# Actions of the braid group on derived categories of representations of simple Lie algebras.

## EPSRC Grant GR/S35387/01

This project ran from 1/6/2004 to 30/11/07, and was funded by the UK Engineering and Physical Sciences Research Council, 144018 pounds.

Co-investigator: Steffen Koenig. Research Asssociates: Karin Baur and Intan Muchtadi-Alamsyah.

Summary of Project (non-specialist)

A braid is a way of twisting a collection of strings around each other. For example, if there are three strings, you could twist the first two, then the second two, and then the first two again, and the ordering of the strings will have been reversed. It is possible to combine two braids in order to get a third - i.e., apply the first braid, and then the second. The collection of all possible braids on a fixed number, n, of strings, together with this combination rule, forms a group - there is a braid which does no twisting; for every braid, there is an "inverse" braid which undoes it, and the bracketing of a long combination of braids is irrelevant. The braid group and its variations, are prevalent throughout mathematics, as they describe such fundamental operations. In particular, there are some known examples in which the braid group acts (by twisting, in some sense), on the derived category of an algebra. Algebras are the main object of study in representation theory, and the derived category is an important invariant which can be associated to an algebra. These examples are very intriguing, and it is not really understood how they come about. In this project, we aim to use the "Category O" (a well known mathematical object from Lie theory) to construct such actions, employing Marsh's experience in Lie theory, Koenig's expertise in derived categories, and the expertise of the research assistant in category O, and at the same time deepening the understanding of the Category O itself.

Objectives

To give a geometric description of m-cluster categories.
• To find a new example of an Artin braid group action on the derived category of representations of an algebra.
• To construct frieze patterns of integers from numbers of matchings on graphs arising from triangulations.
• To give constructions of Richardson elements of simple algebraic groups.
• To compute dimensions of higher secant varieties of minimal orbits.
• To find a connection between cluster algebras and Temperley-Lieb algebras.
• To find a determinantal relation on a cluster algebra of finite simply-laced type.

Papers

[ University of Leeds ] [ School of Mathematics ] [ Department of Pure Maths ]