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1.
A general framework for cluster tilting is set up by showing that any quotient of a triangulated category modulo a tilting
subcategory (i.e., a maximal 1-orthogonal subcategory) carries an induced abelian structure. These abelian quotients turn
out to be module categories of Gorenstein algebras of dimension at most one.
相似文献
2.
Marco Porta 《Advances in Mathematics》2010,225(3):1669-1715
The Popescu-Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differential graded categories. An analogous result for topological triangulated categories has recently been obtained by A. Heider. 相似文献
3.
4.
XiaoJuan Zhao 《中国科学 数学(英文版)》2014,57(11):2329-2334
Let A and B be finite-dimensional algebras over a field k of finite global dimension. Using some results of Gorsky in “Semi-derived Hall algebras and tilting invariance of Bridgeland-Hall algebras”, we prove that if A and B are derived equivalent, then the corresponding m-periodic derived categories are triangulated equivalent. 相似文献
5.
A Jordan–Hölder theorem is established for derived module categories of piecewise hereditary algebras, in particular for representations of quivers and for hereditary abelian categories of a geometric nature. The resulting composition series of derived categories are shown to be independent of the choice of bounded or unbounded derived module categories, and also of the choice of finitely generated or arbitrary modules. 相似文献
6.
Sota Asai 《Algebras and Representation Theory》2018,21(3):635-681
We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable module categories have a structure of triangulated categories. Our main result determines the Grothendieck groups of these stable module categories. As an application, we give a complete classification of the mesh algebras up to stable equivalences. 相似文献
7.
《Journal of Pure and Applied Algebra》2022,226(4):106886
We describe the structure and properties of the finite-dimensional symmetric algebras over an algebraically closed field K which are socle equivalent to the general weighted surface algebras of triangulated surfaces, investigated in [11]. In particular, we prove that all these algebras are tame periodic algebras of period 4. The main results of this paper form an essential step towards a classification of all symmetric tame periodic algebras of period 4. 相似文献
8.
Zhi-Wei Li 《代数通讯》2013,41(9):3725-3753
Beligiannis and Marmaridis in 1994, constructed the one-sided triangulated structures on the stable categories of additive categories induced from some homologically finite subcategories. We extend their results to slightly more general settings. As an application of our results, we give some new examples of one-sided triangulated categories arising from abelian model categories. An interesting outcome is that we can describe the pretriangulated structures of the homotopy categories of abelian model categories via those of stable categories. 相似文献
9.
Triangulated categories and Kac-Moody algebras 总被引:7,自引:0,他引:7
By using the Ringel-Hall algebra approach, we find a Lie algebra arising in each triangulated category with T
2=1, where T is the translation functor. In particular, the generic form of the Lie algebras determined by the root categories, the 2-period
orbit categories of the derived categories of finite dimensional hereditary associative algebras, gives a realization of all
symmetrizable Kac-Moody Lie algebras.
Oblatum 4-XII-1998 & 11-XI-1999?Published online: 21 February 2000 相似文献
10.
Extriangulated category was introduced by H.Nakaoka and Y.Palu to give a unification of properties in exact categories anjd triangulated categories.A notion of tilting(resp.,cotilting)subcategories in an extriangulated category is defined in this paper.We give a Bazzoni characterization of tilting(resp.,cotilting)subcategories and obtain an Auslander-Reiten correspondence between tilting(resp.,cotilting)subcategories and coresolving covariantly(resp.,resolving contravariantly)finite subcatgories which are closed under direct summands and satisfy some cogenerating(resp.,generating)conditions.Applications of the results are given:we show that tilting(resp.,cotilting)subcategories defined here unify many previous works about tilting modules(subcategories)in module categories of Artin algebras and in abelian categories admitting a cotorsion triples;we also show that the results work for the triangulated categories with a proper class of triangles introduced by A.Beligiannis. 相似文献
11.
We classify Nichols algebras of irreducible Yetter–Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known finite-dimensional Nichols algebras of nonabelian group type appear in the result of our classification. We find a new finite-dimensional Nichols algebra over fields of characteristic two. 相似文献
12.
《Journal of Pure and Applied Algebra》2022,226(11):107092
We study the existence of almost split sequences in tri-exact categories, that is, extension-closed subcategories of triangulated categories. Our results unify and extend a number of existence theorems for almost split sequences in abelian categories and exact categories (that is, extension-closed subcategories of abelian categories), and those for almost split triangles in triangulated categories by numerous researchers. As applications, we obtain some new results on the existence of almost split triangles in the derived categories of all modules over an algebra with a unity or a locally finite dimensional algebra given by a quiver with relations. 相似文献
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14.
Robert Laugwitz 《代数通讯》2017,45(8):3653-3666
In this note, we apply classification results for finite-dimensional Nichols algebras to generalizations of Fomin–Kirillov algebras to complex reflection groups. First, we focus on the case of cyclic groups where the corresponding Nichols algebras are only finite-dimensional up to order four, and we include results about the existence of Weyl groupoids and finite-dimensional Nichols subalgebras for this class. Second, recent results by Heckenberger–Vendramin [ArXiv e-prints, 1412.0857 (December 2014)] on the classification of Nichols algebras of semisimple group type can be used to find that these algebras are infinite-dimensional for many non-exceptional complex reflection groups in the Shephard–Todd classification. 相似文献
15.
Dmitry Dubnov 《代数通讯》2013,41(9):4355-4374
We investigate finite dimensional 2-vertex basic algebras of finite global dimension and the derived categories of modules over such algebras. We prove that any superrigid object in the derived category of modules over a “loop-kind” two-vertex algebra is a pure module up to the action of Serre functor and translation. All superrigid objects in the derived categories of modules over two-vertex algebras of global dimension 2 are described. Also we obtain a complete classification of two-vertex basic algebras possessing a full exceptional pair in the derived category of modules. 相似文献
16.
We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair associated with these extensions, we construct a cohomology group, which classifies all the extensions up to equivalence. Moreover, we present a 1–1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points, where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted Lie algebra extensions. 相似文献
17.
We study the problem of lifting and restricting TTF triples (equivalently, recollement data) for a certain wide type of triangulated
categories. This, together with the parametrizations of TTF triples given in Nicolás and Saorín (Parametrizing recollement
data for triangulated categories. To appear in J. Algebra), allows us to show that many well-known recollements of right bounded
derived categories of algebras are restrictions of recollements in the unbounded level, and leads to criteria to detect recollements
of general right bounded derived categories. In particular, we give in Theorem 1 necessary and sufficient conditions for a
right bounded derived category of a differential graded (=dg) category to be a recollement of right bounded derived categories of dg categories.
Theorem 2 considers the case of dg categories with cohomology concentrated in non-negative degrees. In Theorem 3 we consider
the particular case in which those dg categories are just ordinary algebras. 相似文献
18.
Mohammed Guediri 《代数通讯》2013,41(7):2919-2937
We discuss locally simply transitive affine actions of Lie groups G on finite-dimensional vector spaces such that the commutator subgroup [G, G] is acting by translations. In other words, we consider left-symmetric algebras satisfying the identity [x, y]·z = 0. We derive some basic characterizations of such left-symmetric algebras, and we highlight their relationships with the so-called Novikov algebras and derivation algebras. 相似文献
19.
We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponent p, p>17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius-Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. An important step in the classification result is to show that all these Hopf algebras are generated by group-like and skew-primitive elements. 相似文献
20.
The finite dimensional tame hereditary algebras are associated with the extended Dynkin diagrams. An indecomposable module over such an algebra is either preprojective or preinjective or lies in a family of tubes whose tubular type is the corresponding Dynkin diagram. The study of one-point extensions by simple regular modules in such tubes was initiated in [Ri]. We generalise this approach by starting out with algebras which are derived equivalent to a tame hereditary algebra and considering one-point extensions by modules which are simple regular in tubes in the derived category. If the obtained tubular type is again a Dynkin diagram these algebras are called derived Dynkin extensions. Our main theorem says that a representation infinite algebra is derived equivalent to a tame hereditary algebra iff it is an iterated derived Dynkin extension of a tame concealed algebra. As application we get a new proof of a theorem in [AS] about domestic tubular branch enlargements which uses the derived category instead of combinatorial arguments. 相似文献