1、北京师范大学数学科学学院教育部“数学与复杂系统”重点实验室代数年活动之二:表示论研讨会,2014 年 11 月 1-2 日。Mini-workshop on Representation Theory (Nov. 1-2, 2014)ProgramNovember 1st November 2nd8:30-9:20Serge Bouc(Amiens)Bin Zhu(Beijing)9:30-10:20Wei Hu(Beijing)Dengming Xu(Tianjin)10:20-10:50 Tea10:50-11:40Fei Xu(Shantou)Alexander Zimmermann(A
2、miens)12:00-13:30 Lunch14:30-15:20Xianhui Fu(Changchun)15:30-16:20Baolin Xiong(Beijing)16:20-16:50 Tea16:50-17:40Jiaqun Wei(Nanjing)18:00-19:30 DinnerExcursion to Olympic Forest ParkAll lectures will be given in Room 1124 of the New Main Building at Beijing Normal University. (北京师范大学新主楼 1124 报告厅)Tit
3、les and AbstractsSerge Bouc (Universit de Picardie):Correspondence functorsIn this joint work with Jacques Thvenaz, we develop the representation theory of finite sets and correspondences: Let kC the category of finite sets, in which morphisms are k-linear combinations of correspondences (where k is
4、 a given commutative ring), and let F_k be the category of correspondence functors (over k), i.e. the (abelian) category of k-linear functors from kC to k-Modules. An important class of correspondence functors originates in lattices: to each finite lattice T, we associate the functor F_T of “functio
5、ns from a set to T“. We introduce a suitable k-linear category of lattices for which the assignment T - F_T becomes a fully faithful k-linear functor. We also show that the functor F_T is projective in F_k if and only if the lattice T is distributive. Thanks to our previous work on the algebra of es
6、sential relations on a finite set, we obtain a parametrization of the simple correspondence functors by triples (E,R,V) consisting of a finite set E, a partial order relation R on E, and a simple k-linear representation V of the automorphism group of (E,R). The case of a totally ordered lattice T is
7、 of particular interest: the endomorphism algebra of F_T turns out to be naturally isomorphic to a direct product of matrix algebras over k. As a consequence, when k is a field and R is a total order on E, the simple functor parameterized by (E, R, k) is also projective and injective. In general, we
8、 obtain an explicit description of the simple functor S indexed by the triple (E, R, V). In particular, the dimension of S(X) can be explicitly computed, for any finite set X.Xianhui Fu (Northeast Normal University):Powers of a special preenveloping idealLet $R$ be a ring, and let $mathcalJ$ be a sp
9、ecial preenveloping ideal in the category of left $R$-modules $R$-Mod. In the present talk, we will introduce the notion of the $alpha$-th power of $mathcalJ$, where $alpha$ is an arbitrary ordinal. This notion is used to develop further ideal approximation theory. We will prove a generalization of
10、Eklofs Lemma, and use it to prove that the $alpha$-th power of $mathcalJ$ is still special preenveloping. As an application, we show that if $R$ is a left coherent ring, and the class of pure-projective left $R$-modules is closed under extensions, then every $FP$-projective left $R$-module is pure-p
11、rojective.Wei Hu (Beijing Normal University):Derived equivalences and stable equivalences for general finite dimensional algebrasThis talk is based on my joint works with Prof. Changchang Xi. In the framework of general finite dimensional algebras, we will introduce the stable functor of a derived e
12、quivalence, and the almost $nu$-stable derived equivalences, which always induce a stable equivalence of Morita type. We will also give an inductive method to lift a stable equivalence of Morita type to a derived equivalence (also for general finite dimensional algebras). In particular, we show that
13、 every stable equivalence of Morita type between representation-finite algebras (not necessarily self-injective) lifts to a derived equivalence.Jiaqun Wei (Nanjing Normal University):Large support $tau$-tilting modulesWe introduce a large version of support $tau$-tilting modules over any ring. It is
14、 proved that there is a bijective correspondence between the equivalent classes of large support $tau$-tilting modules and two-term large silting complexes. We also show that there are close relations between large support $tau$-tilting modules and star modules. In particular, over a hereditary ring
15、, it is obtained that large support $tau$-tilting modules are equivalent to finendo quasi-tilting modules.Baolin Xiong (Beijing University of Chemical Technology):Gorenstein stable equivalencesIn this talk, we shall introduce a new equivalence relation, which is called Gorenstein stable equivalences
16、. We will give the definition and some examples of Gorenstein stable equivalences.Dengming Xu (Civil Aviation University of China):Homological dimensions and strongly idempotent idealsIn the representation theory of Artin algebras, the well known finitistic dimension conjecture says that every Artin
17、 algebra has finite finitistic projective dimension, so it is one of the main topics to study the finiteness of the finitistic dimension. Let $A$ be an Artin algebra and $e$ an idempotent in $A$. In this talk, we will study the relation among the finitistic dimension of the algebras $A, A/AeA$ and $
18、eAe$ under certain homological conditions on the ideal $AeA$. Precisely, we will provide a proof of the following result:Theorem. Suppose that $AeA$ is a strongly idempotent ideal with finite projective dimension. If the finitistic projective (or injective) dimension of $eAe$ and $A/AeA$ are finite,
19、 then the finitistic projective (or injective) dimension of $A$ is finite.Fei Xu (Shantou University):Local representation theory of finite transporter categoriesFinite transporter categories are considered and studied as generalized groups. Given a transporter category, we shall compare its represe
20、ntations with those of its transporter subcategories. We will carry on our tasks by using transporter category algebras and the Kan extensions among their module categories. The purpose of this work is to find a new way to examine the G-posets, and the local categories such as the orbit categories a
21、nd the fusion systems.In this talk, we shall demonstrate how certain classical setups and tools in the local representation theory of finite groups are generalized. We will show there is a Mackey formula, which leads to a satisfactory theory of relative projectivity and a Green correspondence. If ti
22、mes permits, we shall continue to investigate the block theory of transporter category algebras and establish a Brauer correspondence for blocks. Our approach is based on the work of Boisen, Dade and Miyashita etc on the fully group-graded algebras. In fact, a transporter category algebra is a skew
23、group algebra, and it is fully group-graded.Bin Zhu (Tsinghua University):Cotosion pairs in a 2-Calabi-Yau triangulated category and rooted cluster subalgebrasLet C be a 2-Calabi-Yau triangulated category with a cluster tilting object. We give a classification of (co-)torsion pairs in C, by proving
24、that the trivialness of t-structures in a connected 2-CY triangulated category with a cluster tilting object, and using Iyama-Yoshinos Calabi-Yau reduction. Let A_C be the rooted cluster algebra which is categorified by the cluster structure in C. We give a one-to-one correspondence between the set
25、of co-torsion pairs in C and the set of certain pairs of rooted cluster subalgebras of A_C. This talk bases on jointed works with Yu Zhou, and with Wen Chang.Alexander Zimmermann (Universit de Picardie):Geometrical degeneration in triangulated categoriesLet $A$ be a finite dimensional $k$-algebra ov
26、er a field $k$. An $A$-module of dimension $d$ is then given by a homomorphism of $A$ to the set of $d$ times $d$ matrices over $k$. This set is an affine variety and the general linear group $Gl_d(k)$ acts on it. Orbits correspond to isomorphism classes of modules. A module $M$ degenerates to a mod
27、ule $N$ if the point in the variety corresponding to $N$ belongs to the Zariski closure of the orbit of the point corresponding to $M$. Zwara and Riedtmann showed that $M$ degenerates to $N$ if and only if there is an $A$-module $Z$ and a short exact sequence $0rightarrow Zrightarrow Moplus Zrightar
28、row Nrightarrow 0$.Various attempts were given to generalize this setting and statement to more general situations. Yoshino studied stable categories of maximal Cohen-Macaulay modules.In joint work with Manuel Saorin we consider generally triangulated categories and model Yoshinos approach by a setting of functors between various triangulated categories. We prove for our setting a result analogous to the result of Zwara and Riedtmann. In an earlier paper with Jensen and Su we obtained a result on the partial order property of the so-defined degeneration.