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On the Affine McKay Quiver of Type (A_n|～)

Author: PengLiBo
Tutor: GuoJinYun
School: Hunan Normal University
Course: Basic mathematics
Keywords: McKay quiver Skew group algebras Nakayama translation Affine Dynkin graphs of (A_n) ～
CLC: O157.5
Type: Master's thesis
Year: 2009
Downloads: 38
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Abstract


In 1980, McKay introduces the concept of McKay quiver. He also observes that for a finite subgroup G of SL(2, C), the McKay quiver is one of the extended Dynkin graphs (?) and classical McKay correspondence asserts that there is one to one correspondence between representations of finite Subgroup G of SL(2, C) and the cohomology of the minimal resolution of Klein singularities C~2/G. Affine Dynkin graphs and the Dynkin graphs are very common graphs in math classification, such as complex semi-simple Lie algebras, Quantum Groups, and the representation algebras , especially the type A_n that Plays an important role.Guo[6] has recently pointed out that McKay quiver can be seen as the promotion of the Affine Dynkin graphs and some tectonic approaches. In this paper, the results prove that the affine McKay quiver of (?) can be acquires according to the method of Skew group algebras. From the MaKay quiver of the finite subgroup of SL(1,C) whose MaKay quiver is (?), we obtain the affine McKay quiver of (?) by structuredly covering the finite group of GL(1, C) and embedding the SX(2, C).In fact, the only finite subgroup of SL(1, C) is trivial group with the McKay quiver being (?). By constructing the McKay quiver of the finite subgroup of GL(1, C), we have gotten the cycle quiver covering (?). and we regard it as a G on the quiver of exterior algebra’s skew group algebras. Then we calculated its Nakayama translation. Finally, we have it embedded into SL(2,C) to get the cycle subgroup of SL(2, C). According to Guo[6] we know that its quiver is the McKay quiver of type(?).

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