Cluster-Tilted Algebras of Type D n

Let H be a hereditary algebra of Dynkin type D _n over a field k and 𝒞 _H be the cluster category of H. Assume that n ≥ 5 and that T and T′ are tilting objects in 𝒞 _H . We prove that the cluster-tilted algebra Γ = End_{𝒞 H} (T)^op is isomorphic to Γ′ = End_{𝒞 H} (T′)^op if and only if T = τ ⁱ T′ or T = στ ^j T′ for some integers i and j, where τ is the Auslander–Reiten translation and σ is the automorphism of 𝒞 _H defined in Section 4.     

Relative Group Cohomology and The Orbit Category

Let G be a finite group and ? be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative ?-projective resolution for ? when ? is the family of all subgroups H ≤ G with rk H ≤ rkG ? 1. We answer this question negatively by calculating the relative group cohomology ?H*(G, 𝔽₂) where G = ?/2 × ?/2 and ? is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology ?H*(G, M) can be calculated using the ext-groups over the orbit category of G restricted to the family ?. In second part of the paper, we discuss the construction of a spectral sequence that converges to the cohomology of a group G and whose horizontal line at E ₂ page is isomorphic to the relative group cohomology of G.     

REGULARITY OF ENTROPY SOLUTIONS TO NONCONVEX SCALAR CONSERVATION LAWS WITH MONOTONE INITIAL DATA

We prove that for a given strictly increasing initial datum in Ck, the solution of the initial value problem is piecewise Ck smooth except for flux functions of nonconvex conservation laws in a certain subset of C^k＋1 of first category, defined in the range of the initial datum.     

奇点模型范畴的Quillen等价

令R是左Gorenstein环.我们构造了奇点反导出模型范畴和奇点余导出模型范畴(见文[Models for singularity categories,Adv Math.,2014,254:187-232])之间的Quillen等价.作为应用,给出了投射,内射模的正合复形的同伦范畴之间的一个具体的等价■.     

纯奇点范畴中的Buchweitz定理

我们定义纯奇点范畴D_(psg)~b(R)为有界纯导出范畴D_(pur)~b(R)与纯投射模构成的有界同伦范畴K~b(■)的Verdier商,得到了纯奇点范畴D_(psg)~b(R)三角等价于相对纯投射模的Gorenstein范畴的稳定范畴■的一个充分必要条件.同时,还给出三角等价D_(psg)~b(R)≌D_(psg)~b(S)的充分条件,这里R和S都是环.     

Cartesian Closed Categories of F Z-domains

A subset systemZassigns to each partially ordered setPa certain collectionZ(P) of subsets. In this paper, a new kind of subset systems called directable subset systems is introduced. For a directable subset system Z, the concepts of F Z-way-below relation and F Z-domain are introduced. The well-known Scott topology is naturally generalized to the Z-level and the resulting topology is calledF Z-Scott topology, and the continuous functions with respect to this topology are characterized by preserving the suprema of directed Z-sets. Then, we mainly consider a generalization of the cartesian closedness of the categories DCPO of directed complete posets, BF of bifinite domains and FS ofF Sdomains to the Z-level. Corresponding to them, it is proved that, for a suitable subset system Z, the categories FZCPO ofZ-complete posets,FSFZ of finitely separated FZ-domains andBFFZ of bifiniteF Z-domains are all cartesian closed. Some examples of these categories are given.     

Concretely Locally Presentable and Locally Generated Categories

We give concrete versions of the characterizations of locally -presentable (resp. -generated) categories as categories of models of (resp. weakly) -ary limit-theories and of (resp. weakly) -ary essentially algebraic theories. By concrete we mean that starting with a category of -structures, the theories obtained are extensions of the original ones, and the equivalences of categories are concrete isomorphisms. -presentable and -generated objects are also investigated from this viewpoint.     

三角Hopf代数表示范畴上的Lie余代数

本文在三角Ｈｏｐｆ代数表示范畴上系统地研究了Ｌｉｅ余代数,在此范畴上 的Ｌｉｅ余代数与Ｈｏｐｆ代数之间建立了重要的联系．主要给出了Ｌｉｅ余代数的余包络 余代数的结构．所得结果自然是关于Ｌｉｅ代数的对偶结果,推广了 Ｓｗｅｅｄｌｅｒ Ｍ． Ｅ．, Ｇｕｒｅｖｉｃｈ Ｄ．Ｉ．, Ｍｉｃｈａｅｌｉｓ Ｗ．和 Ｍａｉｉｄ Ｓ．等人的结果．     

关于μ~H中的Lie双代数和余Poisson-Hopf代数(英文)

本文研究余三角 Ｈｏｐｆ代数余模范畴中的 Ｌｉｅ双代数和余 Ｐｏｉｓｓｏｎ－Ｈｏｐｆ代数．我们主要讨论余三角Ｈｏｐｆ代数余模范畴中的Ｌｉｅ双代数和余Ｐｏｉｓｓｏｎ－Ｈｏｐｆ代数之间的关系．     

平衡范畴与半群的双序

研究范畴与半群通过幂等元双序建立的一种自然联系.对每个有幂等元的半群S,其幂等元生成的左、右主理想之集通过双序ω~e,ω~r自然确定两个有子对象、有像且每个包含都右可裂的范畴L(S),R(S),其中态射的性质与S中元素的富足性、正则性有自然对应.利用这个联系,我们定义了"平衡(富足、正规)范畴"概念.对任一平衡(富足、正规)范畴■,我们构造其"锥半群"■,证明■左富足(富足、正则),且每个平衡(富足、正规)范畴■都与某左富足(富足、正则)半群S的左主理想范畴L(S)(作为有子对象的范畴)同构.