Selected Topics in Representation Theory

 WS 2005/06: Wednesday, 13-14 in V2-216 and 14-16 in V3-204 SS 2006: Wednesday, 13-16 in R2-155 WS 2007/08: Wednesday, 14-16 in T2-208

WS 2005/06

• October 19, 2005.
• R.F.: Self injective algebras I: 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.
• A.H.: What are tilting modules? - A very brief introduction to tilting theory.
The main topic will be the Brenner-Butler theorem: If B = 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.
• C.M.R.: 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 with all but at most one Ci zero, and the only non-zero Ci being projective.)
(A stronger assertion: If T is a bounded complex of free modules of finite rank and any map f : T → T[n] with n ≠ 0 is homotopic to zero, then T is homotopy equivalent to a trivial complex.)
• October 26, 2005.
• R.F.: Self injective algebras II: Self injective algebras vs. Frobenius algebras.
• L.H.: Mutations of exceptional sequences
• C.M.R.: Some tilting complexes for Artin algebras
(given by a partition of the vertex set into two disjoint sets I and I', with the requirement that for x ∈ I any simple submodule of P(x) is of the form S(i) with i ∈ I)
(Here is the construction: Let P(I) be the projective cover of the direct sum of all S(i) with i ∈ I, and similarly, P(I') ...
Let f : M → P(I') be a right (add P(I))-approximation of P(I') such that M has a direct summand M' isomorphic to P(I) with M' contained in the kernel of f. Then ... 0 → M → P(I') → 0 ... is a tilting complex.)
• November 2, 2005.
• November 9, 2005.
• R.F.: 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.
• A.H.: Modules with standard filtration 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.
• L.H.: Quadratic algebras and Koszul algebras
• November 16, 2005.
• C.M.R.: What is the meaning of Ext2?
Non-existence of certain modules (certain diamonds = modules with simple socle and simple top).
• A.H.: Modules with standard filtration II: The subcategories F(Δ) and F(∇) of mod A have (relative) almost split sequences.
• L.H.: Koszul algebras and distributive lattices: Koszul algebras are characterized by the fact that the relations yield distributive lattices.
• November 23, 2005.
• R.F.: Burnside's Theorem for Hopf Algebras
• A.H.: Examples:
• Covariantly finite, but not contravariantly finite full subcategories of a module category
• Right F(Δ)-approximations
• C.M.R.: The simplicial complex of cluster tilting objects, corresponding to a hereditary artin algebra A. We provide a description in terms of pairs (T,U), where U is a Serre subcategory of mod(A) and T is a partial tilting A-module which belongs to U.
See Handbook, Part 3.
• November 30, 2005.
• December 7, 2005.
• C.M.R.: The Auslander problem: Let R be left and right noetherian ring. Consider the class W(R) of all (left) R-modules of the form Ext1(NR,RR) where NR is a finitely generated R-module. Is W(R) closed under submodules?
Huang gave a counter example (A3 with 2 sources).
We will show: For a hereditary artin algebra, W(R) is closed under submodules if and only if R is a Nakayama algebra.
It will be shown that for hereditary artin algebra, the problem concerns tilting theory: just consider an injective cogenerator as tilting module.
See Handbook, Part 2.
• R.F.: The Theorem of Wedderburn-Malcev: Conjugacy of maximal separable subalgebras.
• L.H.: Koszul algebras and distributive lattices II (continuation)
• December 14, 2005.
• C.M.R.: Examples of simplicial complexes of tilting modules
• L.H.: 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.
• A.H.: Torsion theories and tilting modules I: Let (TF) 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))=(TF).
• December 21, 2005.
• January 11, 2006.
• C.M.R.: Cluster tilted algebras with three simple modules
• R.F.: 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.
• L.H.: Affine toric geometry and cones (see "Fulton: Introduction to toric geometry")
• January 18, 2006.
• C.M.R.: Tubular algebras with three simple modules
• R.F.: Stable Representation Quivers: 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.
• L.H.: Projective toric geometry and fans (see "Fulton: Introduction to toric geometry")
• February 1, 2006.
• C.M.R.: Basic properties of hereditary length categories
• R.F.: Stable Representation Quivers: 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.
• A.H.: The Auslander-Reiten formula (This talk was based on Chapter IV.4 of ARS, but there is a more comprehensive reference which also includes the functoriality behind the Auslander-Reiten formula by H. Krause: A short proof for Auslander's defect formula.)
• February 8, 2006.

WS 2007/08

• October 24, 2007.
• October 31, 2007.
• November 7, 2007.
• November 14, 2007.
• November 21, 2007.
• November 28, 2007.
• December 5, 2007.
• December 21, 2007.
• C.M.R.:
• L.H.: Slopes for representations of Dynkin quivers (There exists a slope with the property: a module over a path algebra of a quiver is stable precisely if it is indecomposable.)
• January 9, 2008
• January 16, 2008
• January 23, 2008
• January 30, 2008
• February 6, 2008

List of topics to be further considered

C.M.R.: What is Ext2?
(Ext2 corresponds to relations, to the non-existence of modules, ...)
1. Triangulated and derived categories. Related to 1: the stable category and the Heller operator
2. Highest weight categories and quasi-hereditary algebras
3. The Wedderburn-Malcev Theorems (and the use of quivers and species for describing finite-dimensional algebras).

Wish list:

• The lecture notes of Karin Erdmann

• What are "almost projective" modules (ARS, p.157, and open problems Nr.6)
• Calabi-Yau categories with finitely many objects

• Quivers with relations
• Further topics from tilting theory and tame concealed algebras
• Working with special biserial algebras, a lot of examples
• Quivers and geometry:
• Moduli spaces and stability
• Modality
• The representation space of a preprojective algebra
• Lie algebras and representation theory
• Lusztig's nilpotent variety