International Conference on Representations of Algebras and
Related Topics
April 23, 25 - 28, 2009
Woods Hole, Massachusetts, USA
Titles and Abstracts
Claire Amiot, A generalization of cluster categories.
Abstract:
In 2005 Buan, Marsh, Reineke, Reiten and Todorov have introduced the
cluster category associated with an acyclic quiver. Their aim was to
categorify acyclic cluster algebras. In this talk I will define these
categories and show how it is possible to generalize this construction
replacing an acyclic quiver by a finite dimensional algebra of global
dimension 2, or by a quiver with potential.
Frauke Bleher, Finiteness theorems for deformations of complexes.
Abstract: This talk is about joint work with Ted Chinburg on a
generalization to complexes of Mazur's deformation theory for modules
for a profinite group. A new question arises when deforming complexes:
Can the versal deformation be specified by a finite amount of linear
algebra information with coefficients in the versal deformation ring?
In this talk, I will describe some evidence that this is the case when
the complex arises from arithmetic in a suitable sense.
Thomas Bruestle, From Christoffel words to Markoff numbers.
Abstract:
For a pair (a,b) of relatively prime natural numbers, the Christoffel
word C(a,b) is defined by the path with integral vertices which is
closest to the line segment from (0,0) to (a,b). Viewing this line
segment as an arc in the once-punctured torus, we define a J-module M(a,b) for
each Christoffel word. Here J is the Jacobian algebra of the
once-punctured torus. We show that one obtains the Markoff number
associated with C(a,b) by counting submodules of M(a,b).
Aslak Buan, Cluster structures from tubes.
Abstract:
A tube is a uniserial abelian category which e.g. can be realized as
the nilpotent representations over a quiver which is an oriented cycle.
We study the cluster category of a tube. It does not have tilting
objects. However, we show that the set of maximal rigid object has a
nice combinatorial structure, namely that of a cluster algebra of type
B. We point out that this is a special case of a more general
construction of cluster structures from sets of maximal rigid objects
in cluster categories. Based on joint work with Marsh and Vatne.
Giovanni Cerulli, Euler-Poincaré characteristic of quiver grassamannians associated with thinly graded modules.
Abstract:
We discuss a class of quiver representations called thinly graded. For
some thinly graded representaions we consider the associated quiver
grassmannians and we produce a cellular decomposition of them. As a
consequence of this, we give a technique to compute combinatorially
their Euler-Poincaré characteristic. The generalization of this construction to every thinly graded module is a work in progress with Francesco Esposito.
Calin Chindris, Cluster fans for quivers.
Abstract: The cluster fan of a quiver without oriented cycles is the
(possibly infinite) fan on the set of almost positive real Schur roots
whose cones are generated by the so-called compatible subsets. In this
talk, I will present a description of the cluster fan of a quiver in
terms of certain stability conditions of the quiver in question. I will
also explain how our results can be used to derive the
Igusa-Orr-Todorov-Weyman's description of the (N-1)-skeleton of the
cluster fan of a Dynkin quiver with N vertices.
Ernst Dieterich, Real division algebras, restricted quiver representations,and Euclidean configurations.
Abstract: A real division algebra is a non-zero real vector space A, endowed with an R-bilinear multiplication
A × A → A such that the left and right multiplications
by nonzero elements are invertible. A famous theorem of Hopf
(1940) and Bott, Milnor, Kervaire(1958) states that every finite
dimensional real division algebra has dimension 1, 2, 4, or 8. The
problem of classifying all finite dimensional real division algebras up
to isomorphism is solved in the dimensions 1 and 2, but only
partially solved in the dimensions 4 and 8.
While these partial solutions historically emerged from diverse
approaches and techniques, developed by numerous specialists during
half a century, they are at present being understood to follow a common
pattern that ``locally'' relates real division algebras in a
first step to modules over an associative algebra, and in a second step
to configurations in a Euclidean space, in terms of equivalences of
categories. Thus unforeseen connections between non-associative
algebras, modules over associative algebras, and Euclidean geometry
emerge from the attempt to classify real division algebras. In return,
these connections actually enable the classification of certain types
of real division algebras.
I will explain this common pattern and exemplify it by revisiting some
of the known partial classifications of real division algebras under
its unifying perspective.
Audrey Doughty, The Auslander and Ringel-Tachikawa Theorem for submodules embeddings.
Abstract: Auslander and Ringel-Tachikawa have shown that for an
artinian ring R of finite representation type, every R-module is the
direct sum of finitely generated indecomposable R-modules. In
this talk, we will adapt this result to finite representation type full
subcategories of the module category of an artinian ring which are
closed under subobjects and direct sums and contain all projective
modules. In particular, the results in this paper hold for
subspace representations of a poset, in case this category is of finite
representation type.
Christof Geiss, Categorification of the Chamber Ansatz.
Abstract: For an adaptable element w of the Weyl group W the
cluster algebra structure on the coordinate ring of the unipotent
cell Uw is categorified by a subcategory Cw
of the modules over the corresponding preprojective algebra. Under the
cluster character the initial seed consisting of certain generalized
minors corresponds to a canonical cluster tilting object Tv in Cw. In order to solve for Cw
Berenstein, Fomin and Zelevinsky introduced twisted minors. We show
that these twisted minors correspond essentially to the inverse of
Auslander-Reiten translate of the summands of Tv. (joint work with B. Leclerc and Jan Schroeer)
Ellen Kirkman, Invariant Subrings of Regular Algebras under Hopf Algebra Actions.
Abstract: The Shephard-Todd-Chevalley Theorem states that if a finite
group G acts on a commutative polynomial ring A = k[V] as elements of GLn(V ), then the ring of invariants AG
is a polynomial ring if and only if G is generated by reflections. In
the same context Watanabe's Theorem states that if G acts on A as
elements of SLn(V), then the ring of invariants AG
is a Gorenstein ring. We consider generalizations of these theorems to
the noncommutative setting where A is a noetherian Artin-Schelter
regular algebra with a finite group G acting linearly on A. More
generally, we consider actions on A by a finite dimensional semi-simple Hopf algebra H, where each homogeneous component Aj is an H-module and A is an H-module algebra. (with James Kuzmanovich and James Zhang)
Helmut Lenzing, Stable categories of vector bundles on weighted projective lines.
Abstract: TBA
Gregg Musiker, Positivity results for cluster algebras from surfaces.
Abstract: We give combinatorial formulas for cluster algebras
with principal coefficients coming from triangulated surfaces (with or
without punctures), as well as some cluster algebras obtained by
``folding''. In particular, this proves the positivity conjecture
of Fomin and Zelevinsky for such cluster algebras, including those of
classical type. This is joint work with Ralf Schiffler and Lauren
Williams.
Markus Schmidmeier, The entries in the LR-Tableau.
Abstract: Let Γ be the Littlewood-Richardson tableau
corresponding to an embedding M of a subgroup in a finite
abelian p-group. Each individual entry in Γ yields
information about the module structure of subquotients of M, and
about the position of M within the category of embeddings.
Jeanne Scott, Laurent expansions for twisted Pluecker coordinates via perfect matchings.
Abstract:
I will explain how to compute Laurent expansions of twisted Pluecker
coordinates with respect to a cluster of the homogeneous coordinate
ring of the Grassmannian Gr{k,n} associated to a
Postnikov diagram. This expansion formula is described in terms of
(weighted) perfect matchings in an appropriate bipartite graph dual to
the Postnikov diagram.
Hugh Thomas, Higher Auslander algebras, cyclic polytopes, and analogues of tropical cluster algebras.
Abstract: Consider two simple models for the An cluster complex: triangulations of an (n + 3)-gon, and tilting objects for the path algebra of a linearly-oriented An+1
quiver. We show that there are higher-dimensional analogues of both
these sets of objects, and that they are naturally in bijection. These
higher dimensional analogues are: triangulations of a cyclic polytope
of dimension 2d with n + 2d + 1 vertices, and basic tilting
objects over the (d-1)-fold higher Auslander algebra of the path
algebra of the linearly-oriented An+1
quiver (satisfying an additional condition). The analogue of the
cluster variables in the two models are the internal d-dimensional
simplices of the polytope and the non-projective-injective summands of
the tilting objects. While we do not have anything like a cluster
algebra on this set of variables, we show the existence of an analogue
of the tropical cluster algebra structure associated to a
lamination. This is joint work with Steffen Oppermann.
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