Last updated: October 1, 2003

International Conference on Representations of Algebras and

Related Topics


Northeastern University
Boston, Massachusetts

October 3 - 4, 2003


This conference is a follow up to the Maurice Auslander Distinguished Lectures. Its goal is to give an opportunity to a diverse group of mathematicians attending the lectures to exchange ideas and help establish new contacts.


Speakers

T. Bruestle (Sherbrooke)
M. Butler (Liverpool)
A.V.V. Fonseca (Leicester)
H.-B. Foxby (Copenhagen)
A. Frankild (Copenhagen)
W.L. Gan (Cambridge, MA)
M. Kleiner (Syracuse, NY)
E. Marcos (Sao Paolo)
M. Neusel (Lubbock, TX)
I. Platzeck (Bahia Blanca)
M. J. Redondo (Bahia Blanca)
G. Todorov (Boston, MA)
H. Tyler (Manhattan College)
A. Vlassov (Minsk)
J. Weyman (Boston, MA)



For further information, contact Alex Martsinkovsky <alexmart@neu.edu>.



Schedule
Friday, October 3

Location:  325 Behrakis


2:00 - 2:20
J. Weyman
Some remarks on Rees algebras
2:30 - 2:50
A. Fonseca
Tilting Harish-Chandra modules
3:00 - 3:20
A. Vlassov
Deformations of the 2-category of inner categories


Coffee break
3:40 - 4:00
H. Tyler
Admissible sequences and the preprojective component of a quiver
4:10 - 4:30
I. Platzeck Trivial extensions of artin algebras
4:40 - 5:00
M.J. Redondo Hochschild cohomology of trivial extensions
5:10 - 5:30
M. Neusel Inseparable extension of algebras over the Steenrod algebra


Saturday, October 4

Location: 325 Behrakis


9:00 - 9:20
H.-B. Foxby
Auslander categories
9:30 - 9:50
A. Frankild
Adjoint functors and Gorenstein dimensions


Coffee break
10:20 - 10:40
M. Butler
On infinite rank integral representations of groups and orders of finite lattice type
10:50 - 11:10
G. Todorov
Tilting modules and exchange pairs
11:20 - 11:40
T. Bruestle On the representation dimension of tubular algebras


Lunch break
2:00 - 2:20
E. Marcos
N-Koszul algebras
2:30 - 2:50
W. L. Gan
Dioperads and Koszul duality
3:00 - 3:20
M. Kleiner
Hom-finite abelian categories as categories of comodules



Titles and Abstracts of Talks

  • T. Bruestle (University of Sherbrooke), On the representation dimension of tubular algebras.
Abstract: This is joint work with Thorsten Holm. In the talk, we recall Auslander's notion of the representation dimension of an algebra, explain our motivation to study this dimension for particular classes of algebras and outline some results for tubular algebras.
  • M. Butler (University of Liverpool), On infinite rank integral representations of groups and orders of finite lattice type.
Abstract: Let R be the integer group ring of a group of prime order. I will outline a proof (joint work with J.M. Campbell and L.G. Kovacs) that each R-module with FREE underlying abelian group is a direct sum of copies of the well-known indecomposable finite rank modules (R-lattices). We first show that it suffices to consider only countably generated modules, using an argument which works in a far wider context of suitably restricted modules over a quite general type of order of finite lattice type. However, for countably generated R-modules, we seemingly need to use the classical theory of R-lattices, and have been unable to obtain a similar description of (for example) countable rank integral representations of groups of composite squarefree order.
  • A.V.V. Fonseca (University of Leicester), Tilting Harish-Chandra modules.
Abstract:  We will establish the main properties of tilting modules over some special classes of  (finite dimensional) stratified algebras which appear in the classical theory of complex semisimple Lie algebras. These algebras are generalizations of the blocks of the famous Bernstein-Gelfand-Gelfand category.

  • H.-B. Foxby (University of Copenhagen), Auslander categories.
Abstract: Finiteness of Auslander's G-dimension of a finitely generated module over a commutative Noetherian ring is tantamount to membership in a certain category, the Auslander category. This has applications within the theory of homomorphisms of rings. The G in the G-dimension is for Gorenstein; there is also a Cohen-Macaulay version of most of this.

  • A. Frankild (University of Copenhagen), Adjoint functors and Gorenstein dimensions.
Abstract: For a large class of rings, including all those encountered in algebraic geometry, we establish the conjectured Morita-like equivalence, known as Foxby equivalence, between the full subcategory of complexes of finite Gorenstein flat dimension and that of complexes of finite Gorenstein injective dimension. This functorial description meets the expectations and delivers a series of new results, which allows us to establish a well-rounded theory for Gorenstein dimensions. This is joint work with Lars W. Christensen and Henrik Holm.

  • W. L. Gan (MIT), Dioperads and Koszul duality.
Abstract: The theory of Koszul duality for operads is analogous to the theory of Koszul duality for algebras. I will explain the notion of dioperads which generalizes the notion of operads. The reason for introducing this notion is to extend the theory of Koszul duality for operads to the situation involving both operations and cooperations.

  • M. Kleiner (Syracuse University), Hom-finite abelian categories as categories of comodules.
Abstract: The following are equivalent for a skeletally small abelian Hom-finite category over a field with enough injectives and each simple object being an epimorphic image of a projective object of finite length:
(a) Each indecomposable injective has a non-zero socle.
(b) The category is equivalent to the category of socle-finitely copresented right comodules over a right semiperfect and right cocoherent coalgebra such that each simple right comodule is socle-finitely copresented.
(c) The category has left almost split sequences.
The work is joint with Idun Reiten.


  • E. Marcos (University of Sao Paolo), N-Koszul algebras.
Abstract: This is joint work with, E. Green, R. Martinez and Pu Zhang. I will describe a class of algebras called by us N-Koszul algebras. They are the first natural generalization of the notion of Koszul algebra. There are various equivalent definition. We will stress the existence of a complex called the generalized Koszul complex. One of the statements is that an algebra is N-Koszul if and only if it is a quotient of a quiver algebra by an ideal generated by homogeneous elements of degree N and its Yoneda algebra is generated in degrees 0, 1 and 2. Also we show how to regrade its Yoneda algebra to make it a usual Koszul algebra.

  • M. Neusel (Texas Tech University), Inseparable extension of algebras over the Steenrod algebra.
Abstract: We will introduce algebras over the Steenrod algebra; what it means to be an inseparable extension in this category; and what we can say about them.

  • M.-I. Platzeck (Universidad Nacional del Sur), Trivial extensions of artin algebras.
Abstract: Let A be an artin algebra and T(A) the trivial extension of A by its injective cogenerator D(A). We will discuss the general problem of determining when two artin algebras have isomorphic trivial extensions, which is of  interest in tilting theory. The answer to this problem in the particular case of triangular schurian algebras has been one of the main tools in the classification of trivial extensions of finite representation type made by E. Fernández. This application required a description of the algebras having trivial extension B in terms of the quiver and relations for B.
In 1984 Wakamatsu proved that two artin algebras A and A' have isomorphic trivial extensions if and only if there exist an algebra S and an S-bimodule M such that  A  and A' are  the trivial extensions of S by M and D(M) respectively. Using this result we generalize the description given in the triangular schurian case to a wider class of algebras.

  • M. J. Redondo (Universidad Nacional del Sur), Hochschild cohomology of trivial extensions.
Abstract: We describe a long exact sequence computing the Hochschild cohomology of a trivial extension. We study the connecting homomorphism using the cup product, and we infer several results, in particular the first Hochschild cohomology group of a trivial extension never vanishes. We apply the results obtained to determine the dimension of the first Hochschild cohomology group of the trivial extension of a monomial algebra.

  • G. Todorov (Northeastern University), Tilting modules and exchange pairs.
Abstract: We show for hereditary artin algebras of Dynkin type that two nonisomorphic indecomposable modules M and M* form an exchange pair, i.e. there is a module T such that M + T and M* + T are both tilting modules, precisely when dim Ext(M, M*) + dim Ext(M*, M) = 1.

  • H. Tyler (Manhattan College), Admissible sequences and the preprojective component of a quiver.
Abstract: This talk concerns indecomposable preprojective modules over the path algebra of a finite connected quiver without oriented cycles. For each such module, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the module. An efficient way to compute the module is to recover it from its shortest (+)-admissible sequence. The set of equivalence classes of the above sequences has a natural structure of a partially ordered set. For a large class of quivers, the Hasse diagram of the partially ordered set is isomorphic to the preprojective component of the Auslander-Reiten quiver. The techniques of (+)-admissible sequences yield a new result about slices in the preprojective component. This is joint work with M. Kleiner.

  • A. Vlassov (Belarus State University), Deformations of the 2-category of inner categories.
Abstract: In the definition of a category there are a multiplication and, implicitly, a comultiplication. The latter is just the diagonal map on the objects. To deform this comultiplication we consider inner categories in a monoidal category with some special properties. Then we prove that the corresponding deformed structures form a 2-category. Thus we get a class of deformations of 2-categories. These structures may be used to describe quantum systems appearing in topological quantum field theory (TQFT).

  • J. Weyman (Northeastern University), Some remarks on Rees algebras.
Abstract: I will discuss the notion of Rees algebra R(M) of a module M over a commutative ring A. The main result is that if M is a module of cycles in the Koszul complex, then the algebra R(M) is Cohen-Macaulay, with rational singularities. This answers some questions posed by J. Herzog.
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