Joel Kamnitzer

Abstract:   Actions of the Lie algebra sl(2) on vector spaces arise naturally in combinatorics, geometry, and algebra. Such an action consists of a sequence of vector spaces with linear maps between them satisfying certain relations. From this perspective, one can define an action of sl(2) on a category to be a sequence of categories with functors between them satisfying certain relations. Such actions were studied by Chuang-Rouquier in the context of representations of the symmetric group in positive characteristic. They defined such an action and used it to build an equivalence of categories which settled a case of Broue's conjecture. More recently, Cautis, Licata, and the speaker studied an action of sl(2) where the categories involved were derived categories of coherent sheaves on cotangent bundles to Grassmannians. Following the ideas of Chuang-Rouquier, we used this sl(2) action to construct an equivalence of derived categories between different cotangent bundles of Grassmannians.