Let be a connected finite quiver without oriented cycle, A=k（?） the corresponding path algebra with k being an algebraically closed field, A<sup>T</sup> a preprojective tilting module. B=End<sub>A&...Let be a connected finite quiver without oriented cycle, A=k（?） the corresponding path algebra with k being an algebraically closed field, A<sup>T</sup> a preprojective tilting module. B=End<sub>A</sub>T. Then B is called a tame （resp. wild）concealed algebra provided is an Euclidean （resp. wild ） graph. The following result is important in the representation theory of tame concealed algebras （see [1,4.9]）: if A is tame concealed, T= T<sub>0</sub>⊕ T<sub>1</sub> a tilting module with T<sub>0</sub> nonzero preprojective and T<sub>1</sub> regular, then End<sub>A</sub>T<sub>0</sub> is tame concealed. The main purpose of this note is to generalize it to the 'wild' case. For this we generally consider the endomorphism algebra of preprojective partial tilting modules over a concealed algebra. For the notations the readers can refer to Ref.[1].展开更多
Let (Г, I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group ? d . In this paper, we list all indecomposable representations of (θ, I) and give the conditions that those represent...Let (Г, I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group ? d . In this paper, we list all indecomposable representations of (θ, I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Пλ(Г, I). It is shown that those representations given by extending indecomposable representations of (Г, I) are all simple representations of Пλ(Г, I). Therefore, it is concluded that all simple representations of restricted quantum group ū q (sl 2) are realized in terms of deformed preprojective algebra.展开更多
It is well known that Hall polynomials as structural coefficients play an important role in the structure of Lie algebras and quantum groups. By using the properties of representation categories of affine quivers, the...It is well known that Hall polynomials as structural coefficients play an important role in the structure of Lie algebras and quantum groups. By using the properties of representation categories of affine quivers, the task of computing Hall polynomials for affine quivers can be reduced to counting the numbers of solutions of some matrix equations. This method has been applied to obtain Hall polynomials for indecomposable representations of quivers of type Am（m≥1）展开更多
We prove that the transition matrix between a special Poincaré-Birkhoff-Witt(PBW)basis and the semicanonical basis of U+(sln(C))is upper triangular and unipotent under any order which is compatible with the parti...We prove that the transition matrix between a special Poincaré-Birkhoff-Witt(PBW)basis and the semicanonical basis of U+(sln(C))is upper triangular and unipotent under any order which is compatible with the partial order deg.展开更多
基金Project supported by the National Natural Science Foundation of China
文摘Let be a connected finite quiver without oriented cycle, A=k（?） the corresponding path algebra with k being an algebraically closed field, A<sup>T</sup> a preprojective tilting module. B=End<sub>A</sub>T. Then B is called a tame （resp. wild）concealed algebra provided is an Euclidean （resp. wild ） graph. The following result is important in the representation theory of tame concealed algebras （see [1,4.9]）: if A is tame concealed, T= T<sub>0</sub>⊕ T<sub>1</sub> a tilting module with T<sub>0</sub> nonzero preprojective and T<sub>1</sub> regular, then End<sub>A</sub>T<sub>0</sub> is tame concealed. The main purpose of this note is to generalize it to the 'wild' case. For this we generally consider the endomorphism algebra of preprojective partial tilting modules over a concealed algebra. For the notations the readers can refer to Ref.[1].
基金supported by the National Natural Science Foundation of China (Grant Nos. 10671016, 10771014)the Beijing Natural Science Foundation (Grant No. 1062003)Science and Technology Program of Beijing Education Committee (Grant No. KM200710005013)
文摘Let (Г, I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group ? d . In this paper, we list all indecomposable representations of (θ, I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra Пλ(Г, I). It is shown that those representations given by extending indecomposable representations of (Г, I) are all simple representations of Пλ(Г, I). Therefore, it is concluded that all simple representations of restricted quantum group ū q (sl 2) are realized in terms of deformed preprojective algebra.
文摘It is well known that Hall polynomials as structural coefficients play an important role in the structure of Lie algebras and quantum groups. By using the properties of representation categories of affine quivers, the task of computing Hall polynomials for affine quivers can be reduced to counting the numbers of solutions of some matrix equations. This method has been applied to obtain Hall polynomials for indecomposable representations of quivers of type Am（m≥1）
基金supported by National Natural Science Foundation of China (Grant Nos.11171183 and 11371165)Program for Changjiang Scholars and Innovative Research Team in University (Grant No.IRT1264)
文摘We prove that the transition matrix between a special Poincaré-Birkhoff-Witt(PBW)basis and the semicanonical basis of U+(sln(C))is upper triangular and unipotent under any order which is compatible with the partial order deg.
