McKay箭图在群表示论、奇点理论和弦理论中起着重要作用，我们发现它在有限复杂度Koszul自入射代数分类中也十分重要，证明它具有仿射Dynkin图一些性质，可视为其推广。证明一般线性群的有限子群的McKay箭图是它与特殊线性群的交子群的McKay箭图的正则覆盖，且以其商群为群。证明一般线性群的有限子群在高一维空间特殊线性群嵌入的McKay箭图由原McKay箭图将Nakayama平移变为一条箭向得到。得到与一些cluster倾斜代数的联系。对分次自入射代数引入τ-mutation和τ-slice概念，这在Koszul情形推广了BGP反射或APR倾斜。我们研究一些代数和余代数的Hochschild 上同调，得到余代数C与代数A的张量积是内射的C-双余模的充要条件，给出拟缠绕结构的1次Hochschild上同调群平凡的充要条件。引入(*)-序列余代数，给出了余代数是(*)-序列的充要条件，并刻画其一般quiver。我们还研究了项链李代数，引入项链字的左右指标数组，并用其刻画一些有趣的子代数和一些特殊项链李代数的可解性和幂零性，给出sl(n)在项链李代数中的实现并讨论项链李代数同构的条件。

McKay quiver is very important in group representation, the theory of singularities and string theory. We find it is also important in the classification of selfinjective Koszul algebras of finite complexity, prove that it shares certain properties of affine Dynkin diagram, so can be regarded as a generalization of the later. We prove that the McKay quiver of a finite subgroup of a general linear group is a regular covering of that of its intersection with the special linear group, with the quotient group as the group. When a finite group of a general linear group is embedded in the special linear group with one more dimension, we show that the new McKay quiver is obtained from the old one by adding an arrow from each vertex to its Nakayama translation. We introduce τ-mutation and τ-slice for graded selfinjective algebra, these generalized BGP reflection or APR tilting in the Koszul case. We also study the Hochschild cohomology of some algebras and coalgebras, get sufficient and necessary condition for the tensor product of a coalgebra and an algebra to be injective bicomodule over the coalgebra, and we give a sufficient and necessary condition for the first Hochschild cohomology group to be trivial for a quasi entwining structure. We introduce a (*)-serial coalgebra, and find a sufficient and necessary condition for a coalgebra to be (*)-serial and furthermore characterize its ordinary quiver. In studying the necklace Lie algebra, we introduce the left and right index sets of a necklace word and using it to characterize some interesting subalgebras, especially, when a necklace algebra is solvable and nilpotent. We also show how to realize sl(n) in a necklace Lie algebra and discuss when necklace Lie algebras are isomorphic.

McKay箭图在群表示论、奇点理论和弦理论中起着重要作用，我们发现它在有限复杂度Koszul自入射代数分类中也十分重要，证明它具有仿射Dynkin图一些性质，可视为其推广。证明一般线性群的有限子群的McKay箭图是它与特殊线性群的交子群的McKay箭图的正则覆盖，且以其商群为群。证明一般线性群的有限子群在高一维空间特殊线性群嵌入的McKay箭图由原McKay箭图将Nakayama平移变为一条箭向得到。得到与一些cluster倾斜代数的联系。对分次自入射代数引入τ-mutation和τ-slice概念，这在Koszul情形推广了BGP反射或APR倾斜。我们研究一些代数和余代数的Hochschild 上同调，得到余代数C与代数A的张量积是内射的C-双余模的充要条件，给出拟缠绕结构的1次Hochschild上同调群平凡的充要条件。引入(*)-序列余代数，给出了余代数是(*)-序列的充要条件，并刻画其一般quiver。我们还研究了项链李代数，引入项链字的左右指标数组，并用其刻画一些有趣的子代数和一些特殊项链李代数的可解性和幂零性，给出sl(n)在项链李代数中的实现并讨论项链李代数同构的条件。