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Representation type of local and two-vertex bocses

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Science China Mathematics 2009年第5期52卷 949-958页

作者： ZHANg XueYing ZHANg YingBo ZHAO ShuangMei School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China

Let k be an algebraically closed field. It has been proved by Zhang and Xu that if a bocs is of tame representation type, then the degree of the differential of the first solid arrow must be less than or equal to 3. We will prove in the present paper that: The bocs is still wild when the degree of the differential of the first arrow is equal to 3. Especially, the bocs with only one solid arrow is of tame type if and only if the degree of the differential of the arrow is less than or equal to 2. Moreover, we classify in this case the growth problems of the representation category of the bocs and layout the sufficient and necessary conditions when the bocs is of finite representation type, tame domestic and tame exponential growth respectively.

关键词： bocs reduction tame representation type wild representation type domestic 15A21 16g20 16g60 16g70

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Unified tame theorem

Science China Mathematics 2009年第9期52卷 2036-2068页

作者： ZHANg YingBo XU Yunge School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China School of Mathematics and Computer Science Hubei UniversityWuhan 430062China

The well-known tame theorem tells that for a given tame bocs and a positive integer n there exist finitely many minimal bocses, such that any representation of the original bocs of dimension at most n is isomorphic to the image of a representation of some minimal bocses under a certain reduction functor. In the present paper we will give an alternative statement of the tame theorem in terms of matrix problem, by constructing a unified minimal matrix problem whose indecomposable matrices cover all the canonical forms of the indecomposable representations of dimension at most n for each non-negative integer n.

关键词： matrix problem canonical form representation tame type indecomposable 15A21 16g20 16g60 16g70

Compact Dg modules and gorenstein Dg algebras

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Science China Mathematics 2009年第4期52卷 648-676页

作者： MAO XueFeng WU QuanShui School of Mathematical Sciences Fudan UniversityShanghai 200433China

When the base connected cochain Dg algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact Dg module and that of the Dg algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular Dg algebras.When A is a regular Dg algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul Dg algebra. For a general regular Dg algebra A, we give some equivalent conditions for the gorensteiness.For a finite connected Dg algebra A, we prove that Dc(A) and Dc(A op) admit Auslander-Reiten triangles if and only if A and A op are gorenstein Dg algebras. When A is a non-trivial regular Dg algebra such that H(A) is locally finite, Dc(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D lf b (A) and D lf b (A op) instead, when A is a regular Dg algebra.

关键词： differential graded algebra gorenstein Dg algebra regular Dg algebra Koszul Dg algebra compact Dg module Auslander-Reiten triangles amplitude projective dimension 16E10 16E45 16g70

Tubular algebras and affine Kac-Moody algebras

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Science China Mathematics 2007年第4期50卷 521-532页

作者： Zheng-xin CHEN & Ya-nan LIN School of Mathematics and Computer Science, Pujian Normal University, Fuzhou 350007, China School of Mathematical Sciences, Xiamen University, Xiamen 361005, China School of Mathematics and Computer Science Fujian Normal University Fuzhou China School of Mathematical Sciences Xiamen University Xiamen China

The purpose of this paper is to construct quotient algebras L(A) 1 ? /I(A) of complex degenerate composition Lie algebras L(A) 1 ? by some ideals, where L(A 1 ? is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A) 1 ? /I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra L re(A) 1 ? generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A) 1 ? generated by simple A-modules.

关键词： tubular algebras affine Kac-Moody algebras Hall algebras degenerate composition Lie algebras modules with real root 16g70 17B67