Cycle sets were introduced to reduce non-degenerate unitary Yang-Baxter maps to an algebraic system with a single binary operation. Every finite cycle set extends uniquely to a finite cycle set with a compatible abeli...Cycle sets were introduced to reduce non-degenerate unitary Yang-Baxter maps to an algebraic system with a single binary operation. Every finite cycle set extends uniquely to a finite cycle set with a compatible abelian group structure. Etingof et al. introduced affine Yang-Baxter maps. These are equivalent to cycle sets with a specific abelian group structure. Abelian group structures have also been essential to get partial results for the still unsolved retraction problem. We introduce two new classes of cycle sets with an underlying abelian group structure and show that they can be transformed into each other while keeping the group structure fixed. This leads to a proper extension of the retractibility conjecture and new evidence for its truth.展开更多
In this paper, we study the category H (ρ) of semi-stable coherent sheaves of a fixed slope ρ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category....In this paper, we study the category H (ρ) of semi-stable coherent sheaves of a fixed slope ρ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of H (ρ) and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.展开更多
Creative telescoping is the method of choice for obtaining information about definite sums or integrals. It has been intensively studied since the early 1990s, and can now be considered as a classical technique in com...Creative telescoping is the method of choice for obtaining information about definite sums or integrals. It has been intensively studied since the early 1990s, and can now be considered as a classical technique in computer algebra. At the same time, it is still a subject of ongoing research.This paper presents a selection of open problems in this context. The authors would be curious to hear about any substantial progress on any of these problems.展开更多
文摘Cycle sets were introduced to reduce non-degenerate unitary Yang-Baxter maps to an algebraic system with a single binary operation. Every finite cycle set extends uniquely to a finite cycle set with a compatible abelian group structure. Etingof et al. introduced affine Yang-Baxter maps. These are equivalent to cycle sets with a specific abelian group structure. Abelian group structures have also been essential to get partial results for the still unsolved retraction problem. We introduce two new classes of cycle sets with an underlying abelian group structure and show that they can be transformed into each other while keeping the group structure fixed. This leads to a proper extension of the retractibility conjecture and new evidence for its truth.
基金supported in part by NSF of China (Grant No. 10631010)NKBRPC (Grant No. 2006CB805905)
文摘In this paper, we study the category H (ρ) of semi-stable coherent sheaves of a fixed slope ρ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of H (ρ) and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.
基金supported by the National Natural Science Foundation of China under Grant No.11501552the President Fund of the Academy of Mathematics and Systems Science,CAS(2014-cjrwlzx-chshsh)+1 种基金a Starting Grant from the Ministry of Education of Chinasupported by the Austrian FWF under Grant Nos.F5004,Y464-N18,and W1214
文摘Creative telescoping is the method of choice for obtaining information about definite sums or integrals. It has been intensively studied since the early 1990s, and can now be considered as a classical technique in computer algebra. At the same time, it is still a subject of ongoing research.This paper presents a selection of open problems in this context. The authors would be curious to hear about any substantial progress on any of these problems.