BIREP – Representations of finite dimensional algebras at Bielefeld

The ICRA 2012 conference was held from Monday, 13 August, to Friday, 17 August 2012, in the lecture hall H7 in the main university building (UHG) of Bielefeld University (maps are available on the Travel Information page). It consisted of talks given by the invited speakers and additional talks selected shortly before the conference on the basis of submitted abstracts.

Invited speakers were as follows:

- Sergey Fomin (Ann Arbor): Cluster structures in rings of SL
_{3}invariants - Hiraku Nakajima (Kyoto): Monoidal categorification, revisited
- Julia Pevtsova (Seattle): Elementary subalgebras of modular Lie algebras and vector bundles on projective varieties
- Raphaël Rouquier (Oxford/Los Angeles): 2-Hopf algebras
- Michel Van den Bergh (Hasselt): Non-commutative resolutions of determinantal varieties

There was also a special session dedicated to Ragnar-Olaf Buchweitz on the occasion of his 60th birthday. Another session was devoted to the development of computational methods in representation theory.

Detailed information on many of the talks is available via the list of talks and the detailed time tables.

For a better overview and easy reference, time tables containing all talks together with the respective talk titles are available as a pdf document (68 KB).

All lectures were held in the lecture halls of the main university building (UHG).

Monday 13 August |
Tuesday 14 August |
Wednesday 15 August |
Thursday 16 August |
Friday 17 August |
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8:30 | – | 9:00 | Registration | ||||

9:00 | – | 9:50 | Plenary | Plenary | Special session dedicated to Ragnar-Olaf Buchweitz |
Plenary | Plenary |

10:00 | – | 10:30 | Plenary | Plenary | Plenary | Plenary | |

Coffee break | Coffee break | ||||||

11:00 | – | 11:20 | Parallel | Parallel | Parallel | Parallel | |

11:30 | – | 11:50 | Parallel | Parallel | Parallel | Parallel | |

12:00 | – | 12:20 | Parallel | Parallel | Parallel | Parallel | |

Lunch break | |||||||

Excursion | |||||||

14:00 | – | 14:50 | Plenary | Plenary | Plenary | Plenary | |

15:00 | – | 15:30 | Plenary | Plenary | Plenary | Plenary | |

Coffee break | Coffee break | ||||||

16:00 | – | 16:20 | Parallel | Parallel | Parallel | Parallel | |

16:30 | – | 16:50 | Parallel | Parallel | Parallel | Parallel | |

17:00 | – | 17:20 | Parallel | Parallel | Parallel | Parallel | |

17:30 | – | 18:20 | Plenary | Plenary | Plenary | Plenary | |

18:30 | – | Welcome reception |
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19:00 | – | Computer algebra session |
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20:00 | – | Conference dinner |
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Cluster structures in rings of SL

The rings of polynomial SL(V)-invariants of configurations of vectors and linear forms in a k-dimensional complex vector space V have been explicitly described by Hermann Weyl in the 1930s. Each such ring conjecturally carries a natural cluster algebra structure (typically, many of them) whose cluster variables include Weyl's generators. In joint work with Pavlo Pylyavskyy, we describe and explore these cluster structures in the case k=3.

Monoidal categorification, revisited

In my previous 'monoidal categorification of cluster algebras' via perverse sheaves on graded quiver varieties, the definition of a coproduct is missing, except finite type cases. Thus the construction does not give a true monoidal category, though it gives a well-defined multiplication on the quantum Grothendieck group. I explain how to define the true monoidal category.

Elementary subalgebras of modular Lie algebras and vector bundles on projective varieties

Let g be a p-restricted Lie algebra. We call a subalgebra E of g "elementary" of rank r if it is an abelian Lie algebra with trivial p-restriction of dimension r. For a fixed r we consider a projective variety E_{r}(g) that parameterizes all elementary subalgebras of g of rank r. This variety is a natural generalization of the rank variety introduced by Carlson for elementary abelian p-groups and the support variety for Lie algebras of Friedlander and Parshall.

I'll identify this projective variety in various classical cases. I'll also show how representations of g with special properties lead to constructions of families of vector bundles on E_{r}(g), thereby extending the study of "modules of constant Jordan type" and their geometric applications to this more general context.

This is a joint work with Jon Carlson and Eric Friedlander.

2-Hopf algebras

We will give a definition of tensor categories with a coproduct, and construct such structures for 2-Kac-Moody algebras. This is done in the A-infinity setting. As a consequence, the 2-category of A-infinity 2-representations of Kac-Moody algebras is equipped with a closed monoidal structure.

Non-commutative resolutions of determinantal varieties

During the lecture I will discuss joint work with Ragnar Buchweitz and Graham Leuschke on the non-commutative resolution of determinantal varieties. We will show that such resolutions can be constructed in a characteristic free manner and we will discuss their properties. In particular we will give a explicit presentation by generators and relations.