GoingToMeet

Lie algebras and their representations form an extremely rich and important field of mathematics. Geometric representation theory is a relatively new field which has attracted much attention. The general idea is to use geometric methods to construct classically algebraic objects, such as representations of Lie groups and Lie algebras. Geometric techniques have proven to be particularly well suited to establishing positivity and integrality results, as these are often easy consequences of the geometric nature of the objects involved. One is also often able to use the representation theory of objects such as (affine) Kac-Moody algebras to organize and better understand the homology of various interesting spaces appearing in the constructions, like Hilbert schemes, flag varieties, Steinberg varieties, affine Grassmannians, and quiver varieties. In addition to its obvious connections to representation theory, geometric representation theory has been found to be intimately related to combinatorics (crystals, quivers), cluster algebras, mathematical physics, and many other subjects.

Within the theory of Kac-Moody algebras, it is the finite-dimensional and affine Kac-Moody algebras that are most important, not only for Lie theory itself, but also for applications to number theory, soliton equations and mathematical physics. Extended affine Lie algebras are a class of Lie algebras that encompasses these two important types of Kac-Moody algebras, as well as toroidal Lie algebras. Contrary to arbitrary (non-symmetrizable) Kac-Moody algebras, whose general structure is still somewhat mysterious despite three decades of intensive research, the structure theory of extended affine Lie algebras has just recently been worked out.

The summer school will consist of three mini-courses per week for two weeks. These courses will serve as an introduction to geometric representation theory, quantum groups and combinatorial representation theory (i.e. the theory of crystal bases), the structure theory of extended affine Lie algebras and the representation theory of affine and toroidal Lie algebras. They will be aimed at the graduate student / postdoctoral fellow level. Following the summer school there will be a one week conference featuring speakers presenting recent developments in areas related to the material of the summer school.