Actions of Algebraic Groups, Fans and Tilting Modules

• Applicant: Privatdozent Dr. Lutz Hille
• Subject Area: Mathematics
• Term: from 2009 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 124910424

In the project we study various geometric questions (actions of algebraic groups, exceptional sequences of sheaves, structure of quotients and moduli spaces) in connection with algebraic questions (classification of tilting bundles, classification of modules, the structure of triangulated categories, representations theory of groups and algebras, and Hochschild (co)homology). Various geometric and algebraic problems are motivated by problems in physics. The project connects, inside the priority program, algebraic groups with representation theory of finite dimensional algebras and has close connections also to mathematical physics and finite group theory. Roughly the project consists of three main parts: actions of algebraic groups, equivalences of derived (or triangulated) categories and the Auslander algebra of the truncated polynomial ring. In our first part we consider actions of algebraic groups. Here we are mainly interested in a classification of orbits, the existence of a dense orbit and quotients. Mark Blume's work is part of this project. One main question investigates actions of parabolic groups with a dense orbit. In a different part we investigate moduli spaces of quiver representations. Eventually, in the last part we consider stacks associated to root systems. Equivalences of derived categories are often obtained from tilting objects (tilting modules, tilting sheaves) and offer a bridge between algebra and geometry. One main issue of the project is the classification and construction of tilting sheaves and tilting modules. Using the corresponding equivalence one obtains often surprising new insight in geometric questions on one side and also algebraic questions on the other side. In the classification of exceptional sequences of line bundles on rational surfaces occure reflexive polygons (they are not neccessarily convex) and the corresponding smooth toric surfaces. If one considers exceptional sequences of vector bundles then one needs toric stack. In the study of actions of reductive algebraic groups and their parabolic subgroups appears an algebra that plays in important role for the classification of orbits for parabolic group actions, it is the so-called Auslander algebra of the truncated polynomial ring. This algebra occurs in various different context. It appears in the work of Seidel and Thomas on spherical twists, in the joint work with Brüstle, Ringel and Röhrle, in the joint work with Karin Baur, and also in our recent work with Perling on tilting bundles. Moreover, it appears in two new projects, one with David Ploog and another one with Dieter Vossieck. On the other side the project also extends to another direction motivated by the work of Mark Blume on toric stacks.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1388: Representation Theory