Tilt-Critical Algebras of Tame Type

Let A be a finite dimensional k-algebra over an algebraically closed field. Assume A = kQ/I where Q is a quiver without oriented cycles. We say that A is tilt-critical if it is not tilted but every proper convex subcategory of A is tilted. We describe the tilt-critical algebras which are strongly simply connected and tame.     

Tame and Wild Coordinates of

Let be a field of characteristic zero. We characterize coordinates and tame coordinates in , i.e. the images of respectively under all automorphisms and under the tame automorphisms of . We also construct a new large class of wild automorphisms of which maps to a concrete family of nice looking polynomials. We show that a subclass of this class is stably tame, i.e. becomes tame when we extend its automorphisms to automorphisms of .

     

On Ringel–Hall Algebras of Tame Hereditary Algebras

In this paper, the double Ringel–Hall algebras of tame hereditary algebras are decomposed as the quantized enveloping algebras of the infinite-dimensional Lie algebras, which are the central extensions of the affine loop algebras and the infinite-dimensional Heisenberg algebras. The numbers of the generators of the Heisenberg algebras are explicitly given at each dimensional level.     

Derived-tame Tree Algebras

In this note we classify the derived-tame tree algebras up to derived equivalence. A tree algebra is a basic algebra A = kQ/I whose quiver Q is a tree. The algebra A is said to be derived-tame when the repetitive category  of A is tame. We show that the tree algebra A is derived-tame precisely when its Euler form _A is non-negative. Moreover, in this case, the derived equivalence class of A is determined by the following discrete invariants: The number of vertices, the corank and the Dynkin type of _A . Representatives of these derived equivalence classes of algebras are given by the following algebras: the hereditary algebras of finite or tame type, the tubular algebras and a certain class of poset algebras, the so-called semichain-algebras which we introduce below.     

Standard Modules for Tabular Algebras

We introduce cell modules for the tabular algebras defined in a previous work; these modules are analogous to the representations arising from left Kazhdan–Lusztig cells. The standard modules of the title are constructed in an elementary way by suitable tensoring of the cell modules. We show how a certain extended affine Hecke algebra of type A equipped with its Kazhdan–Lusztig basis is an example of a tabular algebra, and verify that in this case our standard modules coincide with other standard modules defined in the literature.     

Extensions of Modules over Schur Algebras, Symmetric Groups and Hecke Algebras

We study the relation between the cohomology of general linear and symmetric groups and their respective quantizations, using Schur algebras and standard homological techniques to build appropriate spectral sequences. As our methods fit inside a much more general context within the theory of finite-dimensional algebras, we develop our results first in that general setting, and then specialize to the above situations. From this we obtain new proofs of several known results in modular representation theory of symmetric groups. Moreover, we reduce certain questions about computing extensions for symmetric groups and Hecke algebras to questions about extensions for general linear groups and their quantizations.     

Geometry of Periodic Modules over Tame Concealed and Tubular Algebras

Let A be a tame concealed or tubular algebra and d the dimension-vector of a periodic module with respect to the action of the Auslander–Reiten translation. We prove that the affine variety mod _A (d) of all A-modules of dimension-vector d is a normal complete intersection. Moreover, we show that a module M in mod _A (d) is nonsingular if and only if Ext _A ²(M,M)=0.     

Modules Over Cluster-Tilted Algebras Determined by Their Dimension Vectors

We prove that indecomposable transjective modules over cluster-tilted algebras are uniquely determined by their dimension vectors. Similarly, we prove that for cluster-concealed algebras, rigid modules lifting to rigid objects in the corresponding cluster category are uniquely determined by their dimension vectors. Finally, we apply our results to a conjecture of Fomin and Zelevinsky on denominators of cluster variables.     

Verma-Type Modules for Quantum Affine Lie Algebras

Let be an untwisted affine Kac–Moody algebra and M_J() a Verma-type module for with J-highest weight P. We construct quantum Verma-type modules M_J^q() over the quantum group , investigate their properties and show that M_J^q() is a true quantum deformation of M_J() in the sense that the weight structure is preserved under the deformation. We also analyze the submodule structure of quantum Verma-type modules. Presented by A. VerschorenMathematics Subject Classifications (2000) 17B37, 17B67, 81R50.The first author is a Regular Associate of the ICTP. The third author was supported in part by a Faculty Research Grant from St. Lawrence University.     

The Category of Firm Modules for Nonunital Monomial Algebras

Let k be a field and X a set and P be a set of words over X. Consider the free nonunital k-algebra over X generated by the nonempty words over X and let R be the quotient of this algebra modulo the ideal generated by the words in P. R is called a “nonunital monomial algebra”. A right R-module M is said to be “firm” if M? _R R → M given by m ? r? mr is an isomorphism. In this article we prove that if R is a nonunital monomial algebra, the category of firm modules is Grothendieck.     

Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( ) containing u( ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G _r ) u( ) have also been constructed, and the results are stated in this setting.     

Simple Weight Modules of Non-Noetherian Generalized Down-Up Algebras

We classify all simple weight modules of non-Noetherian generalized down-up algebras.     

Homological Properties of Modules Over Group Algebras

Let G be a locally compact group, and let L¹ (G) be the Banachalgebra which is the group algebra of G. We consider a varietyof Banach left L¹ (G)-modules over L¹ (G), and seek to determineconditions on G that determine when these modules are eitherprojective or injective or flat in the category. The answerstypically involve G being compact or discrete or amenable. Forexample, in the case where G is discrete and 1 < p < ,we find that the module ^p (G) is injective whenever G is amenable,and that, if it is amenable, then G is ‘pseudo-amenable’,a property very close to that of amenability. 2000 MathematicsSubject Classification 46H25, 43A20.     

Universal Deformation Rings of Modules over Self-injective Cluster-Tilted Algebras Are Trivial

Let k be a fixed algebraically closed field of arbitrary characteristic,let Λ be a finite dimensional self-injective k-algebra,and let V be an indecomposable non-projective left Λ-module with finite dimension over k.We prove that if τΛV is the Auslander-Reiten translation of V,then the versal deformation rings R(Λ,V)and R(Λ,τΛV)(in the sense of F.M.Bleher and the second author)are isomorphic.We use this to prove that if Λ is further a cluster-tilted k-algebra,then R(Λ,V)is universal and isomorphic to k.     

仿射李代数的水平零的Whittaker模

对任意的仿射李代数(g),作者构造了一类水平为零的imaginary Whittaker(g)-模.同时证明了这类模在某些给定条件下是单的.     

Dn型路代数倾斜模与AR箭图边缘

代数表示理论是上个世纪七十年代初兴起的代数学的—个新的分支,而倾斜理论是研究代数表示理论的重要工具之一．本文主要对Dn型路代数倾斜模在其对应的AR-箭图上的结构特点进行研究．通过对Dn型路代数A的AR-箭图ΓA分析,证明了：Dn型路代数倾斜模T的—个必要条件是。〈T〉中至少有三个边缘点．     

Row and Column Removal Theorems for Homomorphisms of Specht Modules and Weyl Modules

We prove a q-analogue of the row and column removal theorems for homomorphisms between Specht modules proved by Fayers and the first author [16]. These results can be considered as complements to James and Donkin’s row and column removal theorems for decomposition numbers of the symmetric and general linear groups. In this paper we consider homomorphisms between the Specht modules of the Hecke algebras of type A and between the Weyl modules of the q-Schur algebra.This research was supported by ARC grant DP0343023. The first author was also supported by a Sesqui Research Fellowship at the University of Sydney.     

Computing the Hochschild Cohomology Groups of Some Families of Incidence Algebras

The purpose of this article is to present some computations of Hochschild cohomology groups of particular classes of incidence algebras using one-point extensions and one-point coextensions.