Selected Topics in Representation Theory
The selected topics started as part of the internal seminar of the BIREP group during 2005–2008. The aim was to create resource and make them available for the everyone interested. This was reinstated in the winter semester 2020/21.
Below you can find a list of topics and short documents about those topics, which include a list of references. For now, they are ordered by the date the corresponding talks were given.
Claus M. Ringel: Degenerations using Riedtmann-Zwara sequences
Claus M. Ringel: The ladder construction of Prüfer modules
Reinhard Waldmüller: The orthogonality relations
Claus M. Ringel: Prüfer modules of finite type (If there are no generic modules, then all Prüfer modules are of finite type,)
Reinhard Waldmüller: Burnside's $p^a q^b$-theorem
Reinhard Waldmüller: Symmetries of flat manifolds
Angela Holtmann: General representations of quivers and canonical decompositions:
Reinhard Waldmüller: l-adic cohomology
Claus M. Ringel: Pillars, see also Brauer-Thrall I (A module which is not of finite type contains indecomposable submodules of arbitrarily large finite length)
Reinhard Waldmüller: Deligne-Lusztig characters
Rolf Farnsteiner: Nakayama algebras: Kupisch series and Morita type
Claus M. Ringel: Gabriel-Roiter inclusions and irreducible monomorphisms
If there is a Gabriel-Roiter inclusion $X \to Y$, there is an irreducible monomorphism $X \to M$ and an epimorphism $M \to Y$ such that the composition $X \to M \to Y$ is a monomorphism.
Rolf Farnsteiner: Simple modules and p-regular classes
Claus M. Ringel: Take-off subcategories: Any infinite cogeneration-closed subcategory contains a minimal infinite cogeneration closed subcategory.
Rolf Farnsteiner: Complexity and Krull dimension
Claus M. Ringel: The real root modules for some quivers
Rolf Farnsteiner: Separated quivers and representation type
Philipp Fahr: Infinite-dimensional Modules over Wild Algebras:
Rolf Farnsteiner: Self injective algebras:
Angela Holtmann: What are tilting modules? - A very brief introduction to tilting theory.
The main topic will be the Brenner-Butler theorem: If $B = {\rm End}({}_AT)$ is the endomorphism ring of an $A$-tilting module ${}_AT$, then "certain subcategories" of the module categories of $A$ and $B$ are equivalent.
Claus M. Ringel: If R is a local artinian ring, then there are no non-trivial tilting complexes.
A tilting complex is trivial if it is homotopy equivalent to a complex $C^\bullet$ with all but at most one $C^i$ zero, and the only non-zero $C^i$ being projective.
A stronger assertion: If $T$ is a bounded complex of free modules of finite rank and any map $f \colon T \to T[n]$ with $n \neq 0$ is homotopic to zero, then $T$ is homotopy equivalent to a trivial complex.
Lutz Hille: Mutations of exceptional sequences
Angela Holtmann: Resolution of simple modules over the polynomial ring k[X1,...,Xn] (and the exterior algebra of a vector space)
Lutz Hille: The braid group action on the set of exceptional sequences
Rolf Farnsteiner: The Burnside theorem
If G is a finite group, and M is a faithful representation, then any irreducible representation occurs as a subfactor of some tensor product of copies of M.
Angela Holtmann: Modules with standard filtration:
- Part I
Let A be an Artin algebra, mod A the category of finitely generated A-modules, and Θ={Θ(1),...,Θ(n)} a sequence of A-modules with Ext1A(Θ(j),Θ(i))=0 for all j≥i. Denote by F(Θ) the full subcategory of mod A of modules filtered by modules from Θ. Then F(Θ) is functorially finite in mod A.
- Part II
The subcategories F(Δ) and F(∇) of mod A have (relative) almost split sequences.
Lutz Hille: Quadratic algebras and Koszul algebras
Lutz Hille: Koszul algebras and distributive lattices
Koszul algebras are characterized by the fact that the relations yield distributive lattices.
Rolf Farnsteiner: Burnside's Theorem for Hopf Algebras
Angela Holtmann: Examples:
- Covariantly finite, but not contravariantly finite full subcategories of a module category
- Right F(Δ)-approximations
Rolf Farnsteiner: The Theorem of Wedderburn-Malcev:
Lutz Hille: The volume of a tilting module
This notion allows to show easily that an almost complete partial tilting module has at most two complements, and at most one in case the module is not sincere.
Angela Holtmann: Torsion theories and tilting modules
Let (T, F) be a torsion theory on mod A such that D(AA)∈T and either T or F contain only finitely many isomorphism classes of indecomposable A-modules. Let T be the sum of all Ext-projectives in T. Then T is a tilting module such that (T(T), F(T))=(T, F).
Rolf Farnsteiner: Stable Representation Quivers:
- The Riedtmann Structure Theorem.
Using covering techniques we shall show that stable representation quivers are determined by certain trees and certain subgroups of the universal coverings defined by these trees.
- Growth Numbers and Zhang's Theorem.
If Q is a non-periodic, tree infinite, regular component of the AR-quiver of an Artin algebra whose growth number is small, then the tree class of Q is an infinite Dynkin diagram.
- Subadditive Functions and Webb's Theorem.
We shall show that the tree classes of the components of the stable Auslander-Reiten quiver associated to the group algebra of a finite group are finite or infinite Dynkin diagrams, or Euclidean diagrams.
Lutz Hille: The volume of a tilting module for $A_n$
