomposable coalgebras, simple comodules and pointed Hopf algebras, Proceedings of the AMS,123 (1995) 2343-2351.
[P]L. Peng, Some Hall polynomials for representation-finite trivial extension algebras. J. Algebra,197 (1997) 1-13.
[PX1]L. Peng, J. Xiao, A realization of affine Lie algebras of type An-1 via the derived categories of cycle quivers, Canadian Math. Soc. Conference Proc,8 (1996) 539-554.
[PX2]L. Peng, J. Xiao, Root categories and simple Lie algebras, J. Algebra,198 (1997) 19-56.
[PX3]L. Peng, J. Xiao, Triangulated categories and Kac-Moody algebras. Invent. math.,140 (2000)563-603.
[R]M. Rosso, Quantum groups and quantum shuffles, Invent.Math.,133(1998) 399-416.
[Rie]C.H. Riedtmann, Lie algebras generated by indecomposables. J. Algebra,170(1994) 526-546.
[Rin1]C M. Ringel, Tame algebras and integral quadratic forms. Springer Lecture Notes in Math., (1099),1984
[Rin2]C.M. Ringel, Hall algebra. In:Topics in Algebras. Banach Centre Publ. Warszawa.1990,26,433-447.
[Rin3]C.M. Ringel, Hall algebras and quantum groups. Invent. Math.,101 (1990) 583-592.
[Rin4]C.M. Ringel, Hall polynomials for the representation-finite hereditary algebras. Adv. Math.,84 (1990) 137-178.
[Rin5]C.M. Ringel, Green's theorem on Hall algebras. Canad. Math. Soc. Conf. Proc. Vol 19,185-245.
[Rin6]C.M. Ringel, Lie algebra arising in representation theory, London Math Soc, 168, Cambridge University Press,1992,284-291.
[Rin7]C.M. Ringel, The composition algebra of a cyclic quiver. Proc.London Math. Soc,1993,66(3):507-537.
[Rin8]C.M. Ringel, PBW-basis of quantum group. J.rein angew. Math.,470(1996) 51-88.
[S]M. Sweedler, Hopf Algebras, W. A. Benjamin, Inc., New York,1969.
[SS1]S. Shnider, S. Sternberg, Quantum Groups:From Coalgebras to Drinfeld Alge-bras, International Press Inc.,Boston,1993.
[SS2]S. Shnider, S. Sternberg, The cobar resolution and a restricted deformation theory for Drinfeld algebras, J. Algebra,169 (1994)343-366.
[VZ]F. Van Oystaeyen, P. Zhang, Quiver Hopf algebras, J. Algebra,280 (2004) 577-589.
[X]J. Xiao, Drinfeld double and Ringel-Green theory of Hall algebras, J. Algebra,190 (1997) 100-144.
[ZG]Y.Z. Zhang, M.D. Gould, Quasi-Hopf superalgebras and elliptic quantum super-groups, J. Math. Phys.40, no.10,5264(1999).
[Zp1]P. Zhang, Composition algebras of affine type. J. Algebra,206 (1998)505-540.
[Zp2]P. Zhang, Representations as elements in affine composition algebras, Trans. Amer. Math. Soc,353 (2000)1221-1249.
[Zp3]P. Zhang, Indecomposables in the composition algebra of the Kronecker algebra. Comm. Algebra,27 (1999) 4633-4639.
[Zp4]P. Zhang, PBW-basis for the composition algebra of the Kronecker algebra. J. Reine Angew. Math.,527 (2000) 97-116.
[Zp5]P. Zhang, Triangular decomposition of the composition algebra of the Kronecker algebra. J. Algebra,184(1996)159-174.
[Zr]R.B. Zhang, Three-manifold invariants arising from Uq(osp(1|2)),Modern Phys. Lett. A 9 (1994) 1453-1465.
[Zs1]S. Zhang, The Hall polynomials for tame quiver algebras. J. Algebra,239 (2001) 606-614.
[Zs2]S. Zhang, Triangular decomposition of tame non-simply-laced composition alge-bras. J. Algebra,241 (2001) 548-577.
[Zs3]S. Zhang, Degenerate Ringel-Hall algebras of finite connected valued Auslander-Reiten quivers. Comm. Algebra,29(2) (2001)477-492.
[Zs4]S. Zhang, Lie algebra determined by tame hereditary algebras (In Chinese). Chinese Ann. Math.,21(5)(A) (2000) 609-612.
[Zs5]S. Zhang, Tilting mutation and duplicated algebras. Comm. Algebra,37 (2009) 3516-3524.
[ZZ]P. Zhang, S. Zhang, Indecomposables as elements in affine composition algebras. J. Algebra,210 (1998) 614-629.