The quotient Z_n of the variety of n-dimensional modules of the free algebra in g generators is singular. M.V.Nori has in 1978 developed an approach to desingularise Z_n. This construction is connected to the variety of n^2-dimensional algebras. Unfortunately, it is smooth only if n=2. We discuss this case and determine the singular locus if n=3.
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Reading course on
"Littelmann paths"
Reading course on the paper "Closures of Conjugacy Classes of Matrices are Normal" by H-P. Kraft and C. Procesi. |
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Representations, Cohomology and Support Spaces 29 April - 1 May, 2007 |
| There has been a proposal to change the order of the lecture. This will be decided as soon as possible! |
A *-quiver is a quiver together with a partition of its set of vertices into
two subsets. To get a representation of the *-quiver we attach to an arrow
originating in one subset and terminating in the other a bilinear form of
the vector space attached to the starting point and a linear map between the
corresponding vector spaces if an arrow originates and terminates in the
same subset.
Thus representations of the *-quiver generalize the notion of
representations of quivers by enabling to consider bilinear forms together
with linear maps. The study of *-quivers was initiated by Zubkov, who
calculated the generators and relations of its algebra of invariants, and by
Derksen and Weyman, who described finite, tame and wild *-quivers. In
contrast to the case of quivers, the orbits of the semisimple
representations of *-quivers are not necessarily closed. This talk deals
with the problem of describing closed orbits and the determination of the
dimension of the categorical quotient, which parametrizes the closed
orbits.
We discuss in detail the Laurent expansions of cluster monomials in
rank 2 cluster algebras. They turn out to be closely related with the
root systems of the corresponding rank 2 Kac-Moody algebras. This is a
joint work with Paul Sherman.
It is a classical well known theorem, that a cocomplete abelian category
with a small projective generator is equivalent to the category of modules
over the endomorphismring of that generator. The equivalence is induced by
the functor represented by the generator. Morita theory states that the
converse is true for module categories over rings, i.e. if two module
categories are equivalent, then the equivalence is given as above. If in
this case one weakens the notion of equivalence to derived equivalence one
obtains an analogous result, which is called the generalized tilting theorem.
In this talk I want to one step further and explain how the above results
apply in homotopy theoretic situations and therefore generalize to non-abelian
settings. Tilting theory is contained as a special case as well as a Morita
theorem for ring spectra. Most of the talk is devoted to explain the concepts
needed to state and prove the main results. These concepts are mainly stable
model categories and symmetric spectra.
The class of finite-dimensional, local group algebras consists precisely of
the group rings of finite p-groups G over fields k of characteristic p, where
p is an arbitrary prime. In a certain sense, these rings are the most
non-semisimple group algebras.
In this talk we will try to understand the structure of such a local group
algebra kG, focusing on the close interaction between the group G and the
algebra kG. Our starting point will be radical filtrations of kG and the
normal series of G which they determine, and our way will lead us through
restricted Lie algebras and their restricted enveloping algebras until we
reach PBW- or rather BZJ-bases of kG. These bases may at least be considered
as a convenient tool for doing calculations efficiently in kG.
Based on the so-called Zassenhaus-ideals of kG we will show which sections of
the group G are (at any rate) completely determined by the algebra kG. If time
permits, we will collect additional invariants of kG which restrict the
structure of possible group bases of kG.
Given an artin algebra A and a subalgebra B, we want to
compare the finiteness of the finitistic dimension of B with that of A. In
general, this is impossible. However, if we assume certain conditions on
the redicals of B and A, it turns out that some nice relationships
between the finitistic dimensions of the two algebras can be obtained. In
particular, we
have the following: Suppose the radical of B coincides with that of A. If
the global dimension of A is upper bounded by 3, then the finitistic
dimension of B is finite. A similar result can be obtained if global
dimension is replaced by representation dimension.
We give an elementary proof of the structure of the quantum affine groups
via the composition algebras of affine quivers. We also consider Reineke's
composition monoid and make some conjectures concerning the relationship
between the two objects.
Affine Lie algebras are a special class
of infinite dimensional Kac-Moody algebras
which has numerous applications to
other branches of mathematics.
For instance the simplest affine Lie algebra
\hat{sl}(2) can be used to study
representations of all general linear groups
and the symmetric groups of arbitary finite letters.
This is achieved via the basic representation
of the affine Lie algebra and vertex operators.
It is also related with the so-called
boson-fermion correspondence in conformal field
theory.
In these talks we will first
review this example and then generalize
it to all simply laced affine Lie algebras.
The latter is a new approach of McKay correspondence
and we will study the interplay between
affine Lie algebras, vertex operators, symmetric
functions and finite group theory.
We will also briefly talk about further
generalization of this to q-deformation
as well as twisted picture.
We show that any maximal exceptional module over a preprojective
algebra of Dynkin type is maximal 1-orthogonal in the sense of
Iyama. This has many interesting consequences. For example it
follows that the representation dimension of a preprojective
algebra is three. It also allows to study maximal exceptional
modules in terms of classical tilting theory.
Furthermore, we define a mutation operation on maximal exceptional
modules and show how this is linked to the mutation of clusters
in the theory of cluster algebras.
The determinant of the Cartan matrix of a finite
dimensional algebra is an invariant of the derived category
and can be very useful for derived equivalence classifications.
In the talk we show how to compute the determinants of
the Cartan matrices for all gentle algebras. This is a class
of algebras of tame representation type which occurs naturally
in various places in representation theory. The definition
of these algebras is of a purely combinatorial nature, and
so are our formulae for the Cartan determinants.
Abstract: I will present a number results of work with the following`
coauthors, Ragnar Buchweitz, Gregory Hartman, Eduardo Marcos,
Dag Madsen, Nicole Snashall and Oeyvind Solberg (in various combinations).
For a Koszul algebra, there is a `comultiplicative' structure to a
minimial resoluton of the semisimple part of the algebra. The coefficients
of the comultiplicative structure have truly wonderful properties which
I will try to describe. As an application, I will give a negative answer to
Happel's question: If R is a finite dimensional algebra of infinite global
dimension, then must the Hochschild cohomology groups, HH^n(R), be
different from 0 for an infinite number if n>=0 ?
Let R be a hereditary, indecomposable, left pure semisimple ring. Inspired
by a recent paper of Reiten and Ringel,
we investigate the perfect cotorsion pair generated by the preinjective
component.
We show that there is a finitely generated product-complete tilting and
cotilting left module W
inducing this cotorsion pair. The module W stores important information on
R.
For example, if we assume R of infinite representation type, then, by a
result of Zimmermann-Huisgen, there are non-preinjective
indecomposable modules occurring as direct summands of products of
preinjective modules, and
it turns out that W is precisely the direct sum of such modules. Moreover,
we prove that R has finite representation type if and only if
every indecomposable summand of W is source of a left almost split map in
R-mod.
Finally, we address the question when W is endofinite.
Abstract: Let Λ be a finite dimensional algebra over an
algebraically closed field k. The Gabriel-Roiter measure for a
Λ-module (not necessarily of finite length) M is, by definition,
a subset of the natural numbers. It was first introduced by Roiter and then by
Gabriel under the name "Roiter measure". To determine the GR measure of a
module M, it is necessary to know the so called GR submodule T
of M. On the other hand, A. Schofield has shown that for hereditary
algebras, any indecomposable exceptional module can be written as an
extension of orthogonal modules. So the question is the following: can we find
such orthogonal modules for an indecomposable Λ-module M,
where Λ is representation directed, not necessarily
hereditary. (Note that all indecomposable Λ-modules are
exceptional.) With this question, I would like to give some properties
of the GR measure and the connection between the GR measure and representation
finite (directed) algebras. Also, I will give some examples and new
questions.
We describe a normal form of Richardson elements in the classical case.
This generalizes a construction for $\mathfrak{gl}_N$ of Brüstle,
Hille, Ringel and Röhrle to the other classical Lie algebras and extends
our construction of admissibile elements in the sense of Lynch to all
parabolic subalgebras of the classical Lie algebras.
As an application we obtain a description of the support of the
Richardson elements and recover the Bala-Carter label of the orbit
of Richardson elements.
References:
We work over a field of arbitrary characteristic. At first we mention some
known results on generators of (semi-)invariants of quiver
representations. Then we give the definition of *-representations of a
quiver, which can be get by changing some vertex vector spaces to the dual
ones, and explain the reason of its introduction. We describe generators
of semi-invariants of quiver *-representations. This is joint work with
A.N. Zubkov.
Donovan's conjecture states that for each finite $l$-group $D$ the
number of Morita equivalence classes of blocks of finite groups with
defect group $D$ is finite. Olaf Duevel has reduced a similar conjecture
to the case of quasi-simple groups. Although this has not yet been
achieved for Donovan's conjecture, one might hope it is possible and
consider quasi-simple groups in more detail. A theorem of Bonnafe and
Rouquier allows a reduction to unipotent and isolated blocks for finite
groups of Lie type. Hiss and Kessar have studied the unipotent blocks
of classical groups. In this talk I will present their results as well
as a generalization to isolated blocks, explaining the relevant concepts
of the representation theory of finite reductive groups underway.
Let G be a connected reductive linear algebraic group and let H be a
closed subgroup of G. According to Serre, H is called G-completely
reducible in G if whenever H lies in a parabolic subgroup P of G, then
it lies in a Levi subgroup of P. In the special case when G = GL(V) we
have that H is GL(V)-completely reducible if and only if V is a
semisimple H-module. So this concept faithfully generalizes the
standard notion of semisimplicity from group representation theory to
algebraic groups.
I shall discuss this concept by Serre and will demonstrate some new
results. In a recent paper with Bate and Martin it was shown that this
"algebraic" notion of Serre is equivalent to an older "geometric" one
due to Richardson. In turn Richardson's notion and this equivalence
allow the use of methods from geometric invariant theory in the study
of G-completely reducible subgroups of G. I shall illustrate this
with some examples.
Abstract: The talk is to give a survey on my recent work (some with my student) on the imprimitive complex reflection groups G(m,p,n). It contains three aspects:
Let R be a commutative local uniserial ring of length n,
p a generator of the maximal ideal, and k the radical factor
field. The pairs (B,A) where B is a finitely generated
R-module and A ⊂ B a
submodule such that pm A = 0 form the objects in the
category Sm(R).
We show that in case m=2 the categories Sm(R)
are in fact quite similar to each other: If also R' is a commutative
local uniserial ring of length n and radical factor field k,
then the categories S2(R) / NR and S2(R') / NR' are equivalent for certain nilpotent
categorical ideals NR and NR'.
As an application, we recover the known classification of all
indecomposable pairs (B,A), up to isomorphy, where B is a
finitely generated abelian p-group and A ⊂ B a
p2-bounded subgroup.
This is a report on joint work with Carla Petroro.
Da Quantengruppen als Algebren mit Erzeugern und Relationen definiert sind, ist ihre konkrete Untersuchung mit Methoden der nichtkommutativen Algebra sehr schwierig. Man ist daher daran interessiert, explizite Realisierungen zu finden.
Für die Quantengruppen U_v(gl_n) wurde eine solche Realisierung von Beilinson/Lusztig/McPherson angegeben, und zwar als Faltungsalgebra auf der Menge der Untervektorräume eines endlich-dimensionalen k-Vektorraums über einem endlichen Körper k. Dabei ist das Produkt der Algebra durch eben diese Faltung gegeben. Eine Verallgemeinerung auf beliebige Kac-Moody-Algebren ist bislang unbekannt. Der positive Teil der Quantengruppen lässt sich (unter einer kanonischen Dreieckszerlegung) mittels der sog. Hall-Algebra realisieren (Ringel, 1990). Hall-Algebren werden mit Hilfe der Theorie der Köcherdarstellungen definiert.
Markus Reineke hat 2004 eine partielle Verallgemeinerung der erwähnten Realisierung angegeben: Der positive Teil einer Quantengruppe lässt sich mit Hilfe sog. gerahmter Köchermodulräume in genau der gleichen Weise realisieren. Entscheidend für dieses Resultat ist die oben genannte Konstruktion mittels Hall-Algebren.
Ich werde der Frage nachgehen, ob sich auf diese Weise auch die gesamte Quantengruppe erhalten lässt. Dabei werde ich mich auf einen Spezialfall der gerahmten Modulräume, die sog. Konfigurationsräume, beschränken.
This is joint work with C. Geiss and B. Leclerc. We generalize our previous results on preprojective algebras of Dynkin type and the relation to Fomin and Zelevinsky's theory of cluster algebras to arbitrary preprojective algebras. In particular, this leads to a new categorification of all acyclic cluster algebras.
"Representations of differential biquivers and coherent sheaves on degenerations of elliptic curves"
Abstract: In my talk I shall discuss the problem of classification of simple coherent sheaves on certain degenerations of elliptic curves. Indecomposable vector bundles on smooth elliptic curves were classified in 1957 by Atiyah. An approach to study coherent sheaves on singular projective curves was suggested in works of Burban, Drozd and Greuel. In particular, it was shown that the category of coherent sheaves on a nodal cubic curve and on cycles of projective lines is tame, and all other degenerations of elliptic curves are wild. However, it turns out that all plane cubic curves are brick-tame. As a main technical tool to prove this result we use representation theory of boxes. Moreover, our approach leads to an interesting class of wild matrix problems which are brick-tame.