BIREP – Representations of finite dimensional algebras at Bielefeld

Helmut Lenzing (Photo: Lutz Hille)

There was a Colloquium in honour of the 75th birthday of Helmut Lenzing at Bielefeld University.

Location: Bielefeld University, all talks were given in room V2-210/216.

Date: Thursday 15 May 2014 and Friday 16 May 2014

Programme: There was a talk on Thursday at 5pm by Ragnar-Olaf Buchweitz (Colloquium of the Faculty of Mathematics) and five talks on Friday. A social dinner was had on Friday evening. See below for details.

Support: DFG CRC 701 "Spectral Structures and Topological Methods in Mathematics"

- Ragnar-Olaf Buchweitz (Toronto)
- Osamu Iyama (Nagoya)
- Dirk Kussin (Chemnitz)
- Steffen Oppermann (Trondheim)
- Idun Reiten (Trondheim)
- Atsushi Takahashi (Osaka)

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17:15 | – | 18:15 | Ragnar-Olaf Buchweitz (Toronto) Matrix Factorizations over Elliptic Curves |

10:00 | – | 11:00 | Osamu Iyama (Nagoya) Cohen-Macaulay representations of Geigle-Lenzing complete intersections |

Coffee break | |||

11:30 | – | 12:30 | Steffen Oppermann (Trondheim) Recollements for Geigle-Lenzing weighted projective varieties |

Lunch break | |||

14:00 | – | 15:00 | Atsushi Takahashi (Osaka) Weighted projective lines and extended cuspidal Weyl groups |

15:15 | – | 16:15 | Dirk Kussin (Chemnitz) On tubes and multiplicities |

Coffee break | |||

16:45 | – | 17:45 | Idun Reiten (Trondheim) My work with Helmut |

19:00 | – | Dinner at Wernings Weinstube (web site and map in German) |

Matrix Factorizations over Elliptic Curves

Given a nonzero polynomial P in n variables, a matrix factorization of P consists
of a pair of square matrices A,B of same size with entries from the polynomial ring such
that AB = P Id, where Id stands for the appropriate identity matrix. If the polynomial is
homogeneous one might further require that the entries of the matrices are homogeneous as
well.

A fundamental result by Orlov implies as a special case that equivalence classes of such
homogeneous matrix factorizations for a cubic polynomial that defines an elliptic curve in
the projective plane are in a natural, though still largely mysterious bijection with the
isomorphism classes of indecomposable objects in the derived category of coherent sheaves on
that elliptic curve. The structure of the latter is known since Atiyah's classification of
such sheaves in 1957.

After recalling the background just described, I will present results by my student Sasha
Pavlov who uses this machinery to determine all possibilities for the degrees and sizes of
the entries of such matrix factorizations and how this will enable us to find all such
matrix factorizations eventually.

Time permitting, I will outline how these results generalize to cones over elliptic curves
embedded in higher projective spaces on the one hand and how, on the other, they relate to
work of Lenzing et al. on weighted projective lines.

Cohen-Macaulay representations of Geigle-Lenzing complete intersections

As a generalization of weighted projective lines, we introduce a class of commutative rings R graded by abelian groups L, which we call Geigle-Lenzing complete intersections. We will study L-graded Cohen-Macaulay R-modules, and show that there always exists a tilting object in the stable category. As an application we study when (R,L) is d-representation finite in the sense of higher dimensional Auslander-Reiten theory. This is a joint work with Herschend, Minamoto and Oppermann.

On tubes and multiplicities

We study three certain natural numbers associated with a tube of regular representations of a tame hereditary (or canonical) algebra over a perfect field. We present a general formula with these numbers and discuss implications for the Auslander-Reiten translation.

Recollements for Geigle-Lenzing weighted projective varieties

Abstract: In my talk I will discuss Geigle-Lenzing weighted projective varieties, which
generalize Geigle and Lenzing's weighted projective lines to higher dimensional situations.

First I will investigate how to "reduce weights" in the case of Geigle-Lenzing weighted P^n.
It will turn out that reducing a weight by one will give rise to a recollement between three
categories of coherent sheaves on GL-projective spaces, where the "quotient" is given by a
hyperplane in the original space.

I will then proceed to investigate the converse construction, that is I will build a new
abelian category as the middle term of a recollement. It will turn out that this converse
construction does not depend too much on the original varieties, so that it gives a method
to introduce Geigle-Lenzing type weights also on hypersurfaces in other projective
varieties.

My work with Helmut

I have written two joint papers with Helmut, one on hereditary categories over arbitrary fields, and one on generalizations of additive functions. I will discuss some of the results and the relationship to other work, especially for the first paper.

Weighted projective lines and extended cuspidal Weyl groups

Abstract: We report on our recent study on a correspondence among weighted projective lines, cusp singularities and cuspidal Weyl groups. In particular, we discuss an isomorphism of Frobenius manifolds between the one from the Gromov-Witten theory for a weighted projective line and the one associated to the invariant theory of an extended cuspidal Weyl group.