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.
- Bill Crawley-Boevey: Sylvester rank functions for rings and universal localization
- Julia Sauter: Exact structures on exact categories
- Henning Krause: Chase's Lemma and its Context
- Marc Stephan: Quillen's Stratification Theorem
- Andrew Hubery: Projective modules over hereditary tensor algebras
- Rolf Farnsteiner: Algebras of finite global dimension:
- Philipp Fahr: Algebraic aspects of stability
- Angela Holtmann: Finitistic dimension conjecture: a reduction formula
- Rolf Farnsteiner: Induced modules: The Mackey decomposition theorem
- Philipp Fahr: The Harder-Narasimhan filtration
- Philipp Fahr: Torsion Theory and Stability
- Rolf Farnsteiner: Induced modules:
- Claus M. Ringel: Artin algebras of dominant dimension ≥ 2
- Angela Holtmann: Families of representations for wild quivers
- Claus M. Ringel: The torsionless modules of an artin algebra
- Rolf Farnsteiner: Cartier's Theorem
- Rolf Farnsteiner: Quasi-hereditary algebras:
- Philipp Fahr: Elementary Modules
- Angela Holtmann: Properties and use of the characteristic tilting module
- Reinhard Waldmüller: The Theorems of Maschke and Artin-Wedderburn
- Rolf Farnsteiner: Hopf modules and integrals: Maschke's Theorem for Lie algebras
- Examples
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.
- The structure of the projective indecomposable modules
For group algebras of finite groups, the socles and the tops of the principal indecomposable modules are isomorphic. We shall look at Nakayama's generalization concerning self injective algebras. - Self injective algebras vs. Frobenius algebras.
- Examples and Morita equivalence
We are going to give examples for symmetric, weakly-symmetric and Frobenius algebras, and discuss stability of these classes of algebras under Morita equivalence. - Frobenius algebras and coalgebras
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.
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.
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.
- 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.
Koszul algebras are characterized by the fact that the relations yield distributive lattices.
- Covariantly finite, but not contravariantly finite full subcategories of a module category
- Right F(Δ)-approximations
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.
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).
- 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.